Weak convergence of balanced stochastic Runge–Kutta methods for stochastic differential equations

ABSTRACT

In this paper, weak convergence of balanced stochastic one-step methods and especially balanced stochastic Runge–Kutta (SRK) methods for Itô multidimensional stochastic differential equations is analyzed. Generalizing a corresponding result obtained by H. Schurz for the standard Euler method, it is shown that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods. As an application, this allows to prove the weak convergence order of the balanced SRK methods presented in earlier work by A. Rathinasamy, P. Nair and D. Ahmadian.

Keywords:

• balanced methods
• stochastic Runge–Kutta methods
• weak convergence

1 Introduction

Determining numerical approximations to solutions of stochastic differential equations is of great interest, as easily evaluable analytical solutions to stochastic differential equations do rarely exist (Kloeden & Platen, 1999; Mao, 2008; Milstein & Tretyakov, 2004). The development of approximation algorithms has therefore been an active field of research during the last decades. Weak convergence of stochastic Runge–Kutta (SRK) methods is studied in literature (Buckwar et al., 2010; Debrabant & Rößler, 2008; Milstein, 1985; Tocino & Vigo-Aguiar, 2002). Efficient SRK methods that deliver second-order weak approximations for stochastic differential equations with multiple stochastic integrals have been introduced in Debrabant & Rößler, 2009; Rößler (2009). Tang and Xiao (2017, 2018, 2019) proposed and analyzed new classes of SRK methods to further reduce the computational effort.

Stiff stochastic differential equations are studied, for example, in Abdulle et al. (2013), Higham (2000) and Kloeden and Platen (1999). To easily handle stiff stochastic differential equation systems, Milstein, Platen and Schurz introduced the balanced method, a class of quasi-implicit numerical schemes (Milstein et al., 1998). Alcock and Burrage (2006) examined its stability and the choice of parameters. Balanced SRK methods are studied by Amiri and Hosseini (2015). Further, Mardones and Mora (2019) discussed first-order weak balanced schemes to stochastic differential equations. Finally, strong convergence of two classes of second-order balanced SRK methods applied to multidimensional Itô stochastic differential equations has been discussed in Rathinasamy et al. (2020).

The main contribution of the current paper is to prove, generalizing a corresponding statement in Schurz (2005) for the standard Euler method, that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods, and to apply this result to derive a corresponding compact result for Runge–Kutta methods. As a direct consequence, this proves the weak convergence order of the methods presented in Rathinasamy et al. (2020) and Rathinasamy and Nair (2018).

The paper is organized as follows: In Section 2, some basic results and definitions are gathered. In Section 3, the weak convergence of balanced one-step methods and balanced SRK methods is proven.

2 Preliminaries

Let $(\mathrm{\Omega },F,\mathbb{P})$ be a complete probability space with a filtration $(F_t{)}_{t\ge 0}$ satisfying the usual conditions and $I=[t_0,T]$ for some $0\le t_0<T<\mathrm{\infty }$. Consider the system of Itô stochastic differential equations of the form

(1) $\mathrm{d}X\left(t\right)=g_0(X(t))\mathrm{d}t+\sum _{k=1}^m{g}_k(X(t))\mathrm{d}{W}_k(t),X(t_0)=X_0,t\in [t_0,T],$(1)

where $X(t)\in {\mathbb{R}}^d$, $g_0:{\mathbb{R}}^d\to {\mathbb{R}}^d$ is the drift vector, $g=(g_1,g_2,\dots ,g_m):{\mathbb{R}}^d\to {\mathbb{R}}^{d,m}$ is called diffusion matrix, $W=(W_1,\dots ,W_m{)}^T$ is an $m$-dimensional Wiener process adapted to the filtration, and $X_0$ is assumed to be $F_{t_0}$-measurable.

Throughout this paper, we consider discrete time approximations $Y_n=Y(t_n)$ with equidistant time discretization $I_h=\{t_0,t_1,\dots ,t_N\}$ with $t_0<t_1<\dots <t_N=T$ of the time interval $I=[t_0,T]$ with step sizes $h_n=t_{n+1}-t_n$ for $n=0,1,\dots ,N-1$ and maximal step size $h_{max}={max}_{n=0,1,\dots ,N-1}{h}_n$.

