Optimal investment strategy for a family with a random household expenditure under the CEV model. (English) Zbl 07571109
Summary: 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.
MSC:
| 62-XX | Statistics |
Keywords:
optimal investment strategy; stochastic household expenditure; constant elasticity of variance; Feynman-Kac formulaReferences:
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