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.
Notes
1 The Feynman-Kac formula (cf. Theorem 4.2 and it proof in Karatzas and Shreve (Citation1988)) shows that the solution to the partial differential equation
with boundary condition
can be written as a conditional expectation
under probability measure
and X is driven by the equation
where
is a Brownian motion under
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
explicitly, we can further verify that the candidates are indeed the solutions of the parabolic pdes and we omit the detail here.