An integral equation approach for pricing American put options under regime-switching model

Abstract

Regime-switching models have been heavily studied recently, as they have some clear advantages of over other non-constant volatility model to resolve the so-called smirk effect displayed when constant volatility models are used to price financial derivatives such as options. However, due to the increased model complexity, the associated computational effort usually increases as well, particularly when they are used to price American-style options. In this paper, a novel computational approach based on integral equations is presented. A distinctive feature of our approach, in comparison with other numerical approaches, is that the coupled partial differential equations (PDEs) in a PDE system have been decoupled in the Fourier space, resulting in a completely decoupled integral equation for each economical states, and thus has greatly reduced computational effort. Some examples with preliminary results for a two-state regime-switching model are used to demonstrate our approach.

Keywords:

• American put option
• integral equation
• regime-switching model
• free boundary problem

AMS SUBJECT CLASSIFICATIONS:

• 00A69
• 35A22
• 39-08

1. Introduction

Options and other financial derivatives have been popularly used in financial markets ever since a simple and analytical formula for pricing European options was derived by Black and Scholes [5] and Merton [45]. A fundamental issue of option pricing is to address how a model can be adopted to price vanilla European-style [6,43] and American-style [4,32] options. However, pricing the latter is far more challenging than the former [32,40], due to the additional right that the holder of the latter has to exercise the option earlier rather than being restricted on the expiry. Mathematically, such an additional right has cast the problem of pricing American option into a moving boundary problem [44,45] with the so-called ‘optimal exercise price’ to be determined as part of the solution. As a result, closed-form solutions to price American-style options are very difficult to obtain, if not impossible [32]. Hence, many researchers have resorted to other solution techniques, especially when more complicated models than the Black–Scholes model are adopted.

There are two main approximate techniques, numerical methods and analytical approximations, to price American options. Among numerical methods, the fundamental methods of finite difference methods in pricing options were presented by Merton et al. [46] and Schwartz [52]. Wu and Kwok [54] later used the Landau transform to convert a moving boundary problem into a fixed boundary problem, before adopting the finite difference methods to the latter. In contrast to finite difference methods used to directly solve the Black–Scholes equation for both time and stock price discretization, the lattice tree approach is essentially a continuous asset price process modelled by a discrete random walk model [37]. Despite that, the trinomial tree method, as a special kind of lattice methods, has the same analytic forms as the explicit finite difference scheme. Monte Carlo simulations, which provide probabilistic solutions to the option pricing problems by simulating the random process of the asset price, was introduced by Boyle [10] and subsequently developed by Boyle et al. [8] and Broadie et al. [11] for the derivatives securities pricing. Longstaff and Schwartz [40] further proposed the well-known least-square Monte Carlo (LSM) approach for pricing American options under simulation approaches. By using the least squares in the simulated paths, the conditional expectation function, which is later used to identify the optimal stopping rule that maximizes the option value at each discrete data, can be estimated backward with a much higher computational efficiency. Each of these numerical methods has its pros and cons. For instance, finite difference methods and lattice methods are easy to understand and implement, so they are popular for practitioners [32]. However, some of them such as the explicit finite difference method and the binomial method may not converge in option prices, as pointed out by Omberg [50]. Even in some cases, finite difference methods and binomial methods have a time-step limitation due to the stability conditions [33].

On the other hand, analytical approximation methods were popularly used at some stages of the research in this area. Using an incomplete Fourier transform, an analytical approximation for the American price involving the free boundary was obtained by MacMillan [42]. The Black–Scholes PDEs were directly solved using the quadratic approximation method by Barone–Adesi and Whaley [4]. Extending the method of Geske and Johnson [24], Kim [36] derived an integral expression for the American put price, along with an integral equation for its free boundary. In contrast to numerical methods, analytical approximations can capture some essential features of American options, especially those associated with the free boundary. Unfortunately, they may induce large errors for long-tenor options.

In addition to continuously searching for suitable numerical approaches for a particular option pricing model, research effort has also been spent on improving pricing models themselves. For example, it is well known that a pricing model followed by a stochastic process with a constant volatility, such as the Black–Scholes model is not the best model to produce results closely verifiable by market prices. Empirical evidence has shown that models based on stochastic volatility, jump diffusion and regime switching processes would better fit market-observed smirk [1–3,22]. More recently, regime-switching models have drawn much attention to option valuation [7,18,19,41,58]. In a regime-switching model, the volatility of the underlying, instead of being assumed to be a constant, is assumed to randomly vary with the economical states, which are closely related to market economic status, investors' sentiment as well as the current government fiscal policies [3].

The regime-switching model was first introduced by Hamilton [27], who later also provided some convincing empirical evidence to demonstrate the advantage of the concept of regime-switching under some special economic and political environments, such as a war [28]. Guo [26] further proposed to let both the drift and volatility of underlying to depend on the state of a hidden Markov chain, to represent different economical states as well as to better describe various changing economical factors. Since then, regime-switching models have been popularly adopted by many researchers and market practitioners.

In terms of regime-switching models to price options, pricing European-style options has been well studied. For example, Bollen [6] employed the lattice method to price European options in a two-regime model. A simple recursive approach for pricing European options in the regime-switching framework was derived by Hardy [29]. Buffington and Elliott [12] used another framework developed by Di Masi et al. [43] and derived a set of PDEs for option prices. A coupled PDE approach for pricing exotic options was later proposed by Boyle and Draviam [9]. In 2012, an exact solution for European options has been presented by Zhu et al. [62]. Since then, some more complex European-style options, which could be better fit the financial market, started to be studied by researchers. Noorani et al. [48], in 2021, developed the Monte Carlo simulation with the generation of K-correlated standard normal random vectors and an equivalent martingale measure for pricing an arithmetic Asian option was derived by them. However, a huge number of paths, which need to be generated through Monte Carlo simulations, led to reducing the computational efficiency. In 2022, to characterize structural changes under market conditions, a spread and exchange European option under a regime-switching model has been considered by Ramponi [51]. Utilized the Fourier-based method proposed in [15], a joint characteristic function of the process and a representation of it based on the sojourn times of the underlying Markov chain are obtained.

On the other hand, due to the existence of a moving boundary, pricing American options under regime-switching has significantly hindered processing of finding an analytical solution, except for the perpetual case presented by Zhang and Guo [59], various numerical solution approaches must be resorted to.

Various numerical solution approaches were developed in the past. Liu et al. [39] extended Carr and Madan [17]'s fast Fourier transform (FFT) applied non-regime switching models to the case of regime switching and obtained an explicit form of the joint characteristic function for a two-state Markov chain in the Fourier space, before using the inverse FFT obtain the option price. However, they did not present the corresponding optimal exercise prices at all. To avoid solving a dense system of linear equations, Jackson et al. [34] proposed a Fourier Space Time-stepping (FST) algorithm with an implicit scheme as the core to deal with the integral and diffusion terms of a partial-integro differential equation (PIDE) over a time step. Then, with the PIDE being transformed into a coupled system of ODEs in the Fourier space and further use of the FST for the boundary conditions, their algorithm can be used to efficiently compute the option prices under regime switching. In 2009, Boychenko and Levendorskii generalized Carr's randomization procedure [16] so that it can be used for regime-switching models. Under the assumption that the value functions in all other states are known, a system of the optimal exercise prices in each state can be derived by using the generalized Carr's procedure. Recently, in 2019 the equations for calculating the optimal exercise prices and option values analytically in the Laplace space were obtained by Lu and Putri [41], who developed a semi-analytic method with a pseudo-steady state approximation of the moving boundary proposed by Zhu [61]. In 2021, extending the same work, Chan and Zhu [18] presented an explicit formula for pricing finite-horizon regime-switching American options by means of the homotopy analysis method (HAM). However, they did not present any concrete results to verify the correctness of their method.

Furthermore, Buffington and Elliott [13] decomposed their value into the corresponding European option and an early exercise premium associated exclusively with the early exercise right of American options under a regime-switching model. Unfortunately, they did not present any numerical results to demonstrate their method. Khaliq and Liu [35] furthermore pointed out that the method proposed by Buffington and Elliot [13] cannot be extended beyond two regimes. By adding a proper penalty term, Khaliq and Liu [35] converted a moving boundary problem into a fixed boundary problem with a PDE defined over a fixed domain. They adopted the so-called θ-method to solve the PDE system implicitly except the nonlinear penalty terms, which are treated explicitly. Similarly, in 2010, Yang [55] developed a combined forward Euler (for the time variable) and finite element (for the space variable) method for pricing American options under a regime-switching model. Later, an explicit finite difference scheme for pricing American options under regime-switching was proposed by Egorova et al. [21] in 2016, utilizing the Landau transform to form a new PDE system on a fixed domain with the moving boundaries included into the PDE. Recently, in 2019, Nwankwo et al. [49] replaced the infinite boundary with the far estimate boundary proposed by Egorova et al. [21] while adopting the logarithmic transformation to map the free boundary for each regime to multi-fixed intervals to obtain second-order accurate solutions. Later, in 2022, Song et al. [53] obtained an approximation for pricing model by utilizing a coupled linear complementarity problem (LCP) on a bounded domain. However, all these methods still suffer a common drawback in terms of numerical efficiency as a result of the presence of multiple optimal exercise prices within a regime-switching model; the calculation of them is characterized by their coupling status. Although some methods [21,49] could be extended to multiple dimensions, these numerical approaches are still coupled in a finite dimension, i.e. each economical state is presented sequentially, which means the calculation of the option price and the optimal exercise price need to be solved sequentially as shown before. Ideally, any new approach that offers a potential to decouple them would immediately be very attractive as far as improving computational efficiency is concerned.

In this paper, we present a novel computational approach based on the integral equation to enhance the computation efficiency when the price of an American option needs to be numerically computed. A distinctive feature of our approach, in comparison with other numerical approaches, is that the coupled PDEs in a PDE system have been decoupled in the Fourier space, resulting in an integral equation for each economical state to be solved sequentially and thus has greatly reduced computational effort. In the same framework, our approach can be extended to pricing American options with any finite number of regimes and with finite maturity, which has been discussed by Zheng and Zhu [60]. However, the current approach proposed in this paper has its unique merits; there is no need to resort to a pre-calculation analysis involving a complicated computation on matrices and eigenvalues and thus has a higher computational speed, especially for a lower number of regime states. Moreover, the two-state regime-switching model is the one that has been most popularly used for a great balance between simplicity and representation of the economical state in the sense that usually there are two distinct states of the economy: growth or recession.

This paper is organized as follows. The regime-switching model is introduced in Section 2. In Section 3, a detailed description of the integral equation approach is provided, together with a formula for the option price and the optimal exercise price. The numerical examples are presented in Section 4 and followed by some concluding remarks in the last section.

2. The model

In this section, we will consider an American put option under regime-switching model described previously by Guo [26] and Buffington and Elliott [13]. Let S be an underlying asset price followed by the stochastic different equation: (1) $\mathrm{d}{S}_t=r{S}_t\mathrm{d}t+{\sigma }_{X_t}{S}_t\mathrm{d}{W}_t,$(1) where r is the risk-free interest rate. $W_t$ is a standard Brownian motion independent of $X_t$. In addition, $X_t\in \{e_1,\dots \cdots ,e_N\}$ is a finite state Markov chain, which could move among a few states corresponding to different values of the volatility ${\sigma }_{X_t}$, as discussed by Buffington and Elliott [13]. Moreover, for each given state $X_t$, the volatility ${\sigma }_{X_t}$ is constant and known.