We consider SRK methods (Burrage & Burrage, 1996; Debrabant & Kværnø, 2008/09; Rößler, 2006, 2009)

(2) $\begin{array}{cc}{Y}_{n+1}& =Y_n+\sum _{k=0}^m\sum _{\mu \in \bar{M}}\sum _{i=1}^s{z}_i^{k,\mu ;n}{g}_k(H_i^{(k,\mu ;n)}),\\ H_i^{(k,\mu ;n)}& =Y_n+\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{j=1}^s{Z}_{i,j}^{k,\mu ;l,\nu ;n}{g}_l(H_j^{(l,\nu ;n)}).\end{array}$(2)

Here, $s$ is the stage number of the SRK method, $\bar{M}\mathrm{a}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{o}\mathrm{f}$ multi-indices (e.g., $\overline{M}=\left\{\right\{j_1\},\{j_1,j_2\}:0\le j_1,j_2\le m\}$), and the $z_i^{k,\mu ;n}$ and $Z_{i,j}^{k,\mu ;l,\nu ;n}$ are some random variables depending on the step size $h_n$ that are independent of $F_{t_n}$, $n=0,1,\dots ,N-1$.

Generalizing the ideas in Alcock and Burrage (2006), Amiri and Hosseini (2015), Kahl and Schurz (2006); Mardones and Mora (2019), Milstein et al. (1998), Rathinasamy and Nair (2018) and Schurz (2005), a balanced version of Equation (2)(2) $\begin{array}{cc}{Y}_{n+1}& =Y_n+\sum _{k=0}^m\sum _{\mu \in \bar{M}}\sum _{i=1}^s{z}_i^{k,\mu ;n}{g}_k(H_i^{(k,\mu ;n)}),\\ H_i^{(k,\mu ;n)}& =Y_n+\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{j=1}^s{Z}_{i,j}^{k,\mu ;l,\nu ;n}{g}_l(H_j^{(l,\nu ;n)}).\end{array}$(2) is given as

(3) $\begin{array}{cc}{Y}_{n+1}& =Y_n+\sum _{k=0}^m\sum _{\mu \in \bar{M}}\sum _{i=1}^s{z}_i^{k,\mu ;n}{g}_k(H_i^{(k,\mu ;n)})+C_n(Y_n-Y_{n+1}),\\ H_i^{(k,\mu ;n)}& =Y_n+\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{j=1}^s{Z}_{i,j}^{k,\mu ;l,\nu ;n}{g}_l(H_j^{(l,\nu ;n)}),\end{array}$(3)

where for an underlying SRK method of weak order $p$ using approximations $\mathrm{\Delta }{W}_n^j$ to the Brownian increments $W_{t_{n+1}}^j-W_{t_n}^j$ the matrices $C_n$ are chosen as

(4) $C_n=h_n^{p+\frac{1}{2}}\left[c_0(t_n,Y_n)+\sum _{j=1}^m(\frac{{(\mathrm{\Delta }{W}_n^j)}^{2p}}{h_n^p}-1)c_j(t_n,Y_n)\right],$(4)

in which the $d\times d$-matrix valued functions $c_j$, $j=0,\dots ,m$, satisfy the following assumption adapted from (Milstein et al., 1998). Here, we denote by $\mathrm{\Xi }$ the set of possible values for $\frac{{(\mathrm{\Delta }{W}_n^j)}^{2p}}{h_n^p}-1$. For example, if $\mathrm{\Delta }{W}_n^j=W_{t_{n+1}}^j-W_{t_n}^j$ and $p=2$, then $\mathrm{\Xi }=[-1,\mathrm{\infty })$, and if $p=1$ and $\mathrm{\Delta }{W}_n^j=\pm \sqrt{h_n}$, then $\mathrm{\Xi }=\{0\}$.

Assumption 1 For any sequence of real numbers ${\alpha }_i$ with ${\alpha }_0\in [0,\bar{\alpha }]$, ${\alpha }_i\in \mathrm{\Xi }$, $i=1,\dots ,m$ and $\bar{\alpha }\ge h_n$ for all step sizes $h_n$ considered and $(t,x)\in [0,\mathrm{\infty }]\times {\mathbb{R}}^d$, the matrix

$M(t,x)=I_d+{\alpha }_0^{p+\frac{1}{2}}\left[c_0(t,x)+\sum _{j=1}^m{\alpha }_j{c}_j(t,x)\right]$

has an inverse fulfilling $∥M{(t,x)}^{-1}∥\le M<\mathrm{\infty }$.

The constant $M$ relates to the individual bounds on the norm of the $c_j$ matrices.

