Remarks on the Optimal Portfolio Problem in Discrete Variables with Multiple Stochastic Processes

Abstract

Abstract-We are concerned with the optimal portfolio problem under stochastic environment; in particular, we deal with the case of two independent stochastic processes in discrete variables. One process is typical random walk, which is regarded as a discrete version of the standard Brownian motion, and the other is the Poisson process. We derive a discrete Hamilton-Jacobi-Bellman (HJB) equation for the value function and try to solve it. Examples are also discussed.Index Terms-Discrete hamilton-jacobi-bellman equation, discrete processes, multiple stochastic processes, optimal portfolio problem.

On the Convergence of Discrete Processes With Multiple Independent Variables

“…One example is the optimal portfolio problem. We have derived a discrete Hamilton-Jacobi-Bellman equation for the value function to characterize the extremals (see [5,10]). The convergence of discrete to continuous versions is worth further investigation.…”

“…Yoshida and Ishimura [10] established the following theorem, which may be interpreted as a discrete analogue of Ito's formula with jump process. Theorem 2.1.…”