To simplify our discussion, in the following we focus on a special case that $X_t$ is a two-state Markov chain, in which $X_t$ alternates between 1 and 2 such that (2) $X_t=\{\begin{array}{ll}1,& \text{when the economy is in a state of growth,}\\ 2,& \text{when the economy is in a state of recession.}\end{array}$(2) Moreover, as a regime generator of the Markov chain $X_t$, Q is assumed to be given as (3) $Q=\left(\begin{array}{cc}{\lambda }_1& -{\lambda }_1\\ -{\lambda }_2& {\lambda }_2\end{array}\right),$(3) where ${\lambda }_i$ is the transition rate from regime i to the other regime. Actually, as a hidden Markov chain generated by Q [57], the transition probability $P_i$ from regime i to the other regime is represented as a Poisson process with (4) $P_i(t_i>t)={\text{e}}^{-{\lambda }_{i}t},i=1,2,$(4) where $t_i$ is the time spent in regime i before moving to the other regime.

Under the above assumption, the two American put options $V_1(S,t)$ and $V_2(S,t)$ under a two-state regime-switching model satisfy the following coupled PDEs: $\begin{array}{rl}\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_1^2{S}^2\frac{{\mathrm{\partial }}^2{V}_1}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}-r{V}_1& ={\lambda }_1(V_1-V_2),\\ \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2& ={\lambda }_2(V_2-V_1).\end{array}$These above option values $V_1(S,t)$ and $V_2(S,t)$ have two optimal exercise prices, $S_{f_1}$ and $S_{f_2}$, corresponding to ${\sigma }_1$ and ${\sigma }_2$, respectively. Under the assumption ${\sigma }_1<{\sigma }_2$, Yi [56] proved that $V_1<V_2$ and $S_{f_1}>S_{f_2}$ for each $t\in [0,T]$, where T is the expiration time of the option. Therefore, as shown in Figure 1, the optimal exercise boundaries $S_{f_1}$ and $S_{f_2}$ divide the pricing domain into two regions: a common continuation and a transition region, as discussed by Buffington and Elliott [13]. In other words, the stock price S is in the common continuation region if $S>S_{f_1}$, and in the transition region if $S_{f_2}<S<S_{f_1}$. Later, we will describe how to determine the computed results obtained from a regime-switching model as the American options based on Figure 1. Now, in the following it is important to present the PDE systems for each region.

Figure 1. The option pricing domain with two optimal exercise boundaries.

Firstly, we discuss the case of S in the common continuation region. When $S>S_{f_1}$, the two coupled PDE systems for the value of the American put option with the boundary conditions can be derived as (5) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_1^2{S}^2\frac{{\mathrm{\partial }}^2{V}_1}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}-r{V}_1={\lambda }_1(V_1-V_2),}\\ V_1(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_1(S,t)=0,\\ V_1(S_{f_1}(t),t)=K-S_{f_1}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}(S_{f_1}(t),t)=-1,}\end{array}\end{array}$(5) (6) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}\end{array}$(6) where $V_j(S,t)(j=1,2)$ is the American put option price in state j, r is the risk-free interest rate, K is the strike price and T is the expiration time of the option. $S_{f_j}(j=1,2)$ is the optimal exercise value. It should be noted that each of the above systems contains a second-order inhomogeneous PDE for the option price of one state, while the other state are ‘coupled’ in the differential operator applied to them.

When S in the transition region $(S_{f_2}<S<S_{f_1})$, the value of $V_1$ is the intrinsic value indeed. Thus the PDE system can be obtained as (7) $\{\begin{array}{l}{V}_1(S,t)=K-S,\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=K-S,\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}$(7) with the same stated parameters.

To solve Systems (5(5) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_1^2{S}^2\frac{{\mathrm{\partial }}^2{V}_1}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}-r{V}_1={\lambda }_1(V_1-V_2),}\\ V_1(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_1(S,t)=0,\\ V_1(S_{f_1}(t),t)=K-S_{f_1}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}(S_{f_1}(t),t)=-1,}\end{array}\end{array}$(5) )–(7(7) $\{\begin{array}{l}{V}_1(S,t)=K-S,\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=K-S,\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}$(7) ), we shall first make all variables dimensionless by the following dimensionless variables: $x=\ln \left[\frac{S}{K}\right],{\tau }_j=\frac{{\sigma }_j^2}{2}(T-t),u_j(x,{\tau }_j)=\frac{V_j(S,t)}{K},p_j(\tau )=\ln \left[\frac{S_{f_j}(t)}{K}\right],\text{for}j=1,2.$With the dimensionless, Systems (5(5) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_1^2{S}^2\frac{{\mathrm{\partial }}^2{V}_1}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}-r{V}_1={\lambda }_1(V_1-V_2),}\\ V_1(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_1(S,t)=0,\\ V_1(S_{f_1}(t),t)=K-S_{f_1}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}(S_{f_1}(t),t)=-1,}\end{array}\end{array}$(5) ) and (6(6) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}\end{array}$(6) ) in the common continuation region become (8) $\begin{array}{rl}& \{\begin{array}{l}{\displaystyle -\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }{\tau }_1}+\frac{{\mathrm{\partial }}^2{u}_1}{\mathrm{\partial }{x}^2}+({\alpha }_1-1)\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }x}-({\alpha }_1+{\beta }_1)u_1=-{\beta }_1{u}_2,}\\ u_1(x,0)=max(1-{\text{e}}^x,0),\\ \underset{x\to \mathrm{\infty }}{lim}{u}_1(x,{\tau }_1)=0,\\ u_1(p_1({\tau }_1),{\tau }_1)=1-{\text{e}}^{p_1({\tau }_1)},\\ {\displaystyle \frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }x}(p_1({\tau }_1),{\tau }_1)=-{\text{e}}^{p_1({\tau }_1)},}\end{array}\end{array}$(8) (9) $\begin{array}{rl}& \{\begin{array}{l}{\displaystyle -\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{\tau }_2}+\frac{{\mathrm{\partial }}^2{u}_2}{\mathrm{\partial }{x}^2}+({\alpha }_2-1)\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}-({\alpha }_2+{\beta }_2)u_2=-{\beta }_2{u}_1,}\\ u_2(x,0)=max(1-{\text{e}}^x,0),\\ \underset{x\to \mathrm{\infty }}{lim}{u}_2(x,{\tau }_2)=0,\\ u_2(p_2({\tau }_2),{\tau }_2)=1-{\text{e}}^{p_2({\tau }_2)},\\ {\displaystyle \frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}(p_2({\tau }_2),{\tau }_2)=-{\text{e}}^{p_2({\tau }_2)},}\end{array}\end{array}$(9) where ${\alpha }_j=\frac{2r}{{\sigma }_j^2},{\beta }_j=\frac{2{\lambda }_j}{{\sigma }_j^2},\text{for}j=1,2.$Additionally, System (7(7) $\{\begin{array}{l}{V}_1(S,t)=K-S,\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=K-S,\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}$(7) ) in the continuation region becomes (10) $\{\begin{array}{l}{\displaystyle -\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{\tau }_2}+\frac{{\mathrm{\partial }}^2{u}_2}{\mathrm{\partial }{x}^2}+({\alpha }_2-1)\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}-({\alpha }_2+{\beta }_2)u_2=-{\beta }_2(1-{\text{e}}^x),}\\ u_2(x,0)=1-{\text{e}}^x,\\ \underset{x\to \mathrm{\infty }}{lim}{u}_2(x,{\tau }_2)=0,\\ u_2(p_2({\tau }_2),{\tau }_2)=1-{\text{e}}^{p_2({\tau }_2)},\\ {\displaystyle \frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}(p_2({\tau }_2),{\tau }_2)=-{\text{e}}^{p_2({\tau }_2)}.}\end{array}$(10) By now, two dimensionless coupled PDE systems have been derived. In the following section, the solution techniques using incomplete Fourier transformation to obtain integral equation approaches for Systems (8(8) $\begin{array}{rl}& \{\begin{array}{l}{\displaystyle -\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }{\tau }_1}+\frac{{\mathrm{\partial }}^2{u}_1}{\mathrm{\partial }{x}^2}+({\alpha }_1-1)\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }x}-({\alpha }_1+{\beta }_1)u_1=-{\beta }_1{u}_2,}\\ u_1(x,0)=max(1-{\text{e}}^x,0),\\ \underset{x\to \mathrm{\infty }}{lim}{u}_1(x,{\tau }_1)=0,\\ u_1(p_1({\tau }_1),{\tau }_1)=1-{\text{e}}^{p_1({\tau }_1)},\\ {\displaystyle \frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }x}(p_1({\tau }_1),{\tau }_1)=-{\text{e}}^{p_1({\tau }_1)},}\end{array}\end{array}$(8) )–(10(10) $\{\begin{array}{l}{\displaystyle -\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{\tau }_2}+\frac{{\mathrm{\partial }}^2{u}_2}{\mathrm{\partial }{x}^2}+({\alpha }_2-1)\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}-({\alpha }_2+{\beta }_2)u_2=-{\beta }_2(1-{\text{e}}^x),}\\ u_2(x,0)=1-{\text{e}}^x,\\ \underset{x\to \mathrm{\infty }}{lim}{u}_2(x,{\tau }_2)=0,\\ u_2(p_2({\tau }_2),{\tau }_2)=1-{\text{e}}^{p_2({\tau }_2)},\\ {\displaystyle \frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}(p_2({\tau }_2),{\tau }_2)=-{\text{e}}^{p_2({\tau }_2)}.}\end{array}$(10) ) will be discussed.

3. An integral equation approach for pricing American put options under regime-switching model

In this section, the integral equation approaches of American options under regime-switching model will be presented with the method of incomplete Fourier transformation. The coupled PDE systems will be decoupled in the Fourier space, resulting a completely decoupled integral equation for each economical states.