3 Weak convergence analysis for balanced Runge–Kutta methods

In this section, for $\kappa \in \mathbb{R}$, let $C_{b(\kappa )}^l({\mathbb{R}}^d,{\mathbb{R}}^n)$ be the set of all $l$-times continuously differentiable functions $f:{\mathbb{R}}^d\to {\mathbb{R}}^n$ with uniformly bounded derivatives up to order $l$, such that for some constant $K_f$

$max\{\parallel f{\parallel }_n,\parallel \mathrm{\nabla }f{\parallel }_{n\times d},\parallel {\mathrm{\nabla }}^{2}f{\parallel }_{n\times d\times d},\dots \}\le K_f(1+\parallel x{\parallel }_d^{\kappa }),$

where $\parallel \cdot {\parallel }_n$ denotes the Euclidean vector norm on ${\mathbb{R}}^n$, $\parallel \cdot {\parallel }_{n\times d}$ the Frobenius norm on ${\mathbb{R}}^{n,d}$, and in general

$\parallel A{\parallel }_{n\times d_1\times \dots \times d_k}=\sqrt{\sum _{i_0=1}^n\sum _{i_1=1}^{d_1}\dots \sum _{i_k=1}^{d_k}{A}_{i_0,i_1,\dots ,i_k}}\mathrm{f}\mathrm{o}\mathrm{r}A\in {\mathbb{R}}^{n,d_1,\dots ,d_k}.$

To reduce the convergence analysis of balanced Runge–Kutta methods to the one of the underlying methods we will first establish the following theorem that generalizes a corresponding statement in Schurz (2005) for the standard Euler method.

Theorem 1 Let $\tilde{Y}$ be a numerical method with one-step representation ${\tilde{Y}}_{[s,y]}(t)$ that is weakly consistent of order $p$, i. e. for $f\in C_{b(\kappa )}^2([t_0,T]\times {\mathbb{R}}^d,\mathbb{R})$ there exists a constant $K_f^P$ depending only on $t_0,T,f,g_0,\dots ,g_m$ such that the local weak error fulfills

$|\mathbb{E}[f(X_{[u,x]}(t))-f({\tilde{Y}}_{[u,x]}(t))]|\le K_f^P(1+\parallel x{\parallel }_d^{4\kappa })(t-u{)}^{p+1}$

where $X_{[u,x]}(t)$ denotes the solution of stochastic differential equation (1)(1) $\mathrm{d}X\left(t\right)=g_0(X(t))\mathrm{d}t+\sum _{k=1}^m{g}_k(X(t))\mathrm{d}{W}_k(t),X(t_0)=X_0,t\in [t_0,T],$(1) with initial value $X_{[u,x]}(u)=x$. Let further $Y$ be a numerical method with one-step representation $Y_{[u,y]}(t)$, the sequence of approximations $Y_k$ defined by $Y_{k+1}=Y_{[t_k,Y_k]}(t_{k+1})$ and

$d_{[t_k,Y_k]}(t_{k+1})={\tilde{Y}}_{[t_k,Y_k]}(t_{k+1})-Y_{[t_k,Y_k]}(t_{k+1})$

and assume that for some $\rho \in \mathbb{N}$

$\sqrt[4]{\mathbb{E}[\parallel \mathbb{E}[d_{[t_k,Y_k]}(t_{k+1})|{\mathcal{F}}_{t_k}]{\parallel }_d^4]}\le K_1\sqrt[4]{1+\mathbb{E}[\parallel Y_k{\parallel }_d^{2\rho }]}(t_{k+1}-t_k{)}^{p+1},$

$\sqrt{\mathbb{E}[\parallel d_{[t_k,Y_k]}(t_{k+1}){\parallel }_d^4]}\le K_2\sqrt{1+\mathbb{E}[\parallel Y_k{\parallel }_d^{2\rho }]}(t_{k+1}-t_k{)}^{p+1}.$

Assume finally that

$\underset{k=0,\dots ,N}{max}\mathbb{E}[\parallel Y_k{\parallel }_d^{max\{4\kappa ,2\rho \}}]\le K_3,\underset{k=0,\dots ,N}{max}\mathbb{E}[\parallel {\tilde{Y}}_{[t_k,Y_k]}(t_{k+1}){\parallel }_d^{4\kappa }]\le K_4.$

Then $Y$ is weakly convergent of order $p$, i.e. there exists $C$ independent of $I_h$ such that

$\underset{k=0,\dots ,N}{max}|\mathbb{E}[f(X(t_k))-f(Y_k)]|\le C{h}_{max}^p.$

Proof The proof follows the one of [22, Theorem 5.2], with a few changes. Suppose that $g\in C_{b(\kappa )}^2({\mathbb{R}}^d,\mathbb{R})$ and consider

