Optimal consumption and investment for markets with random coefficients

TL;DR

This paper investigates an optimal investment and consumption problem in a stochastic market with random coefficients, providing theoretical results on the PDE solution and numerical convergence, with applications to stochastic volatility models.

Contribution

It introduces a novel analysis of the HJB equation with stochastic coefficients, proving solution smoothness and super geometrical convergence rates for numerical schemes.

Findings

Proved uniqueness and smoothness of the HJB solution.

Established super geometrical convergence rate for numerical schemes.

Applied results to stochastic volatility market models.

Abstract

We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamical programming approach leads to an investigation of the Hamilton Jacobi Bellman (HJB) equation which is a highly non linear partial differential equation (PDE) of the second oder. By using the Feynman - Kac representation we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of the iterative numerical schemes for both the value function and the optimal portfolio. We show, that in this case, the optimal convergence rate is super geometrical, i.e. is more rapid than any geometrical one. We apply our results to a stochastic volatility financial market.