To make the computation easier, we will first consider the systems in the common continuation region. Now, using Fourier transform defined as ${\hat{u}}_j(\eta ,{\tau }_j)=\mathcal{F}\{u_j(x,{\tau }_j)\}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{u}_j(x,{\tau }_j){\text{e}}^{-i\eta x}\mathrm{d}x,$System (8(8) $\begin{array}{rl}& \{\begin{array}{l}{\displaystyle -\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }{\tau }_1}+\frac{{\mathrm{\partial }}^2{u}_1}{\mathrm{\partial }{x}^2}+({\alpha }_1-1)\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }x}-({\alpha }_1+{\beta }_1)u_1=-{\beta }_1{u}_2,}\\ u_1(x,0)=max(1-{\text{e}}^x,0),\\ \underset{x\to \mathrm{\infty }}{lim}{u}_1(x,{\tau }_1)=0,\\ u_1(p_1({\tau }_1),{\tau }_1)=1-{\text{e}}^{p_1({\tau }_1)},\\ {\displaystyle \frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }x}(p_1({\tau }_1),{\tau }_1)=-{\text{e}}^{p_1({\tau }_1)},}\end{array}\end{array}$(8) ) and System (9(9) $\begin{array}{rl}& \{\begin{array}{l}{\displaystyle -\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{\tau }_2}+\frac{{\mathrm{\partial }}^2{u}_2}{\mathrm{\partial }{x}^2}+({\alpha }_2-1)\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}-({\alpha }_2+{\beta }_2)u_2=-{\beta }_2{u}_1,}\\ u_2(x,0)=max(1-{\text{e}}^x,0),\\ \underset{x\to \mathrm{\infty }}{lim}{u}_2(x,{\tau }_2)=0,\\ u_2(p_2({\tau }_2),{\tau }_2)=1-{\text{e}}^{p_2({\tau }_2)},\\ {\displaystyle \frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}(p_2({\tau }_2),{\tau }_2)=-{\text{e}}^{p_2({\tau }_2)},}\end{array}\end{array}$(9) ) will be changed to the following ordinary differential equation (ODE) systems: (11) $\begin{array}{rl}& \{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{\hat{u}}_1}{\mathrm{\partial }{\tau }_1}+B_1{\hat{u}}_1={\beta }_1{\hat{u}}_2+f_1(\eta ,{\tau }_1),}\\ {\displaystyle {\hat{u}}_{10}={\hat{u}}_1(0)={\int }_{p_1(0)}^{+\mathrm{\infty }}{\text{e}}^{-i\eta x}(1-{\text{e}}^x{)}^{+}\mathrm{d}x,}\end{array}\end{array}$(11) (12) $\begin{array}{rl}& \{\begin{array}{l}{\displaystyle A\frac{\mathrm{\partial }{\hat{u}}_2}{\mathrm{\partial }{\tau }_1}+B_2{\hat{u}}_2={\beta }_2{\hat{u}}_1+f_2(\eta ,{\tau }_1),}\\ {\hat{u}}_{20}={\hat{u}}_2(0)={\int }_{p_2(0)}^{+\mathrm{\infty }}{\text{e}}^{-i\eta x}(1-{\text{e}}^x{)}^{+}\mathrm{d}x,\end{array}\end{array}$(12) where $A=\frac{{\tau }_1}{{\tau }_2}=\frac{{\sigma }_1^2}{{\sigma }_2^2},B_j(\eta )={\eta }^2-i\eta ({\alpha }_j-1)+{\alpha }_j+{\beta }_j,$and $f_j(\eta ,{\tau }_1)={\text{e}}^{-i\eta p_j({\tau }_1)}\{{\text{e}}^{p_j({\tau }_1)}-[i\eta +({\alpha }_j-1)+p_j^{\prime }({\tau }_1)][1-{\text{e}}^{p_j({\tau }_1)}]\},\text{for}j=1,2.$One should also note that Systems (11(11) $\begin{array}{rl}& \{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{\hat{u}}_1}{\mathrm{\partial }{\tau }_1}+B_1{\hat{u}}_1={\beta }_1{\hat{u}}_2+f_1(\eta ,{\tau }_1),}\\ {\displaystyle {\hat{u}}_{10}={\hat{u}}_1(0)={\int }_{p_1(0)}^{+\mathrm{\infty }}{\text{e}}^{-i\eta x}(1-{\text{e}}^x{)}^{+}\mathrm{d}x,}\end{array}\end{array}$(11) ) and (12(12) $\begin{array}{rl}& \{\begin{array}{l}{\displaystyle A\frac{\mathrm{\partial }{\hat{u}}_2}{\mathrm{\partial }{\tau }_1}+B_2{\hat{u}}_2={\beta }_2{\hat{u}}_1+f_2(\eta ,{\tau }_1),}\\ {\hat{u}}_{20}={\hat{u}}_2(0)={\int }_{p_2(0)}^{+\mathrm{\infty }}{\text{e}}^{-i\eta x}(1-{\text{e}}^x{)}^{+}\mathrm{d}x,\end{array}\end{array}$(12) ) still consist of coupled differential equation with those corresponding initial conditions in the Fourier space, each of which contains an inhomogeneous first-order nonlinear ODE for the option price of one state, while the other state is coupled in the differential operator applied to.

Solving the first-order linear ODEs is relatively straightforward. The result is (13) $\begin{array}{rl}{\hat{u}}_1(\eta ,{\tau }_1)& ={\hat{u}}_{10}[c_{11}(\eta ){\text{e}}^{{\mu }_1{\tau }_1}+c_{12}(\eta ){\text{e}}^{{\mu }_2{\tau }_1}]\\ & +{\int }_0^{{\tau }_1}{f}_1(\eta ,\xi )[c_{11}(\eta ){\text{e}}^{{\mu }_1({\tau }_1-\xi )}+c_{12}(\eta ){\text{e}}^{{\mu }_2({\tau }_1-\xi )}]\mathrm{d}\xi ,\end{array}$(13) (14) $\begin{array}{rl}{\hat{u}}_2(\eta ,{\tau }_2)& ={\hat{u}}_{20}[c_{21}(\eta ){\text{e}}^{A{\mu }_1{\tau }_2}+c_{22}(\eta ){\text{e}}^{A{\mu }_2{\tau }_2}]\\ & +{\int }_0^{A{\tau }_2}\frac{f_2(\eta ,\xi )}{A}[c_{21}(\eta ){\text{e}}^{{\mu }_1(A{\tau }_2-\xi )}+c_{22}(\eta ){\text{e}}^{{\mu }_2(A{\tau }_2-\xi )}]\mathrm{d}\xi ,\end{array}$(14) where $\begin{array}{rl}{\mu }_1(\eta )& =\frac{1}{2A}[\rho (\eta )-2B_2(\eta )+\mathrm{\Delta }(\eta )],{\mu }_2(\eta )=\frac{1}{2A}[\rho (\eta )-2B_2(\eta )-\mathrm{\Delta }(\eta )],\\ \rho (\eta )& =B_2(\eta )-A{B}_1(\eta ),\mathrm{\Delta }(\eta )=\sqrt{{\rho }^2+4A{\beta }_1{\beta }_2},\end{array}$and (15) $\begin{array}{rl}{c}_{11}(\eta )& =\frac{{\beta }_1-B_1(\eta )-{\mu }_2}{{\mu }_1-{\mu }_2}=\frac{1}{2}+\frac{2A{\beta }_1+\rho }{2\mathrm{\Delta }(\eta )},\\ c_{12}(\eta )& =-\frac{{\beta }_1-B_1(\eta )-{\mu }_1}{{\mu }_1-{\mu }_2}=\frac{1}{2}-\frac{2A{\beta }_1+\rho }{2\mathrm{\Delta }(\eta )},\\ c_{21}(\eta )& =\frac{{\beta }_2-B_2(\eta )-A{\mu }_2}{A({\mu }_1-{\mu }_2)}=\frac{1}{2}+\frac{2{\beta }_2-\rho }{2\mathrm{\Delta }(\eta )},\\ c_{22}(\eta )& =-\frac{{\beta }_2-B_2(\eta )-A{\mu }_1}{A({\mu }_1-{\mu }_2)}=\frac{1}{2}-\frac{2{\beta }_2-\rho }{2\mathrm{\Delta }(\eta )},\end{array}$(15) which is obviously $c_{11}+c_{12}=c_{21}+c_{22}=1.$By now, decoupled integral equation formulation in the Fourier space has been derived. Resulting in the integral equations, the coupled Systems (5(5) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_1^2{S}^2\frac{{\mathrm{\partial }}^2{V}_1}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}-r{V}_1={\lambda }_1(V_1-V_2),}\\ V_1(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_1(S,t)=0,\\ V_1(S_{f_1}(t),t)=K-S_{f_1}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}(S_{f_1}(t),t)=-1,}\end{array}\end{array}$(5) ) and (6(6) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}\end{array}$(6) ) in the common continuation region are decoupled in the Fourier space. In other words, the formulation of the original PDE in each regime can be calculated sequentially in the Fourier space. To obtain the formulations in the original space, define the following Fourier inversion transform: (16) $u_j(x,{\tau }_j)={\mathcal{F}}^{-1}\{{\hat{u}}_j(\eta ,{\tau }_j)\}=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\hat{u}}_j(\eta ,{\tau }_j){\text{e}}^{i\eta x}\mathrm{d}x.$(16) Applying the transform to Equations (13(13) $\begin{array}{rl}{\hat{u}}_1(\eta ,{\tau }_1)& ={\hat{u}}_{10}[c_{11}(\eta ){\text{e}}^{{\mu }_1{\tau }_1}+c_{12}(\eta ){\text{e}}^{{\mu }_2{\tau }_1}]\\ & +{\int }_0^{{\tau }_1}{f}_1(\eta ,\xi )[c_{11}(\eta ){\text{e}}^{{\mu }_1({\tau }_1-\xi )}+c_{12}(\eta ){\text{e}}^{{\mu }_2({\tau }_1-\xi )}]\mathrm{d}\xi ,\end{array}$(13) ) and (14(14) $\begin{array}{rl}{\hat{u}}_2(\eta ,{\tau }_2)& ={\hat{u}}_{20}[c_{21}(\eta ){\text{e}}^{A{\mu }_1{\tau }_2}+c_{22}(\eta ){\text{e}}^{A{\mu }_2{\tau }_2}]\\ & +{\int }_0^{A{\tau }_2}\frac{f_2(\eta ,\xi )}{A}[c_{21}(\eta ){\text{e}}^{{\mu }_1(A{\tau }_2-\xi )}+c_{22}(\eta ){\text{e}}^{{\mu }_2(A{\tau }_2-\xi )}]\mathrm{d}\xi ,\end{array}$(14) ) and after some tedious algebraic manipulations (see Appendix A), we obtain (17) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}+c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]+i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(17) Additionally, (18) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_2(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(18) It should be emphasized Equations (17(17) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}+c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]+i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(17) ) and (18(18) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_2(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(18) ) consist of decoupled integral equation formulations, each of which contains only one nonlinear integral equation formulations for the option price of one regime. Equations (17(17) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}+c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]+i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(17) ) and (18(18) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_2(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(18) ) could be directly computed, when $S>S_{f1}$ and $S>S_{f2}$, respectively, to determine the values of American options in the regime-switching model. However, both Equations (17(17) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}+c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]+i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(17) ) and (18(18) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_2(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(18) ) involve their associated optimal exercise values, $S_{f1}(\tau )$ and $S_{f2}(\tau )$, respectively, which still remain unknown. Hence, for the optimal exercise prices, the boundary conditions would be applied [20]. Fortunately, two integral equations for the optimal exercise prices can be derived by using the boundaries conditions (19) $\begin{array}{rl}\frac{1-{\text{e}}^{p_1({\tau }_1)}}{2}& =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(p_1({\tau }_1)-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(p_1({\tau }_1)-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi ,\end{array}$(19) and (20) $\begin{array}{rl}\frac{1-{\text{e}}^{p_2({\tau }_2)}}{2}& =\frac{1}{2\pi }{\int }_{p_2(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(p_2({\tau }_2)-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(p_2({\tau }_2)-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){]}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(20) The factor of $\frac{1}{2}$ appears in the left of the optimal exercise price equations due to the properties of the Fourier transform. As proved by Dettman [20], it can be viewed as the complete Fourier transform of a discontinuous function, and the inverted Fourier transform of a discontinuous function will converge to the midpoint of the discontinuity. Moreover, it should be noted that the first term of Equations (17(17) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}+c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]+i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(17) )–(20(20) $\begin{array}{rl}\frac{1-{\text{e}}^{p_2({\tau }_2)}}{2}& =\frac{1}{2\pi }{\int }_{p_2(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(p_2({\tau }_2)-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(p_2({\tau }_2)-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){]}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(20) ) is missing because of $p_j(0)=\ln \frac{S_{fj}(T)}{K}=0.$Overall, to determine the price of American put options, the optimal exercise prices $p_i({\tau }_i)$ should be computed first from Equations (19(19) $\begin{array}{rl}\frac{1-{\text{e}}^{p_1({\tau }_1)}}{2}& =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(p_1({\tau }_1)-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(p_1({\tau }_1)-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi ,\end{array}$(19) )–(20(20) $\begin{array}{rl}\frac{1-{\text{e}}^{p_2({\tau }_2)}}{2}& =\frac{1}{2\pi }{\int }_{p_2(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(p_2({\tau }_2)-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(p_2({\tau }_2)-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){]}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(20) ). Furthermore, the prices of American options Equations (17(17) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}+c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]+i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(17) ) and (18(18) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_2(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(18) ) could be summarized as (21) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \end{array}$(21) and (22) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(22) In terms of the original dimensional parameters, the optimal exercise prices $S_{f1}$ and $S_{f2}$ can be rewritten as (23) $\begin{array}{rl}{S}_{f1}({\tau }_1)& =K{\text{e}}^{p_1({\tau }_1)}\\ & =K-\frac{K}{\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \end{array}$(23) and (24) $\begin{array}{rl}{S}_{f2}({\tau }_2)& =K{\text{e}}^{p_2({\tau }_2)}\\ & =K-\frac{K}{A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_2^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(24) By now, the integral equation approach for pricing American options in the common continuation region under regime-switching is presented. The solutions of integral equations Equations (23(23) $\begin{array}{rl}{S}_{f1}({\tau }_1)& =K{\text{e}}^{p_1({\tau }_1)}\\ & =K-\frac{K}{\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \end{array}$(23) )–(24(24) $\begin{array}{rl}{S}_{f2}({\tau }_2)& =K{\text{e}}^{p_2({\tau }_2)}\\ & =K-\frac{K}{A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_2^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(24) ) will be the functions of the optimal exercise prices $S_{f1}$ and $S_{f2}$, respectively, which can be carried over into Equations (21(21) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \end{array}$(21) )–(22(22) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(22) ) to calculate the option price.