$M_g(k):=\mathbb{E}|\mathbb{E}[g({\tilde{Y}}_{[t_k,Y_k]}(t_{k+1}))-g(Y_{[t_k,Y_k]}(t_{k+1}))|{\mathcal{F}}_{t_k}]|.$

Then we have for some ${\theta }_{k,1},{\theta }_{k,2}\in [0,1]$ that

$M_g(k)=\mathbb{E}|⟨\mathrm{\nabla }g(Y_k),\mathbb{E}[d_{[t_k,Y_k]}(t_{k+1})|{\mathcal{F}}_{t_k}]{⟩}_d$

$+\mathbb{E}[⟨{\mathrm{\nabla }}^{2}g({\eta }_2(t_{k+1}))(Y_{k+1}-Y_k+{\theta }_{k,1}{d}_{[t_k,Y_k]}(t_{k+1})),d_{[t_k,Y_k]}(t_{k+1})){⟩}_d|{\mathcal{F}}_{t_k}]|$

where ${\eta }_2(t_{k+1})=Y_k+{\theta }_{k,2}(Y_{k+1}-Y_k+{\theta }_{k,1}{d}_{[t_k,Y_k]}(t_{k+1}))$ and $⟨\cdot ,\cdot {⟩}_d$ denotes the Euclidean scalar product in ${\mathbb{R}}^d$. Applying the Cauchy–Schwarz inequality, we obtain that

$M_g(k)=\mathbb{E}|⟨\mathrm{\nabla }g(Y_k),\mathbb{E}[d_{[t_k,Y_k]}(t_{k+1})|{\mathcal{F}}_{t_k}]{⟩}_d$

$+\mathbb{E}[{\theta }_{k,1}⟨{\mathrm{\nabla }}^{2}g({\eta }_2(t_{k+1}))d_{[t_k,Y_k]}(t_{k+1}),d_{[t_k,Y_k]}(t_{k+1}){⟩}_d|{\mathcal{F}}_{t_k}]$

$+\mathbb{E}[⟨{\mathrm{\nabla }}^{2}g({\eta }_2(t_{k+1}))(Y_{k+1}-Y_k),d_{[t_k,Y_k]}(t_{k+1}){⟩}_d|{\mathcal{F}}_{t_k}]|$

$\le \mathbb{E}|\parallel \mathrm{\nabla }g(Y_k){\parallel }_d\parallel \mathbb{E}[d_{[t_k,Y_k]}(t_{k+1})|{\mathcal{F}}_{t_k}]{\parallel }_d$

$+\mathbb{E}[{\theta }_{k,1}\parallel {\mathrm{\nabla }}^{2}g({\eta }_2(t_{k+1})){\parallel }_{d\times d}\parallel d_{[t_k,Y_k]}(t_{k+1}){\parallel }_d^2|{\mathcal{F}}_{t_k}]|$

$\le \sqrt{\mathbb{E}[\parallel \mathrm{\nabla }g(Y_k){\parallel }_d^2]}\sqrt{\mathbb{E}[\parallel \mathbb{E}[d_{[t_k,Y_k]}(t_{k+1})|{\mathcal{F}}_{t_k}]{\parallel }_d^2]}$

$+\sqrt{\mathbb{E}[\parallel {\mathrm{\nabla }}^{2}g({\eta }_2(t_{k+1})){\parallel }_{d\times d}^2]}\sqrt{\mathbb{E}[\parallel d_{[t_k,Y_k]}(t_{k+1}){\parallel }_d^4]}$

$\le [t]\sqrt[4]{\mathbb{E}[\parallel \mathrm{\nabla }g(Y_k){\parallel }_d^4]}\sqrt[4]{\mathbb{E}[\parallel \mathbb{E}[d_{[t_k,Y_k]}(t_{k+1})|{\mathcal{F}}_{t_k}]{\parallel }_d^4]}$

$+\sqrt[4]{\mathbb{E}[\parallel {\mathrm{\nabla }}^{2}g({\eta }_2(t_{k+1})){\parallel }_{d\times d}^4]}\sqrt{\mathbb{E}[\parallel d_{[t_k,Y_k]}(t_{k+1}){\parallel }_d^4]}$

$\le 2^{3/4}{K}_g\sqrt[4]{1+\mathbb{E}[\parallel Y_k{\parallel }_d^{4\kappa }]}\sqrt[4]{\mathbb{E}[\parallel \mathbb{E}[d_{[t_k,Y_k]}(t_{k+1})|{\mathcal{F}}_{t_k}]{\parallel }_d^4]}$

