A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

Abstract

Principle of optimality or dynamic programming leads to derivation of a partial differential equation (PDE) for solving optimal control problems, namely the Hamilton‐Jacobi‐Bellman (HJB) equation. In general, this equation cannot be solved analytically; thus many computing strategies have been developed for optimal control problems. Many problems in financial mathematics involve the solution of stochastic optimal control (SOC) problems. In this work, the variational iteration method (VIM) is applied for solving SOC problems. In fact, solutions for the value function and the corresponding optimal strategies are obtained numerically. We solve a stochastic linear regulator problem to investigate the applicability and simplicity of the presented method and prove its convergence. In particular, for Merton's portfolio selection model as a problem of portfolio optimization, the proposed numerical method is applied for the first time and its usefulness is demonstrated. For the nonlinear case, we investigate its convergence using Banach's fixed point theorem. The numerical results confirm the simplicity and efficiency of our method.

Development of the method of quasi-optimal robust control for periodic operational processes

“…For today, one of the weak points in the field of optimum control is the lack of a theoretical base for the optimal control trajectory search method formalization [7][8][9].…”

Filtering method for linear and non-linear stochastic optimal control of partially observable systems II

“…Many studies have been done about this subject. For example, in [11] Kafash et al used this equation to evaluate this kind of problem. One of the important and useful case of SOC problem is SOC with jumping diffusion.…”

Uncertain Optimal Control Approach for CO2 Mitigation Problem

“…Especially for finance, such as in . Recently, presented a variational iteration method (VIM) for solving stochastic optimal control problems, and then applied the method in Merton's portfolio selection model.…”