On the other hand, the next step is to solve System (10(10) $\{\begin{array}{l}{\displaystyle -\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{\tau }_2}+\frac{{\mathrm{\partial }}^2{u}_2}{\mathrm{\partial }{x}^2}+({\alpha }_2-1)\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}-({\alpha }_2+{\beta }_2)u_2=-{\beta }_2(1-{\text{e}}^x),}\\ u_2(x,0)=1-{\text{e}}^x,\\ \underset{x\to \mathrm{\infty }}{lim}{u}_2(x,{\tau }_2)=0,\\ u_2(p_2({\tau }_2),{\tau }_2)=1-{\text{e}}^{p_2({\tau }_2)},\\ {\displaystyle \frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}(p_2({\tau }_2),{\tau }_2)=-{\text{e}}^{p_2({\tau }_2)}.}\end{array}$(10) ) in the continuation region when $S_{f2}<S<S_{f1}$. Using Fourier transform defined as ${\hat{u}}_j(\eta ,{\tau }_j)=\mathcal{F}\{u_j(x,{\tau }_j)\}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{u}_j(x,{\tau }_j){\text{e}}^{-i\eta x}\mathrm{d}x.$System (10(10) $\{\begin{array}{l}{\displaystyle -\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{\tau }_2}+\frac{{\mathrm{\partial }}^2{u}_2}{\mathrm{\partial }{x}^2}+({\alpha }_2-1)\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}-({\alpha }_2+{\beta }_2)u_2=-{\beta }_2(1-{\text{e}}^x),}\\ u_2(x,0)=1-{\text{e}}^x,\\ \underset{x\to \mathrm{\infty }}{lim}{u}_2(x,{\tau }_2)=0,\\ u_2(p_2({\tau }_2),{\tau }_2)=1-{\text{e}}^{p_2({\tau }_2)},\\ {\displaystyle \frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }x}(p_2({\tau }_2),{\tau }_2)=-{\text{e}}^{p_2({\tau }_2)}.}\end{array}$(10) ) will be changed to the following ODE system: (25) $\{\begin{array}{l}{\displaystyle A\frac{\mathrm{\partial }{\hat{u}}_2}{\mathrm{\partial }{\tau }_1}+B_2{\hat{u}}_2=g(\eta ,{\tau }_1),}\\ {\hat{u}}_{21}={\int }_{p_2(0)}^{p_1(0)}{\text{e}}^{-i\eta x}(1-{\text{e}}^x)\mathrm{d}x,\end{array}$(25) where $A=\frac{{\tau }_1}{{\tau }_2}=\frac{{\sigma }_1^2}{{\sigma }_2^2},B_2(\eta )={\eta }^2-i\eta ({\alpha }_2-1)+{\alpha }_2+{\beta }_2,$and $g(\eta ,{\tau }_1)={\int }_{p_2(0)}^{p_1(0)}{\beta }_2{\text{e}}^{-i\eta x}(1-{\text{e}}^x)\mathrm{d}x+{\text{e}}^{-i\eta p_2({\tau }_1)}\{{\text{e}}^{p_2({\tau }_1)}-[i\eta +({\alpha }_2-1)+p_2^{\prime }({\tau }_1)][1-{\text{e}}^{p_2({\tau }_1)}]\}.$Solving the first-order linear ODEs is a relatively straightforward. The result is (26) ${\hat{u}}_2(\eta ,{\tau }_2)={\int }_0^{{\tau }_1}g(\eta ,{\tau }_1){\text{e}}^{-B_2(\eta )({\tau }_1-\xi )}\mathrm{d}\xi .$(26) To obtain the formulation in the original space, we need to perform the Fourier inversion transform $u_2(x,{\tau }_1)={\mathcal{F}}^{-1}\{{\hat{u}}_2(\eta ,{\tau }_1\}=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\hat{u}}_2(\eta ,{\tau }_1){\text{e}}^{i\eta x}\mathrm{d}x.$Applying the transform to Equation (26(26) ${\hat{u}}_2(\eta ,{\tau }_2)={\int }_0^{{\tau }_1}g(\eta ,{\tau }_1){\text{e}}^{-B_2(\eta )({\tau }_1-\xi )}\mathrm{d}\xi .$(26) ) and after some computations (see Appendix B), we obtain (27) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1-({\text{e}}^x+{\text{e}}^{-{\alpha }_2{\tau }_1}){\text{e}}^{-{\beta }_2{\tau }_1}\mathcal{N}\left(\frac{x+({\alpha }_2-1){\tau }_1-p_2(0)}{2\sqrt{{\tau }_1}}\right)\\ & +{\beta }_2{\int }_0^{{\tau }_1}{\text{e}}^{x-{\beta }_2({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi \\ & -({\alpha }_2+{\beta }_2){\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi .\end{array}$(27) By using the similar boundary condition as Equation (14(14) $\begin{array}{rl}{\hat{u}}_2(\eta ,{\tau }_2)& ={\hat{u}}_{20}[c_{21}(\eta ){\text{e}}^{A{\mu }_1{\tau }_2}+c_{22}(\eta ){\text{e}}^{A{\mu }_2{\tau }_2}]\\ & +{\int }_0^{A{\tau }_2}\frac{f_2(\eta ,\xi )}{A}[c_{21}(\eta ){\text{e}}^{{\mu }_1(A{\tau }_2-\xi )}+c_{22}(\eta ){\text{e}}^{{\mu }_2(A{\tau }_2-\xi )}]\mathrm{d}\xi ,\end{array}$(14) ), the optimal exercise price $S_{f2}({\tau }_2)$ in the continuation region could be computed first, in order to determine the option price, as (28) $\begin{array}{rl}{S}_{f_2}({\tau }_2)& =K-\frac{K}{\pi }{\beta }_2(1-{\text{e}}^x)A{\tau }_2-2K({\text{e}}^x+{\text{e}}^{-{\alpha }_2{\tau }_1}){\text{e}}^{-A{\beta }_2{\tau }_2}\mathcal{N}\left(\frac{x+({\alpha }_2-1)A{\tau }_2-p_2(0)}{2\sqrt{A{\tau }_2}}\right)\\ & +2K{\beta }_2{\int }_0^{A{\tau }_2}{\text{e}}^{x-{\beta }_2(A{\tau }_2-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)(A{\tau }_2-\xi )-p_2(\xi )}{2\sqrt{A{\tau }_2-\xi }}\right)\mathrm{d}\xi \\ & -2K({\alpha }_2+{\beta }_2){\int }_0^{A{\tau }_2}{\text{e}}^{-({\alpha }_2+{\beta }_2)(A{\tau }_2-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)(A{\tau }_2-\xi )-p_2(\xi )}{2\sqrt{A{\tau }_2-\xi }}\right)\mathrm{d}\xi .\end{array}$(28) However, these integral equations are nonlinear and a numerical method is needed to derive the results. Hence, this solution procedure is numerically realized in the following section. In addition, some numerical examples are provided to demonstrate its accuracy and efficiency.

4. Numerical examples and discussion

In this section, we provide some numerical examples to demonstrate that numerical values can be easily produced from our integral equation approaches. Before a meaningful discussion is carried out, in the following, we will provide an outline of our numerical approach.

4.1. Numerical implementation

The discretization processes of the integral equations are shown in this section. The main task of the numerical approach of the integral equations is to find the optimal exercise prices $S_{f1}$ and $S_{f2}$ from Equations (23(23) $\begin{array}{rl}{S}_{f1}({\tau }_1)& =K{\text{e}}^{p_1({\tau }_1)}\\ & =K-\frac{K}{\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \end{array}$(23) ), (24(24) $\begin{array}{rl}{S}_{f2}({\tau }_2)& =K{\text{e}}^{p_2({\tau }_2)}\\ & =K-\frac{K}{A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_2^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(24) ) and (28(28) $\begin{array}{rl}{S}_{f_2}({\tau }_2)& =K-\frac{K}{\pi }{\beta }_2(1-{\text{e}}^x)A{\tau }_2-2K({\text{e}}^x+{\text{e}}^{-{\alpha }_2{\tau }_1}){\text{e}}^{-A{\beta }_2{\tau }_2}\mathcal{N}\left(\frac{x+({\alpha }_2-1)A{\tau }_2-p_2(0)}{2\sqrt{A{\tau }_2}}\right)\\ & +2K{\beta }_2{\int }_0^{A{\tau }_2}{\text{e}}^{x-{\beta }_2(A{\tau }_2-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)(A{\tau }_2-\xi )-p_2(\xi )}{2\sqrt{A{\tau }_2-\xi }}\right)\mathrm{d}\xi \\ & -2K({\alpha }_2+{\beta }_2){\int }_0^{A{\tau }_2}{\text{e}}^{-({\alpha }_2+{\beta }_2)(A{\tau }_2-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)(A{\tau }_2-\xi )-p_2(\xi )}{2\sqrt{A{\tau }_2-\xi }}\right)\mathrm{d}\xi .\end{array}$(28) ), which can be carried over into Equations (21(21) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \end{array}$(21) ), (22(22) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(22) ) and (27(27) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1-({\text{e}}^x+{\text{e}}^{-{\alpha }_2{\tau }_1}){\text{e}}^{-{\beta }_2{\tau }_1}\mathcal{N}\left(\frac{x+({\alpha }_2-1){\tau }_1-p_2(0)}{2\sqrt{{\tau }_1}}\right)\\ & +{\beta }_2{\int }_0^{{\tau }_1}{\text{e}}^{x-{\beta }_2({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi \\ & -({\alpha }_2+{\beta }_2){\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi .\end{array}$(27) ) to calculate the option prices. The procedure for finding the optimal exercise price is as follows:

