Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem

TL;DR

This paper introduces a Riccati transformation-based method to solve the Hamilton-Jacobi-Bellman equation in stochastic optimal allocation, transforming it into a quasi-linear equation, and demonstrates its effectiveness through numerical examples.

Contribution

The paper develops a novel Riccati transformation approach for solving constrained stochastic optimal control problems, including a fully implicit numerical scheme and convergence analysis.

Findings

Successfully transforms nonlinear HJB into a quasi-linear equation

Proves existence, uniqueness, and bounds for solutions

Demonstrates numerical method on portfolio optimization with DAX 30

Abstract

In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical H\"older smooth solutions. We furthermore construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit traveling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index as an example of application of the method.