$+2^{3/4}{K}_g\sqrt[4]{1+\mathbb{E}[\parallel ({\eta }_2(t_{k+1})){\parallel }_d^{4\kappa }]}\sqrt[4]{\mathbb{E}[\parallel d_{[t_k,Y_k]}(t_{k+1}){\parallel }_d^4]}$

$\le 2^{3/4}{K}_g{K}_1\sqrt[4]{1+\mathbb{E}[\parallel Y_k{\parallel }_d^{4\kappa }]}\sqrt[4]{1+\mathbb{E}[\parallel Y_k{\parallel }_d^{2\rho }]}(t_{k+1}-t_k{)}^{p+1}$

$+2^{3/4}{K}_g{K}_2\sqrt[4]{1+3\mathbb{E}[\parallel Y_k{\parallel }_d^{4\kappa }]+3\mathbb{E}[\parallel Y_{k+1}{\parallel }_d^{4\kappa }]+3\mathbb{E}[\parallel {\tilde{Y}}_{[t_k,Y_k]}(t_{k+1}){\parallel }_d^{4\kappa }]}$

$\cdot \sqrt{1+\mathbb{E}[\parallel Y_k{\parallel }_d^{2\rho }]}(t_{k+1}-t_k{)}^{p+1}$

$\le C_g(t_{k+1}-t_k{)}^{p+1}$

with $C_g:=2^{3/4}{K}_g\sqrt{1+K_3}\left(K_1+K_2\sqrt[4]{1+6K_3+3K_4}\right)$. Let

$u(s,x)=\mathbb{E}[f(X_{s,x}(t_{k+1}))].$

For the global weak error

${\epsilon }_0(t_{k+1}):=|\mathbb{E}[f(X(t_{k+1}))-f(Y_{k+1})]|$

it follows then

${\epsilon }_0(t_{k+1})=|\sum _{i=0}^{k-1}\left(\mathbb{E}[u(t_{i+1},X_{t_i,Y_i}(t_{i+1}))]-\mathbb{E}[u(t_{i+1},Y_{t_i,Y_i}(t_{i+1}))]\right)$

$+\mathbb{E}[f(X_{[t_k,Y_k]}(t_{k+1}))]-\mathbb{E}[f(Y_{[t_k,Y_k]}(t_{k+1}))]|$

$\le \sum _{i=0}^{k-1}\mathbb{E}|\mathbb{E}[u(t_{i+1},X_{t_i,Y_i}(t_{i+1}))-u(t_{i+1},{\tilde{Y}}_{t_i,Y_i}(t_{i+1}))|{\mathcal{F}}_{t_i}]|$

$+\sum _{i=0}^{k-1}{M}_u(i)+\mathbb{E}|\mathbb{E}[f(X_{[t_k,Y_k]}(t_{k+1}))-f({\tilde{Y}}_{[t_k,Y_k]}(t_{k+1}))|{\mathcal{F}}_{t_k}]|$

$+M_f(k)$

$\le \left((K_f^P+K_u^P)(1+K_3)+C_g+C_f\right)\sum _{i=0}^k(t_{i+1}-t_i{)}^{p+1}$

$\le \left((K_f^P+K_u^P)(1+K_3)+C_g+C_f\right)(T-t_0)h_{max}^p.$

Applying this theorem to a balanced Runge–Kutta method and its underlying method, we obtain

Theorem 2 Let $Y$ be given by the balanced SRK method (3), and assume that its underlying Runge–Kutta method (2) is weakly consistent of order $p$. Assume further that

a) the coefficients $g_j$ of stochastic differential equation (1)(1) $\mathrm{d}X\left(t\right)=g_0(X(t))\mathrm{d}t+\sum _{k=1}^m{g}_k(X(t))\mathrm{d}{W}_k(t),X(t_0)=X_0,t\in [t_0,T],$(1) fulfill a Lipschitz and linear growth condition, i. e. there exists $K,L\in \mathbb{R}$ such that for all $x,y\in {\mathbb{R}}^d$

(5) $\parallel g_j(x)-g_j(y){\parallel }_d\le L\parallel x-y{\parallel }_d,\parallel g_j(x){\parallel }_d\le K(1+\parallel x{\parallel }_d),$(5)

and

b) that the coefficients of the balanced method fulfill that $c_j\in C_{b(\kappa )}^0({\mathbb{R}}^d,{\mathbb{R}}^{d,d})$, $j=0,\dots ,m$, and that there exists $K_z$ such that

(6) $|z_i^{k,\nu ;n}|\le K_z\sqrt{h_n},|Z_{i,j}^{k,\mu ;l,\nu ;n}|\le K_z\sqrt{h_n}$(6)

and

(7) $\parallel \sum _{\nu \in \bar{M}}\sum _{i=1}^s\mathbb{E}[(I_d+C_n{)}^{-1}{z}_i^{k,\nu ;n}]\parallel \le K_z{h}_n$(7)

for $i,j=1,\dots ,s$ as well as $\mu ,\nu \in \bar{M}$, $k,l=0,\dots ,m$, and $n=0,\dots ,N-1$.