First, Equation (23(23) $\begin{array}{rl}{S}_{f1}({\tau }_1)& =K{\text{e}}^{p_1({\tau }_1)}\\ & =K-\frac{K}{\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \end{array}$(23) ) is used to obtain the values of the function $S_{f1}$ when $\tau \in [0,{\tau }_M]$, where ${\tau }_M=\frac{{\sigma }^2}{2}T$. In this process, we discretize uniformly the time interval $0=s_1<s_2<\cdots <s_n<\cdots <s_M={\tau }_M,$where $s_n=(n-1)\ast s_M/(M-1),\mathrm{d}s=s_M/(M-1)$, and we choose N large enough so that we can comfortably put $-N={\eta }_1<{\eta }_2<\cdots <{\eta }_j<\cdots <{\eta }_J=N,$where ${\eta }_j=(j-1)\ast 2N/(N-1),\mathrm{d}\eta =2N/(N-1)$. Thus we obtain a set of nonlinear algebraic equations for $S_{f1}(s_m)$ (denoted by $S_{f1}^{(m)}$) for $m=1,2,\dots \cdots M,$ (29) $\begin{array}{rl}{S}_{f1}^{(m)}& =K-{\mathrm{\Sigma }}_{n=1}^{m-1}\frac{K}{\pi }\mathrm{d}s{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})(s_m-s_n)-\frac{\{2A[\ln S_{f1}^{(m)}-\ln S_{f1}^{(n)}]+(2{\alpha }_2-A-1)(s_m-s_n){\}}^2}{8A(1+A)(s_m-s_n)}}\\ & \times \{{\mathrm{\Sigma }}_{j=1}^J\mathrm{d}\eta [c_{11}({\eta }_j){\text{ e}}^{\frac{\mathrm{\Delta }({\eta }_j)}{2A}(s_m-s_n)}+c_{12}({\eta }_j){\text{ e}}^{-\frac{\mathrm{\Delta }({\eta }_j)}{2A}(s_m-s_n)}]\\ & \times {\text{e}}^{-\frac{(1+A)(s_m-s_n)}{2A}{\{{\eta }_j-i\frac{2A[\ln S_{f1}^{(m)}-\ln S_{f1}^{(n)}]+(2{\alpha }_2-A-1)(s_m-s_n)}{2(A+1)(s_m-s_n)}\}}^2}\\ & \times \{\frac{S_{f1}^{(n)}}{K}-[{\alpha }_1-1+\frac{S_{f1}^{(m)}-S_{f_1}^{(n)}}{(m-n)\mathrm{d}s}][1-\frac{S_{f1}^{(n)}}{K}]-i{\eta }_j[1-\frac{S_{f1}^{(n)}}{K}]\}\end{array}$(29) Since the terminal value of the optimal exercise price, $S_{f_1}^{(1)}=K$, is known, Equation (23(23) $\begin{array}{rl}{S}_{f1}({\tau }_1)& =K{\text{e}}^{p_1({\tau }_1)}\\ & =K-\frac{K}{\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_1({\tau }_1)-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \end{array}$(23) ) can be calculated using MATLAB built-in root-finding function (fsolve). The above procedure is similar to the one used in [36]. The similar discretization processes of Equations (24(24) $\begin{array}{rl}{S}_{f2}({\tau }_2)& =K{\text{e}}^{p_2({\tau }_2)}\\ & =K-\frac{K}{A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[p_2({\tau }_2)-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_2^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(24) ) and (28(28) $\begin{array}{rl}{S}_{f_2}({\tau }_2)& =K-\frac{K}{\pi }{\beta }_2(1-{\text{e}}^x)A{\tau }_2-2K({\text{e}}^x+{\text{e}}^{-{\alpha }_2{\tau }_1}){\text{e}}^{-A{\beta }_2{\tau }_2}\mathcal{N}\left(\frac{x+({\alpha }_2-1)A{\tau }_2-p_2(0)}{2\sqrt{A{\tau }_2}}\right)\\ & +2K{\beta }_2{\int }_0^{A{\tau }_2}{\text{e}}^{x-{\beta }_2(A{\tau }_2-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)(A{\tau }_2-\xi )-p_2(\xi )}{2\sqrt{A{\tau }_2-\xi }}\right)\mathrm{d}\xi \\ & -2K({\alpha }_2+{\beta }_2){\int }_0^{A{\tau }_2}{\text{e}}^{-({\alpha }_2+{\beta }_2)(A{\tau }_2-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)(A{\tau }_2-\xi )-p_2(\xi )}{2\sqrt{A{\tau }_2-\xi }}\right)\mathrm{d}\xi .\end{array}$(28) ) will not be presented in detail. Once the values of the functions $S_{f1}^{(m)}$ and $S_{f2}^{(m)}$ are obtained, the values of the option prices can be straightforwardly computed through Equations (21(21) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \end{array}$(21) ), (22(22) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(22) ) and (27(27) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1-({\text{e}}^x+{\text{e}}^{-{\alpha }_2{\tau }_1}){\text{e}}^{-{\beta }_2{\tau }_1}\mathcal{N}\left(\frac{x+({\alpha }_2-1){\tau }_1-p_2(0)}{2\sqrt{{\tau }_1}}\right)\\ & +{\beta }_2{\int }_0^{{\tau }_1}{\text{e}}^{x-{\beta }_2({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi \\ & -({\alpha }_2+{\beta }_2){\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi .\end{array}$(27) ).

4.2. Numerical examples

In this section, numerical examples are provided to illustrate various properties of American put options under the regime-switching model and show the difference between Black–Scholes model and the two-state regime-switching model. Furthermore, we focus on accuracy and then efficiency of the regime switching model by using our new integral equation approach. The implementation of the schemes has been done by using Matlab R2020a on Dual-Core Intel Core i5 1.60 GHz.

To verify the accuracy of the integral equation approach, we first check the degeneracy of the regime-switching model to the classical Black–Scholes model by setting the same values of the transition rates ${\lambda }_j$ and the volatilities ${\sigma }_j$ (for j = 1, 2). For instance, if the ${\lambda }_j$ and ${\sigma }_j$ are set to 0 and 0.3, respectively, with K = 100, T = 1, r = 0.1, depicted in Figure 2 is the comparison of optimal exercise prices calculated from our approach, with those obtained from Kim's formula [36]. The real line and the dash lines are indicating Kim's result under Black–Scholes model and the optimal exercise prices $S{f}_j$ (for j = 1, 2) under the regime-switching model, respectively. As can be seen from Figure 2, the curves of two optimal exercise prices become the same one since the systems degenerate the same as well. In addition, an overall agreement is shown particularly for smaller time to expiry. When the time to expiry is increased, the differences between the two become larger. This is expected as the truncation error increases with the time to expiration when the double integration is calculated, whereas the problem with Kim's is to reduce the truncation error by the normal distribution function.

Figure 2. The optimal exercise boundaries when the regime-switching model degenerate to Black–Scholes model.

Another way to validate the accuracy is to compare the integral equation approach with the perpetual American option, when the expiry date T tends very large. The analytical solution of the perpetual American option under the regime-switching model was obtained by Guo and Zhang [59]. In addition, Lu and Putri [41] showed their results for the perpetual American option in a semi-analytic method. As shown in Table 1, our results are compared with those under the two-state regime-switching model, with $r=0.03,{\sigma }_1=0.9,{\sigma }_2=0.5,{\lambda }_1={\lambda }_2=1$ and K = 5. It is evident that our results are very close to the analytical results.

Table 1. Optimal exercise price of perpetual American puts.

Analytic results [59]		Lu and Purti's results [41]		Our results
$S_{f1}$	$S_{f2}$	$S_{f1}$	$S_{f2}$	$S_{f1}$	$S_{f2}$
0.44055	0.61156	0.44059	0.61163	0.44053	0.61129

As far as the computational efficiency is concerned, we make first comparison of the CPU times consumed by computing our integral equation, when the regime-switching model degenerate to the classical Black–Scholes, and Kim's integral equation to obtain an optimal exercise price. As shown in Table 2, it is obvious to see that our approach only takes 0.57 s to compute the optimal exercise prices with the time to expiry T = 1, while it takes 6.74 s with Kim's integral equation. In other words, our approach is nearly 10 times faster than the Kim's. In the iteration of Kim's integral equation, the normal distribution function needs to be recurred repeatedly, which leads to an increase in computational speed. In addition, it is well known that the efficiency of the integral equations is better than that of FDM [14,32] in the classical Black–Scholes model or under the regime-switching model, which could illustrate the efficiency of our approach.

Table 2. Comparison of the CPU time.

	CPU time
Our equation	0.57 s
Kim's equation	6.74 s

Further to verify the efficiency for American options with a finite time horizon under a two-state model, we compare CPU times consumed by computing our integral equation and Monte Carlo simulation to obtain the option prices. As present in Table 3, our approach only takes 6.80 s to compute the option prices, while it takes 15.4 s with Monte Carlo simulation, which means that the time savings through our new approach is about 50$\mathrm{\%}$ over Monte Carlo simulation. Although Monte Carlo simulation is very convenient, it does not stand out when efficiency is considered, especially in the lower dimensions.

Table 3. Comparison of the CPU time.

	CPU time
Our equation	6.80 s
Monte Carlo simulation	15.4 s

Now, for American options with a finite time horizon, we will check accuracy if our approach performs well for evaluating American options. As far as the option prices is concerned, we compare our results with results obtained through Monte Carlo simulations and those published results by using the tree method [38] and the finite difference method [55]. The calculation was carried out with the parameter values: $r=0.1,{\sigma }_1=0.4,{\sigma }_2=0.2,{\lambda }_1=1.375968919,{\lambda }_2=1.031976689,K=100$ and T = 1. From Tables 4 and 5, one can see that all of the results from the integral equation approach agree overall well with the other two sets of results. However, there are slight discrepancies between our solutions and those given at some time values, which seem to be due to the choices of truncation.

Table 4. Comparison of the American put option price in regime 1.

	$V_1$
S	CN	Tree	MC	Our result
90	14.6919	14.5189	14.5025	14.5021
100	9.9245	9.9059	9.9127	9.8985
110	6.7017	6.7010	6.7026	6.6958
120	4.5257	4.5228	4.5286	4.5255

Table 5. Comparison of the American put option price in regime 2.

	$V_2$
S	CN	Tree	MC	Our result
90	11.6126	11.5425	11.5369	11.4804
100	6.7423	6.7423	6.7401	6.7459
110	3.9244	3.9239	3.9824	3.8905
120	2.3083	2.3088	2.3057	2.2992

Depicted in Figure 3 are four different optimal exercise prices as a function of time to expiration, $\tau =T-t$. The dashed lines represent $S_{f1}-rs$ and $S_{f2}-rs$, the optimal exercise prices obtained from the regime-switching model, while the real lines represent $S_{f1}-ns$ and $S_{f2}-ns$, the optimal exercise prices calculated from the classical Black–Scholes model, corresponding to volatilities ${\sigma }_1$ and ${\sigma }_2$, respectively. As can be observed from Figure 3, the optimal exercise prices under regime-switching are bounded between those on the Black–Scholes model, which agree well with the conclusions of Yi [56] and Yang [55]. In other words, if ${\sigma }_1<{\sigma }_2$, then $S_{f2}^0<S_{f2}<S_{f1}<S_{f1}^0$, where the superscript 0 stands for the optimal exercise price without regime switching. This is not surprising at all, as optimal exercise prices are monotonically decreasing functions of volatility. Even in extreme cases, the values are calculated in a certain state for the entire time. The difference between $S_{f1}$ and $S_{f1}^0$ ($S_{f2}$ and $S_{f2}^0$) represents the lower (higher) option premium of having a certain non-zero probability that the underlying will spend time in a state of lower (higher) volatility. As time to expiration decreases, the expected amount of time spent in the other state of the economy decreases. This causes the subtracted (added) option value due to regime switching to decrease, which leads to the option premium to increase (decrease). Of course, as time approaches expiration, all optimal exercise values converge to the strike price.

Figure 3. The optimal exercise boundaries.

