Sequential $\delta$-optimal consumption and investment for stochastic volatility markets with unknown parameters

TL;DR

This paper develops a sequential estimation method for optimal investment and consumption strategies in a stochastic volatility market with unknown parameters, ensuring near-optimal performance.

Contribution

It introduces a $ ext{delta}$-optimal strategy that incorporates sequential estimation of the unknown drift in a stochastic volatility model.

Findings

The strategy achieves $ ext{delta}$-optimality using observed economic factors.

The approach effectively estimates the unknown drift parameter.

The method applies to markets with Ornstein-Uhlenbeck type volatility.

Abstract

We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic volatility and unknown stock appreciation rate. The volatility parameter is driven by an external economic factor modeled as a diffusion process of Ornstein-Uhlenbeck type with unknown drift. We use the dynamical programming approach and find an optimal financial strategy which depends on the drift parameter. To estimate the drift coefficient we observe the economic factor $Y$ in an interval $[0,T_0]$ for fixed $T_0>0$, and use sequential estimation. We show, that the consumption and investment strategy calculated through this sequential procedure is $\delta$-optimal.