Then $\mathrm{Y}$ is weakly convergent of order $p$.

Proof First, note that Equation (3)(3) $\begin{array}{cc}{Y}_{n+1}& =Y_n+\sum _{k=0}^m\sum _{\mu \in \bar{M}}\sum _{i=1}^s{z}_i^{k,\mu ;n}{g}_k(H_i^{(k,\mu ;n)})+C_n(Y_n-Y_{n+1}),\\ H_i^{(k,\mu ;n)}& =Y_n+\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{j=1}^s{Z}_{i,j}^{k,\mu ;l,\nu ;n}{g}_l(H_j^{(l,\nu ;n)}),\end{array}$(3) implies that

$\parallel H_i^{(k,\mu ;n)}{\parallel }_d\le \parallel Y_n{\parallel }_d+\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{j=1}^s|Z_{i,j}^{k,\mu ;l,\nu ;n}|\parallel g_l(H_j^{(l,\nu ;n)}){\parallel }_d.$

Using Equations (5)(5) $\parallel g_j(x)-g_j(y){\parallel }_d\le L\parallel x-y{\parallel }_d,\parallel g_j(x){\parallel }_d\le K(1+\parallel x{\parallel }_d),$(5) and(6)(6) $|z_i^{k,\nu ;n}|\le K_z\sqrt{h_n},|Z_{i,j}^{k,\mu ;l,\nu ;n}|\le K_z\sqrt{h_n}$(6) , we obtain

$\parallel H_i^{(k,\mu ;n)}{\parallel }_d\le \parallel Y_n{\parallel }_d+\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{j=1}^s{K}_z\sqrt{h_n}K(1+\parallel H_j^{(l,\nu ;n)}{\parallel }_d)$

which implies

(8) $\underset{k,\mu ,i}{max}\parallel H_i^{(k,\mu ;n)}{\parallel }_d\le K_5(1+\parallel Y_n{\parallel }_d)$(8)

for all $h_n\in (0,h_{max})$, where

$K_5=\frac{1+(m+1)|\bar{M}|s{K}_{z}K\sqrt{h_{max}}}{1-(m+1)|\bar{M}|s{K}_{z}K\sqrt{h_{max}}},h_{max}=\frac{1}{{(1+(m+1)|\bar{M}|s{K}_{z}K)}^2}.$

Here, $|\bar{M}|$ denotes the number of elements in $\bar{M}$.

Further, Equation (3)(3) $\begin{array}{cc}{Y}_{n+1}& =Y_n+\sum _{k=0}^m\sum _{\mu \in \bar{M}}\sum _{i=1}^s{z}_i^{k,\mu ;n}{g}_k(H_i^{(k,\mu ;n)})+C_n(Y_n-Y_{n+1}),\\ H_i^{(k,\mu ;n)}& =Y_n+\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{j=1}^s{Z}_{i,j}^{k,\mu ;l,\nu ;n}{g}_l(H_j^{(l,\nu ;n)}),\end{array}$(3) implies that

(9) $Y_{n+1}=Y_n+(I_d+C_n{)}^{-1}\sum _{k=0}^m\sum _{\nu \in \bar{M}}\sum _{i=1}^s{z}_i^{k,\nu ;n}{g}_k(H_i^{(k,\nu ;n)}).$(9)