Now, we consider the monotonicity with respect to the volatility and the transition rate, respectively. The values of the optimal exercise prices under the regime-switching model with different parameters are presented in Table 6. For instance, if all other parameters are fixed except ${\sigma }_2$, both the optimal exercises prices, $S_{f1}$ and $S_{f2}$, decrease with an increasing ${\sigma }_2$. This shows that a larger ${\sigma }_2$ leads to a lower option premium. Furthermore, when the ratio of the volatilities, $A={\sigma }_1^2/{\sigma }_2^2$, is greater than 1, the value of $S_{f_1}$ is greater than that of $S_{f_2}$. If the ratio $A\ge 1$, the value of $S_{f_2}$ is greater, as also shown in Figure 4. The real line represents $S_{f2}$, the optimal exercise price in State 2 with volatility value at ${\sigma }_2$. Comparing Table 6 with Figure 4, we could find that the optimal exercise prices are monotonically increasing functions of the ratio of the volatilities A. As the ratio A increases, the weight of the higher-volatility increases, which leads to a greater optimal exercise prices.

Figure 4. The optimal exercise boundaries with different volatilities($A\in (0,\mathrm{\infty })$).

Table 6. Parameters for regime-switching volatilities.

		${\lambda }_1=0.5$	${\lambda }_2=0.5$	${\lambda }_1=0.3$	${\lambda }_2=0.5$	${\lambda }_1=0.5$	${\lambda }_2=0.3$
${\sigma }_1$	${\sigma }_2$	$S_{f1}$	$S_{f2}$	$S_{f1}$	$S_{f2}$	$S_{f1}$	$S_{f2}$
0.3	0.3	76.6394	76.2191	75.7424	75.8513	77.5621	76.5810
0.3	0.5	72.5664	68.0419	71.1661	67.1328	73.5478	69.3678
0.5	0.3	71.1619	82.4703	69.2605	82.0769	73.3150	82.6605
0.5	0.5	70.6750	71.0693	68.5912	70.2763	72.6235	71.8233

On the other hand, the values of optimal exercise prices are also influenced by the transition rate. If all other parameters are fixed, the values of the optimal exercise prices increase as an increasing transition rate ${\lambda }_1$, which is shown in Table 6. The real line represents $S_{f2}$, the optimal exercise price in State 2 with transition rate at ${\lambda }_2$. In addition, if we assume ${\sigma }_1<{\sigma }_2$ as above, the optimal exercise prices are monotonically increasing functions of the ratio of ${\lambda }_1/{\lambda }_2$, as shown in Table 6 and Figure 5. The reason is that a larger ratio of ${\lambda }_1/{\lambda }_2$ implies a shorter period for staying at a low-volatility state and a smaller weight on the low-volatility, ${\sigma }_1$, which leads to a smaller average volatility.

Figure 5. Two optimal exercise boundaries with different transition rates.

Once the efficiency and accuracy of our approach are demonstrated, it is interesting to remark on how the computed results obtained from a regime-switching model are used in pricing American options. There have been two different arguments for the valuation of American options under a regime-switching model in terms of the market state at the current time known or not.

In the literature, some researchers [6,29,34] believe that the economic states at the point when an option contract needs to be priced is known and thus all one needs is to use the information corresponding to that states to price the option contract. In other words, for the option price, one needs to determine which of the three regions shown in Figure 1 the current underlying price is within. For example, at a given time t, if we know that we are currently in a good economic state with a low volatility, then one should use $V_1$, as here we have assumed that ${\sigma }_1<{\sigma }_2$. Otherwise, one use $V_2$. More specifically, if the current stock price S is $\mathrm{\$}40$, then the option price V can be calculated from Equation (21(21) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \end{array}$(21) ), as provided by Table 7.

Table 7. Optimal exercise decision in a known state.

	State 1			State 2
	S=40	S=20	S=10	S=40	S=20	S=10
$S_{f1}=35$	Continue	Exercise	Exercise
	System 5(5) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_1^2{S}^2\frac{{\mathrm{\partial }}^2{V}_1}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}-r{V}_1={\lambda }_1(V_1-V_2),}\\ V_1(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_1(S,t)=0,\\ V_1(S_{f_1}(t),t)=K-S_{f_1}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}(S_{f_1}(t),t)=-1,}\end{array}\end{array}$(5)	K-S
$S_{f2}=15$				Continue	Continue	Exercise
				System 6(6) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}\end{array}$(6)	System 7(7) $\{\begin{array}{l}{V}_1(S,t)=K-S,\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=K-S,\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}$(7)	K-S

On the other hand, some other researchers [25,30,31,47] argued that in practice it is difficult to determine which state the underlying asset price belongs to and thus the state of the financial market is hidden. For example, He and Zhu [30,31] introduced a new parameter ${\pi }_j$, representing the probability of the economic state being in state j (for j = 1, 2) at the time when an option contract needs to be priced, and they proposed that ${\pi }_j$ be determined directly with other model parameters estimated through market historical data. For example, under the same assumption that ${\sigma }_1<{\sigma }_2$ at a given time t, then the function of the option price becomes a weighted function as $V={\pi }_1{V}_1+{\pi }_2{V}_2,$as shown in Table 8.

Table 8. Option value when the state is hidden.

	S=40	S=20	S=10
$S_{f1}=35$	Continue	Exercise	Exercise
$V_1$	System 5(5) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_1^2{S}^2\frac{{\mathrm{\partial }}^2{V}_1}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}-r{V}_1={\lambda }_1(V_1-V_2),}\\ V_1(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_1(S,t)=0,\\ V_1(S_{f_1}(t),t)=K-S_{f_1}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }S}(S_{f_1}(t),t)=-1,}\end{array}\end{array}$(5)	K-S
$S_{f2}=15$	Continue	Continue	Exercise
$V_2$	System 6(6) $\begin{array}{r}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=max(K-S,0),\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}\end{array}$(6)	System 7(7) $\{\begin{array}{l}{V}_1(S,t)=K-S,\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }_2^2{S}^2\frac{{\mathrm{\partial }}^2{V}_2}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}-r{V}_2={\lambda }_2(V_2-V_1),}\\ V_2(S,T)=K-S,\\ \underset{S\to \mathrm{\infty }}{lim}{V}_2(S,t)=0,\\ V_2(S_{f_2}(t),t)=K-S_{f_2}(t),\\ {\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }S}(S_{f_2}(t),t)=-1,}\end{array}$(7)	K-S
V	${\pi }_1{V}_1+{\pi }_2{V}_2$

In addition to the two pricing methodologies shown above, there is actually another way to determine the probability ${\pi }_j$ (for j = 1, 2) indirectly which utilizes more known features of the regime-switching model itself. Guo [26] and Liu et al. [39] showed that once all the parameters have been determined from the historical data available up to time t, the characteristic function of $T_j$ can be computed, where $T_j$ is the occupation time in the state j (for j = 1, 2). With these two total times available, it is more reasonable to directly assume that the probability of the economic state being in state j, ${\pi }_j$, is equal to the weight of the occupation time, $T_j/T$, instead of being estimated from the historical data as He and Zhu provided [30,31].

Furthermore, Fuh and Wang [23] provided another way to determine the probability ${\pi }_j$. The stationary distribution ${\eta }_j=T_j/T$ of being in state j was obtained by solving the balance equation $\eta Q=0$, where Q is the matrix of the transition rates.

Therefore, an approximation formula of the occupation times $T_j$ (for j = 1, 2) is $\frac{{\eta }_1}{{\eta }_2}=\frac{T_1/T}{T_2/T}=\frac{{\lambda }_2}{{\lambda }_1},$with ${\eta }_1+{\eta }_2=1$ and $T_1+T_2=T$. Since we know the weight of the occupation times, the weight of the initial states can be calculated as $\frac{{\pi }_1}{{\pi }_2}=\frac{{\lambda }_2}{{\lambda }_1}.$Once the parameters ${\pi }_j$ (for j = 1, 2) are calculated, the option price V can be (30) $V={\pi }_1{V}_1+{\pi }_2{V}_2,$(30) together with ${\pi }_1+{\pi }_2=1$.

5. Conclusion

In this paper, a novel computational approach based on integral equations is presented, which can enhance the computation efficiency when the price of an American option needs to be numerically computed. The coupled partial differential equations (PDEs) in a PDE system have been decoupled in the Fourier space, resulting a completely decoupled integral equation for each economical states in a regime-switching mode and thus has greatly reduced computational effort. Numerical examples for American options is presented, and the results compare well with those preliminary result. As for the valuation of American options, in terms of the state known or hidden, there are two different methodologies to determine the option price. Moreover, with the occupation time in one state available calculated, a new way is provided to estimate the weight of the current state.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix 1.

