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

Danping Li, Xiaotao Liu and Hailong Liu

Communications in Statistics - Theory and Methods, 2022, vol. 51, issue 17, 5993-6007

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.

Date: 2022
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DOI: 10.1080/03610926.2020.1851718

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