Optimal investment strategy for a family with a random household expenditure under the CEV model

Abstract

This paper considers an optimal investment strategy to maximize the expected constant absolute risk averse (CARA) utility of the terminal wealth for a family in the presence of stochastic household expenditure under the constant elasticity of variance (CEV) model. Since the corresponding Hamilton-Jacobi-Bellman (HJB) equation is difficult to solve for the high dimensionality and nonlinearity, previous work only gives an approximate numerical solution for some special model parameters under the slow-fluctuating regime assumption. In this paper, by directly conjecturing the functional form of the value function, we transform the HJB equation into two one-dimensional parabolic partial differential equations (pdes) and further find their explicit solutions via the Feynman-Kac formula. We prove that the exact and explicit solution for the value function as well as the optimal investment strategy can be expressed as integral of confluent hyper-geometric function. Finally, numerical examples are provided to illustrate the effects of parameters on the optimal strategies.

Keywords:

• Optimal investment strategy
• stochastic household expenditure
• constant elasticity of variance
• Feynman-Kac formula

Notes

1 The Feynman-Kac formula (cf. Theorem 4.2 and it proof in Karatzas and Shreve (1988)) shows that the solution to the partial differential equation $$\frac{\partial u}{\partial t}(t,x)+\mu (t,x)\frac{\partial u}{\partial x}(t,x)+\frac{1}{2}{\sigma }^2(t,x)\frac{{\partial }^{2}u}{\partial x^2}(t,x)-V(t,x)u(t,x)+f(t,x)=0,$$ with boundary condition $u(T,x)=\psi \left(x\right)$ can be written as a conditional expectation $$u(t,X)=E^{\mathbb{Q}}\left[{\int }_t^T{e}^{-{\int }_t^{\nu }V\left(\tau ,X_{\tau }\right)d\tau }f\left(r,X_r\right)dr+e^{-{\int }_t^{T}V\left(\tau ,X_{\tau }\right)d\tau }\psi \left(X_T\right)|X_t=x\right],$$ under probability measure $\mathbb{Q}$ and X is driven by the equation $$dX=\mu (t,X)dt+\sigma (t,X)d{W}^{\mathbb{Q}},$$ where $W^{\mathbb{Q}}\left(t\right)$ is a Brownian motion under $\mathbb{Q}$ with the initial condition X(t) = x. It is worthy to note that some regularity conditions should be satisfied before the application of Feynman-Kac formula. Since here we solve $g(t,S),h(t,S)$ explicitly, we can further verify that the candidates are indeed the solutions of the parabolic pdes and we omit the detail here.

Additional information

Funding

This research was supported by National Natural Science Foundation of China (Grant Nos. 11801179, 71771220, 11971172, 12071147, 71790592, 71873088), the “Chenguang Program”, Shanghai, China (No. 18CG26), the Fundamental Research Funds for the Central Universities (No. 2019ECNU-HWFW028), the 111 Project, China (No. B14019).