In the appendix, the details of the computation from Equations (13(13) $\begin{array}{rl}{\hat{u}}_1(\eta ,{\tau }_1)& ={\hat{u}}_{10}[c_{11}(\eta ){\text{e}}^{{\mu }_1{\tau }_1}+c_{12}(\eta ){\text{e}}^{{\mu }_2{\tau }_1}]\\ & +{\int }_0^{{\tau }_1}{f}_1(\eta ,\xi )[c_{11}(\eta ){\text{e}}^{{\mu }_1({\tau }_1-\xi )}+c_{12}(\eta ){\text{e}}^{{\mu }_2({\tau }_1-\xi )}]\mathrm{d}\xi ,\end{array}$(13) )–(18(18) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_2(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}+c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}{\tau }_1}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2A\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{21}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}\\ & +c_{22}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_2(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_2(\xi )}-[{\alpha }_2-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_2(\xi )}]-i\eta [1-{\text{e}}^{p_2(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(18) ) are presented. To obtain the integral equation formulation in the original space, applying the Fourier Inversion transform into Equations (13(13) $\begin{array}{rl}{\hat{u}}_1(\eta ,{\tau }_1)& ={\hat{u}}_{10}[c_{11}(\eta ){\text{e}}^{{\mu }_1{\tau }_1}+c_{12}(\eta ){\text{e}}^{{\mu }_2{\tau }_1}]\\ & +{\int }_0^{{\tau }_1}{f}_1(\eta ,\xi )[c_{11}(\eta ){\text{e}}^{{\mu }_1({\tau }_1-\xi )}+c_{12}(\eta ){\text{e}}^{{\mu }_2({\tau }_1-\xi )}]\mathrm{d}\xi ,\end{array}$(13) ) and (14(14) $\begin{array}{rl}{\hat{u}}_2(\eta ,{\tau }_2)& ={\hat{u}}_{20}[c_{21}(\eta ){\text{e}}^{A{\mu }_1{\tau }_2}+c_{22}(\eta ){\text{e}}^{A{\mu }_2{\tau }_2}]\\ & +{\int }_0^{A{\tau }_2}\frac{f_2(\eta ,\xi )}{A}[c_{21}(\eta ){\text{e}}^{{\mu }_1(A{\tau }_2-\xi )}+c_{22}(\eta ){\text{e}}^{{\mu }_2(A{\tau }_2-\xi )}]\mathrm{d}\xi ,\end{array}$(14) ), we obtain (A1) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\hat{u}}_{10}{c}_{11}(\eta ){\text{e}}^{{\mu }_1(\eta ){\tau }_1}\mathrm{d}\eta +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\hat{u}}_{10}{c}_{12}(\eta ){\text{e}}^{{\mu }_2(\eta ){\tau }_1}\mathrm{d}\eta \\ & +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{{\tau }_1}{c}_{11}(\eta )f_1(\eta ,\xi ){\text{e}}^{{\mu }_1(\eta )({\tau }_1-\xi )}\mathrm{d}\xi \mathrm{d}\eta \\ & +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{{\tau }_1}{c}_{12}(\eta )f_1(\eta ,\xi ){\text{e}}^{{\mu }_2(\eta )({\tau }_1-\xi )}\mathrm{d}\xi \mathrm{d}\eta \\ & \triangleq I_1+I_2+I_3+I_4,\end{array}$(A1) and additionally, (A2) $\begin{array}{rl}{u}_2(x,{\tau }_2)& =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\hat{u}}_{20}{c}_{21}(\eta ){\text{e}}^{A{\mu }_1(\eta ){\tau }_2}\mathrm{d}\eta +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\hat{u}}_{20}{c}_{22}(\eta ){\text{e}}^{A{\mu }_2(\eta ){\tau }_2}\mathrm{d}\eta \\ & +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{A{\tau }_2}{c}_{21}(\eta )f_2(\eta ,\xi ){\text{e}}^{{\mu }_1(\eta )(A{\tau }_2-\xi )}\mathrm{d}\xi \mathrm{d}\eta \\ & +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{A{\tau }_2}{c}_{22}(\eta )f_2(\eta ,\xi ){\text{e}}^{{\mu }_2(\eta )(A{\tau }_2-\xi )}\mathrm{d}\xi \mathrm{d}\eta \\ & \triangleq I{I}_1+I{I}_2+I{I}_3+I{I}_4.\end{array}$(A2) We solve $I_1$ of $u_1(x,{\tau }_1)$ first: (A3) $\begin{array}{rl}{I}_1& =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\hat{u}}_{10}{c}_{11}(\eta ){\text{e}}^{{\mu }_1(\eta ){\tau }_1}\mathrm{d}\eta =u_1(x,0)\ast g_1(x,{\tau }_1)\\ & =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u,\end{array}$(A3) where (A4) $\begin{array}{rl}{g}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{c}_{11}(\eta ){\text{e}}^{{\mu }_1(\eta ){\tau }_1}\mathrm{d}\eta \\ & =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{c}_{11}(\eta ){\text{e}}^{\frac{{\tau }_1}{2A}[-(B_2(\eta )+A{B}_1(\eta ))+\mathrm{\Delta }(\eta )]}\mathrm{d}\eta \\ & =\frac{1}{2\pi }{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1}{\text{e}}^{-\frac{[2Ax+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2Ax+(2{\alpha }_2-A-1){\tau }_1}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta .\end{array}$(A4) Using the same method, we will get (A5) $\begin{array}{rl}{I}_2& =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u,\end{array}$(A5) (A6) $\begin{array}{rl}{I}_3& =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{{\tau }_1}{f}_1(\eta ,\xi )c_{11}(\eta ){\text{e}}^{{\mu }_1(\eta )({\tau }_1-\xi )}\mathrm{d}\xi \mathrm{d}\eta \\ & =\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{11}(\eta ){\text{e}}^{i\eta x+[\frac{\mathrm{\Delta }(\eta )}{2A}-\frac{(1+A){\eta }^2-i\eta (2{\alpha }_2-A-1)}{2A}-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})]({\tau }_1-\xi )-i\eta p_1(\xi )}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \\ & =\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]+i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(A6) and (A7) $\begin{array}{rl}{I}_4& =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{{\tau }_1}{f}_1(\eta ,\xi )c_{12}(\eta ){\text{e}}^{{\mu }_1(\eta )({\tau }_1-\xi )}\mathrm{d}\xi \mathrm{d}\eta \\ & =\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{12}(\eta ){\text{e}}^{i\eta x+[-\frac{\mathrm{\Delta }(\eta )}{2A}-\frac{(1+A){\eta }^2-i\eta (2{\alpha }_2-A-1)}{2A}-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})]({\tau }_1-\xi )-i\eta p_1(\xi )}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \\ & =\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(A7) so Equation (A1(A1) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\hat{u}}_{10}{c}_{11}(\eta ){\text{e}}^{{\mu }_1(\eta ){\tau }_1}\mathrm{d}\eta +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\hat{u}}_{10}{c}_{12}(\eta ){\text{e}}^{{\mu }_2(\eta ){\tau }_1}\mathrm{d}\eta \\ & +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{{\tau }_1}{c}_{11}(\eta )f_1(\eta ,\xi ){\text{e}}^{{\mu }_1(\eta )({\tau }_1-\xi )}\mathrm{d}\xi \mathrm{d}\eta \\ & +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{{\tau }_1}{c}_{12}(\eta )f_1(\eta ,\xi ){\text{e}}^{{\mu }_2(\eta )({\tau }_1-\xi )}\mathrm{d}\xi \mathrm{d}\eta \\ & \triangleq I_1+I_2+I_3+I_4,\end{array}$(A1) ) will be (A8) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =I_1+I_2+1_3+1_4\\ & =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]+i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi \\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{c}_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]-i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(A8) Now, the value of Equation (13(13) $\begin{array}{rl}{\hat{u}}_1(\eta ,{\tau }_1)& ={\hat{u}}_{10}[c_{11}(\eta ){\text{e}}^{{\mu }_1{\tau }_1}+c_{12}(\eta ){\text{e}}^{{\mu }_2{\tau }_1}]\\ & +{\int }_0^{{\tau }_1}{f}_1(\eta ,\xi )[c_{11}(\eta ){\text{e}}^{{\mu }_1({\tau }_1-\xi )}+c_{12}(\eta ){\text{e}}^{{\mu }_2({\tau }_1-\xi )}]\mathrm{d}\xi ,\end{array}$(13) ) can be expressed as (A9) $\begin{array}{rl}{u}_1(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{p_1(0)}^{+\mathrm{\infty }}(1-{\text{e}}^u{)}^{+}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A}){\tau }_1-\frac{[2A(x-u)+(2{\alpha }_2-A-1){\tau }_1{]}^2}{8A(1+A){\tau }_1}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}+c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta ){\tau }_1}{2A}}]{\text{e}}^{-\frac{(1+A){\tau }_1}{2A}{[\eta -i\frac{2A(x-u)+(2{\alpha }_2-A-1)}{2(A+1){\tau }_1}]}^2}\mathrm{d}\eta \mathrm{d}u\\ & +\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_1+\frac{{\beta }_1+{\beta }_2}{2A})({\tau }_1-\xi )-\frac{\{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi ){\}}^2}{8A(1+A)({\tau }_1-\xi )}}\\ & \times {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[c_{11}(\eta ){\text{e}}^{\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}+c_{12}(\eta ){\text{e}}^{-\frac{\mathrm{\Delta }(\eta )}{2A}({\tau }_1-\xi )}]{\text{e}}^{-\frac{(1+A)({\tau }_1-\xi )}{2A}{\{\eta -i\frac{2A[x-p_1(\xi )]+(2{\alpha }_2-A-1)({\tau }_1-\xi )}{2(A+1)({\tau }_1-\xi )}\}}^2}\\ & \times \{{\text{e}}^{p_1(\xi )}-[{\alpha }_1-1+p_1^{\prime }(\xi )][1-{\text{e}}^{p_1(\xi )}]+i\eta [1-{\text{e}}^{p_1(\xi )}]\}\mathrm{d}\eta \mathrm{d}\xi .\end{array}$(A9) The method of solving Equation (14(14) $\begin{array}{rl}{\hat{u}}_2(\eta ,{\tau }_2)& ={\hat{u}}_{20}[c_{21}(\eta ){\text{e}}^{A{\mu }_1{\tau }_2}+c_{22}(\eta ){\text{e}}^{A{\mu }_2{\tau }_2}]\\ & +{\int }_0^{A{\tau }_2}\frac{f_2(\eta ,\xi )}{A}[c_{21}(\eta ){\text{e}}^{{\mu }_1(A{\tau }_2-\xi )}+c_{22}(\eta ){\text{e}}^{{\mu }_2(A{\tau }_2-\xi )}]\mathrm{d}\xi ,\end{array}$(14) ) is similar.

Appendix 2.