Let

$d_{[t_k,Y_k]}(t_{k+1})={\tilde{Y}}_{[t_k,Y_k]}(t_{k+1})-Y_{k+1}$

where ${\tilde{Y}}_{[t_k,Y_k]}(t_{k+1})$ denotes the numerical approximation given by the underlying Runge–Kutta method (Equation (2))(2) $\begin{array}{cc}{Y}_{n+1}& =Y_n+\sum _{k=0}^m\sum _{\mu \in \bar{M}}\sum _{i=1}^s{z}_i^{k,\mu ;n}{g}_k(H_i^{(k,\mu ;n)}),\\ H_i^{(k,\mu ;n)}& =Y_n+\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{j=1}^s{Z}_{i,j}^{k,\mu ;l,\nu ;n}{g}_l(H_j^{(l,\nu ;n)}).\end{array}$(2) of the balanced method at time point $t_{k+1}$ when starting at time $t_k$ with value $Y_k$. It follows then from Equations (2)(2) $\begin{array}{cc}{Y}_{n+1}& =Y_n+\sum _{k=0}^m\sum _{\mu \in \bar{M}}\sum _{i=1}^s{z}_i^{k,\mu ;n}{g}_k(H_i^{(k,\mu ;n)}),\\ H_i^{(k,\mu ;n)}& =Y_n+\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{j=1}^s{Z}_{i,j}^{k,\mu ;l,\nu ;n}{g}_l(H_j^{(l,\nu ;n)}).\end{array}$(2) and (9)(9) $Y_{n+1}=Y_n+(I_d+C_n{)}^{-1}\sum _{k=0}^m\sum _{\nu \in \bar{M}}\sum _{i=1}^s{z}_i^{k,\nu ;n}{g}_k(H_i^{(k,\nu ;n)}).$(9) that

$d_{[t_k,Y_k]}(t_{k+1})=\underset{=(I_d+C_k{)}^{-1}(I_d+C_k-I_d)}{\underbrace{(I_d-{(I_d+C_k)}^{-1})}}\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{i=1}^s{z}_i^{l,\nu ;k}{g}_l(H_i^{(l,\nu ;k)})$

$=(I_d+C_k{)}^{-1}{C}_k\sum _{l=0}^m\sum _{\nu \in \bar{M}}\sum _{i=1}^s{z}_i^{l,\nu ;k}{g}_l(H_i^{(l,\nu ;k)}),$

implying together with Equation (4)(4) $C_n=h_n^{p+\frac{1}{2}}\left[c_0(t_n,Y_n)+\sum _{j=1}^m(\frac{{(\mathrm{\Delta }{W}_n^j)}^{2p}}{h_n^p}-1)c_j(t_n,Y_n)\right],$(4) , Assumption 1, Equations (6)(6) $|z_i^{k,\nu ;n}|\le K_z\sqrt{h_n},|Z_{i,j}^{k,\mu ;l,\nu ;n}|\le K_z\sqrt{h_n}$(6) ,(5)(5) $\parallel g_j(x)-g_j(y){\parallel }_d\le L\parallel x-y{\parallel }_d,\parallel g_j(x){\parallel }_d\le K(1+\parallel x{\parallel }_d),$(5) and (8)(8) $\underset{k,\mu ,i}{max}\parallel H_i^{(k,\mu ;n)}{\parallel }_d\le K_5(1+\parallel Y_n{\parallel }_d)$(8) that

(10) $\begin{array}{c}\sqrt[4]{\mathbb{E}[\parallel \mathbb{E}[d_{[t_k,Y_k]}(t_{k+1})|{\mathcal{F}}_{t_k}]{\parallel }_d^4]}\le K_1\sqrt[4]{1+\mathbb{E}[\parallel Y_k{\parallel }_d^{4(1+\kappa )}]}{(t_{k+1}-t_k)}^{p+1},\\ \sqrt{\mathbb{E}[\parallel d_{[t_k,Y_k]}(t_{k+1}){\parallel }_d^4]}\le K_2\sqrt{1+\mathbb{E}[\parallel Y_k{\parallel }_d^{4(1+\kappa )}]}{(t_{k+1}-t_k)}^{p+1}.\end{array}$(10)

Similarly, we obtain from Equation (9)(9) $Y_{n+1}=Y_n+(I_d+C_n{)}^{-1}\sum _{k=0}^m\sum _{\nu \in \bar{M}}\sum _{i=1}^s{z}_i^{k,\nu ;n}{g}_k(H_i^{(k,\nu ;n)}).$(9) by using Equation (4)(4) $C_n=h_n^{p+\frac{1}{2}}\left[c_0(t_n,Y_n)+\sum _{j=1}^m(\frac{{(\mathrm{\Delta }{W}_n^j)}^{2p}}{h_n^p}-1)c_j(t_n,Y_n)\right],$(4) , Assumption 1, Equations (6(6) $|z_i^{k,\nu ;n}|\le K_z\sqrt{h_n},|Z_{i,j}^{k,\mu ;l,\nu ;n}|\le K_z\sqrt{h_n}$(6) ),(5)(5) $\parallel g_j(x)-g_j(y){\parallel }_d\le L\parallel x-y{\parallel }_d,\parallel g_j(x){\parallel }_d\le K(1+\parallel x{\parallel }_d),$(5) and (8(8) $\underset{k,\mu ,i}{max}\parallel H_i^{(k,\mu ;n)}{\parallel }_d\le K_5(1+\parallel Y_n{\parallel }_d)$(8) )that

(11) $\parallel Y_{n+1}-Y_n{\parallel }_d\le K_6(1+\parallel Y_n{\parallel }_d)\sqrt{h_n})$(11)

with $K_6=M(m+1)|\bar{M}|s{K}_{z}K(1+K_5)$, and in addition using Equation (7)(7) $\parallel \sum _{\nu \in \bar{M}}\sum _{i=1}^s\mathbb{E}[(I_d+C_n{)}^{-1}{z}_i^{k,\nu ;n}]\parallel \le K_z{h}_n$(7) that

$\parallel \mathbb{E}(Y_{n+1}-Y_n|{\mathcal{F}}_{t_n}){\parallel }_d\le \parallel \sum _{k=0}^m\sum _{\nu \in \bar{M}}\sum _{i=1}^s\mathbb{E}[(I_d+C_n{)}^{-1}{z}_i^{k,\nu ;n}]g_k(Y_n){\parallel }_d$

(12,13) $+\parallel \sum _{k=0}^m\sum _{\nu \in \bar{M}}\sum _{i=1}^s\mathbb{E}[(I_d+C_n{)}^{-1}{z}_i^{k,\nu ;n}[g_k(H_i^{(k,\nu ;n)})-g_k(Y_n)]|{\mathcal{F}}_{t_n}]{\parallel }_d$(12,13)

(14) $\le K_7{h}_n(1+\parallel Y_n{\parallel }_d)$(14)

with

$K_7=(m+1)K_{z}K+(m+1)|\bar{M}|sM{K}_{z}L(m+1)|\bar{M}|s{K}_{z}K(1+K_5).$

With Equations (11)(11) $\parallel Y_{n+1}-Y_n{\parallel }_d\le K_6(1+\parallel Y_n{\parallel }_d)\sqrt{h_n})$(11) and(12)(12,13) $+\parallel \sum _{k=0}^m\sum _{\nu \in \bar{M}}\sum _{i=1}^s\mathbb{E}[(I_d+C_n{)}^{-1}{z}_i^{k,\nu ;n}[g_k(H_i^{(k,\nu ;n)})-g_k(Y_n)]|{\mathcal{F}}_{t_n}]{\parallel }_d$(12,13) , it follows from [15, Lemma 9.1] that for each $r\in \mathbb{N}$, there exists a constant $K_{2r}$ such that for all $N$

(15) $\underset{k=0,\dots ,N}{max}\mathbb{E}[\parallel Y_k{\parallel }_d^{2r}]\le K_{2,r},\underset{k=0,\dots ,N}{max}\mathbb{E}[\parallel {\tilde{Y}}_{[t_k,Y_k]}(t_{k+1}){\parallel }_d^{2r}]\le K_{2,r}.$(15)

With Equations (10)(10) $\begin{array}{c}\sqrt[4]{\mathbb{E}[\parallel \mathbb{E}[d_{[t_k,Y_k]}(t_{k+1})|{\mathcal{F}}_{t_k}]{\parallel }_d^4]}\le K_1\sqrt[4]{1+\mathbb{E}[\parallel Y_k{\parallel }_d^{4(1+\kappa )}]}{(t_{k+1}-t_k)}^{p+1},\\ \sqrt{\mathbb{E}[\parallel d_{[t_k,Y_k]}(t_{k+1}){\parallel }_d^4]}\le K_2\sqrt{1+\mathbb{E}[\parallel Y_k{\parallel }_d^{4(1+\kappa )}]}{(t_{k+1}-t_k)}^{p+1}.\end{array}$(10) and (13(14) $\le K_7{h}_n(1+\parallel Y_n{\parallel }_d)$(14) ), the assertion follows now by Theorem 1.

As a direct consequence, Theorem 2 yields the weak convergence order of the balanced versions of W2Ito1, SRI1, W2Ito2 and SRK4 discussed in Rathinasamy et al. (2020) and Rathinasamy and Nair (2018).

Corollary 1 For sufficiently smooth drift and diffusion, the balanced versions of W2Ito1, SRI1, W2Ito2 and SRK4 are of weak convergence order 2.

Acknowledgements

The authors are grateful to two anonymous referees for their helpful hints which improved the presentation of the material.

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Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

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