In the appendix, the details of the computation from Equations (26(26) ${\hat{u}}_2(\eta ,{\tau }_2)={\int }_0^{{\tau }_1}g(\eta ,{\tau }_1){\text{e}}^{-B_2(\eta )({\tau }_1-\xi )}\mathrm{d}\xi .$(26) ) to (27(27) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1-({\text{e}}^x+{\text{e}}^{-{\alpha }_2{\tau }_1}){\text{e}}^{-{\beta }_2{\tau }_1}\mathcal{N}\left(\frac{x+({\alpha }_2-1){\tau }_1-p_2(0)}{2\sqrt{{\tau }_1}}\right)\\ & +{\beta }_2{\int }_0^{{\tau }_1}{\text{e}}^{x-{\beta }_2({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi \\ & -({\alpha }_2+{\beta }_2){\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi .\end{array}$(27) ) are presented. To obtain the integral equation formulation in the original space, applying the Fourier Inversion transform into Equation (26(26) ${\hat{u}}_2(\eta ,{\tau }_2)={\int }_0^{{\tau }_1}g(\eta ,{\tau }_1){\text{e}}^{-B_2(\eta )({\tau }_1-\xi )}\mathrm{d}\xi .$(26) ), we obtain (A10) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{{\tau }_1}{\int }_{p_2(0)}^{p_1(0)}{\beta }_2{\text{e}}^{-i\eta x}(1-{\text{e}}^x)\mathrm{d}x\mathrm{d}\xi \mathrm{d}\eta \\ & +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{{\tau }_1}{\text{e}}^{-i\eta p_2(\xi )}{\text{e}}^{[{\eta }^2-i\eta ({\alpha }_2-1)+{\alpha }_2+{\beta }_2]({\tau }_1-\xi )}\\ & \times [{\text{e}}^{p_2(\xi )}-(i\eta +{\alpha }_2-1+p_2^{\prime }(\xi ))(1-{\text{e}}^{p_2(\xi )})]\mathrm{d}\xi \mathrm{d}\eta \\ & =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1+\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x-[{\eta }^2-i\eta ({\alpha }_2-1)+{\alpha }_2+{\beta }_2]({\tau }_1-\xi )}{\text{e}}^{-i\eta p_2(\xi )}\\ & \times [{\text{e}}^{p_2(\xi )}-(1-{\text{e}}^{p_2(\xi )})(i\eta +{\alpha }_2-1+p_2^{\prime }(\xi ))]\mathrm{d}\eta \mathrm{d}\xi \\ & =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1+I.\end{array}$(A10) Now, we compute (A11) $\begin{array}{rl}I& ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{-({\tau }_1-\xi ){\eta }^2-i\eta [-x-({\alpha }_2-1)({\tau }_1-\xi )+p_2(\xi )]}\\ & \times [{\text{e}}^{p_2(\xi )}-(1-{\text{e}}^{p_2(\xi )})({\alpha }_2-1+p_2^{\prime }(\xi ))-i\eta (1-{\text{e}}^{p_2(\xi )})]\mathrm{d}\eta \mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi ({\tau }_1-\xi )}}{\text{e}}^{-\frac{[x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi ){]}^2}{4({\tau }_1-\xi )}}\\ & \times \{{\text{e}}^{p_2(\xi )}-(1-{\text{e}}^{p_2(\xi )})[p_2^{\prime }(\xi )+\frac{{\alpha }_2-1}{2}-\frac{x-p_2(\xi )}{2({\tau }_1-\xi )}]\}\mathrm{d}\xi .\end{array}$(A11) Define $H=\frac{[x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi ){]}^2}{4({\tau }_1-\xi )}=\frac{[Y-Q(\xi ){]}^2}{4({\tau }_1-\xi )},$where $Y=x+({\alpha }_2-1){\tau }_1$ and $Q(\xi )=p_2(\xi )+({\alpha }_2-1)\xi$, with $Q^{\prime }(\xi )=p_2^{\prime }(\xi )+({\alpha }_2-1)$. Therefore, Equation (A11(A11) $\begin{array}{rl}I& ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{-({\tau }_1-\xi ){\eta }^2-i\eta [-x-({\alpha }_2-1)({\tau }_1-\xi )+p_2(\xi )]}\\ & \times [{\text{e}}^{p_2(\xi )}-(1-{\text{e}}^{p_2(\xi )})({\alpha }_2-1+p_2^{\prime }(\xi ))-i\eta (1-{\text{e}}^{p_2(\xi )})]\mathrm{d}\eta \mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi ({\tau }_1-\xi )}}{\text{e}}^{-\frac{[x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi ){]}^2}{4({\tau }_1-\xi )}}\\ & \times \{{\text{e}}^{p_2(\xi )}-(1-{\text{e}}^{p_2(\xi )})[p_2^{\prime }(\xi )+\frac{{\alpha }_2-1}{2}-\frac{x-p_2(\xi )}{2({\tau }_1-\xi )}]\}\mathrm{d}\xi .\end{array}$(A11) ) could be rewritten as (A12) $\begin{array}{rl}I& ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi ({\tau }_1-\xi )}}{\text{e}}^{-\frac{[Y-Q(\xi ){]}^2}{4({\tau }_1-\xi )}}\{{\text{e}}^{p_2(\xi )}-(1-{\text{e}}^{p_2(\xi )})[Q^{\prime }(\xi )-\frac{{\alpha }_2-1}{2}-\frac{x-p_2(\xi )}{2({\tau }_1-\xi )}]\}\mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi ({\tau }_1-\xi )}}{\text{e}}^{-\frac{[Y-Q(\xi ){]}^2}{4({\tau }_1-\xi )}}\{{\text{e}}^{p_2(\xi )}-(1-{\text{e}}^{p_2(\xi )})[Q^{\prime }(\xi )-\frac{Y-Q(\xi )}{2({\tau }_1-\xi )}]\}\mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi ({\tau }_1-\xi )}}{\text{e}}^{-\frac{[Y-Q(\xi ){]}^2}{4({\tau }_1-\xi )}}[{\text{e}}^{p_2(\xi )}(1+Q^{\prime }(\xi )-\frac{Y-Q(\xi )}{2({\tau }_1-\xi )})-(Q^{\prime }(\xi )-\frac{Y-Q(\xi )}{2({\tau }_1-\xi )})]\mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi ({\tau }_1-\xi )}}{\text{e}}^{-\frac{[Y-Q(\xi ){]}^2}{4({\tau }_1-\xi )}}\left[{\text{e}}^{p_2(\xi )}(1+Q^{\prime }(\xi )-\frac{Y-Q(\xi )}{2({\tau }_1-\xi )})\right]\mathrm{d}\xi \\ & -{\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi ({\tau }_1-\xi )}}{\text{e}}^{-\frac{[Y-Q(\xi ){]}^2}{4({\tau }_1-\xi )}}[Q^{\prime }(\xi )-\frac{Y-Q(\xi )}{2({\tau }_1-\xi )}]\mathrm{d}\xi \\ & \triangleq Q_1-Q_2.\end{array}$(A12) Now, we compute $Q_1$ and $Q_2$, respectively. (A13) $\begin{array}{rl}{Q}_1& ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi ({\tau }_1-\xi )}}{\text{e}}^{-\frac{[Y-Q(\xi ){]}^2}{4({\tau }_1-\xi )}}\left[{\text{e}}^{p_2(\xi )}(1+Q^{\prime }(\xi )-\frac{Y-Q(\xi )}{2({\tau }_1-\xi )})\right]\mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi }}{\text{e}}^{-\frac{[Y-Q(\xi )+2({\tau }_1-\xi ){]}^2}{4({\tau }_1-\xi )}}{\text{e}}^{(Y-Q(\xi ))+({\tau }_1-\xi )}\\ & \times {\text{e}}^{p_2(\xi )}\frac{2({\tau }_1-\xi )+2Q^{\prime }(\xi )({\tau }_1-\xi )-(Y-Q(\xi ))}{2({\tau }_1-\xi )\sqrt{{\tau }_1-\xi }}\mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{x-{\beta }_2({\tau }_1-\xi )}}{2\sqrt{\pi }}{\text{e}}^{-\frac{[Y-Q(\xi )+2({\tau }_1-\xi ){]}^2}{4({\tau }_1-\xi )}}\\ & \times \frac{(Q^{\prime }(\xi )+2)\sqrt{{\tau }_1-\xi }+\frac{1}{2\sqrt{{\tau }_1-\xi }}[Q(\xi )-Y-2({\tau }_1-\xi )]}{(\sqrt{{\tau }_1-\xi }{)}^2}\mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}{\text{e}}^{x-{\beta }_2({\tau }_1-\xi )}\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }\mathcal{N}\left(\frac{Y-Q(\xi )+2({\tau }_1-\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi .\end{array}$(A13) (A14) $\begin{array}{rl}{Q}_2& ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi ({\tau }_1-\xi )}}{\text{e}}^{-\frac{[Y-Q(\xi ){]}^2}{4({\tau }_1-\xi )}}[Q^{\prime }(\xi )-\frac{Y-Q(\xi )}{2({\tau }_1-\xi )}]\mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi }}{\text{e}}^{-\frac{[Y-Q(\xi ){]}^2}{4({\tau }_1-\xi )}}\left[\frac{Q^{\prime }(\xi )2({\tau }_1-\xi )-(Y-Q(\xi ))}{2({\tau }_1-\xi )\sqrt{{\tau }_1-\xi }}\right]\mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi }}{\text{e}}^{-\frac{[Y-Q(\xi ){]}^2}{4({\tau }_1-\xi )}}\frac{(Q^{\prime }(\xi )\sqrt{{\tau }_1-\xi }-\frac{1}{2\sqrt{{\tau }_1-\xi }}[Y-Q(\xi )]}{(\sqrt{{\tau }_1-\xi }{)}^2}\mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }\mathcal{N}\left(\frac{Y-Q(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi .\end{array}$(A14) Thus, by substituting the above expression into Equations (A11(A11) $\begin{array}{rl}I& ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{-({\tau }_1-\xi ){\eta }^2-i\eta [-x-({\alpha }_2-1)({\tau }_1-\xi )+p_2(\xi )]}\\ & \times [{\text{e}}^{p_2(\xi )}-(1-{\text{e}}^{p_2(\xi )})({\alpha }_2-1+p_2^{\prime }(\xi ))-i\eta (1-{\text{e}}^{p_2(\xi )})]\mathrm{d}\eta \mathrm{d}\xi \\ & ={\int }_0^{{\tau }_1}\frac{{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}}{2\sqrt{\pi ({\tau }_1-\xi )}}{\text{e}}^{-\frac{[x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi ){]}^2}{4({\tau }_1-\xi )}}\\ & \times \{{\text{e}}^{p_2(\xi )}-(1-{\text{e}}^{p_2(\xi )})[p_2^{\prime }(\xi )+\frac{{\alpha }_2-1}{2}-\frac{x-p_2(\xi )}{2({\tau }_1-\xi )}]\}\mathrm{d}\xi .\end{array}$(A11) ), (A10(A10) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{{\tau }_1}{\int }_{p_2(0)}^{p_1(0)}{\beta }_2{\text{e}}^{-i\eta x}(1-{\text{e}}^x)\mathrm{d}x\mathrm{d}\xi \mathrm{d}\eta \\ & +\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x}{\int }_0^{{\tau }_1}{\text{e}}^{-i\eta p_2(\xi )}{\text{e}}^{[{\eta }^2-i\eta ({\alpha }_2-1)+{\alpha }_2+{\beta }_2]({\tau }_1-\xi )}\\ & \times [{\text{e}}^{p_2(\xi )}-(i\eta +{\alpha }_2-1+p_2^{\prime }(\xi ))(1-{\text{e}}^{p_2(\xi )})]\mathrm{d}\xi \mathrm{d}\eta \\ & =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1+\frac{1}{2\pi }{\int }_0^{{\tau }_1}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\text{e}}^{i\eta x-[{\eta }^2-i\eta ({\alpha }_2-1)+{\alpha }_2+{\beta }_2]({\tau }_1-\xi )}{\text{e}}^{-i\eta p_2(\xi )}\\ & \times [{\text{e}}^{p_2(\xi )}-(1-{\text{e}}^{p_2(\xi )})(i\eta +{\alpha }_2-1+p_2^{\prime }(\xi ))]\mathrm{d}\eta \mathrm{d}\xi \\ & =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1+I.\end{array}$(A10) ) can be obtained as (A15) $\begin{array}{rl}{u}_2(x,{\tau }_1)& =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1+Q_1-Q_2\\ & =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1+{\int }_0^{{\tau }_1}{\text{e}}^{x-{\beta }_2({\tau }_1-\xi )}\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }\mathcal{N}\left(\frac{Y-Q(\xi )+2({\tau }_1-\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi \\ & -{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }\mathcal{N}\left(\frac{Y-Q(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi \\ & =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1+{\int }_0^{{\tau }_1}{\text{e}}^{x-{\beta }_2({\tau }_1-\xi )}\mathrm{d}\mathcal{N}\left(\frac{Y-Q(\xi )+2({\tau }_1-\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\\ & -{\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}\mathrm{d}\mathcal{N}\left(\frac{Y-Q(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\\ & =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1+{\text{e}}^x\mathcal{N}\left(\frac{x-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right){|}_{\xi ={\tau }_1}-{\text{e}}^{x-{\beta }_2{\tau }_1}\mathcal{N}\left(\frac{x+({\alpha }_2-1){\tau }_1-p_2(0)}{2\sqrt{{\tau }_1}}\right)\\ & +{\beta }_2{\int }_0^{{\tau }_1}{\text{e}}^{x-{\beta }_2({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi \\ & -\mathcal{N}\left(\frac{x-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right){|}_{\xi ={\tau }_1}+{\text{e}}^{-({\alpha }_2+{\beta }_2){\tau }_1}\mathcal{N}\left(\frac{x+({\alpha }_2-1){\tau }_1-p_2(0)}{2\sqrt{{\tau }_1}}\right)\\ & -({\alpha }_2+{\beta }_2){\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi \\ & =\frac{1}{2\pi }{\beta }_2(1-{\text{e}}^x){\tau }_1-({\text{e}}^x+{\text{e}}^{-{\alpha }_2{\tau }_1}){\text{e}}^{-{\beta }_2{\tau }_1}\mathcal{N}\left(\frac{x+({\alpha }_2-1){\tau }_1-p_2(0)}{2\sqrt{{\tau }_1}}\right)\\ & +{\beta }_2{\int }_0^{{\tau }_1}{\text{e}}^{x-{\beta }_2({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi \\ & -({\alpha }_2+{\beta }_2){\int }_0^{{\tau }_1}{\text{e}}^{-({\alpha }_2+{\beta }_2)({\tau }_1-\xi )}\mathcal{N}\left(\frac{x+({\alpha }_2-1)({\tau }_1-\xi )-p_2(\xi )}{2\sqrt{{\tau }_1-\xi }}\right)\mathrm{d}\xi \\ & +{\mathit{1}}_{x=p_2(\xi )}({\text{e}}^x-1),\end{array}$(A15) where ${\mathit{1}}_{x=p_2(\xi )}=\{\begin{array}{ll}{\displaystyle \frac{1}{2},}& x=p_2(\xi ),\\ 0,& x<p_2(\xi ).\end{array}$