Sequential $\delta$-Optimal Consumption and Investment for Stochastic Volatility Markets with Unknown Parameters

Abstract

We consider an optimal investment and consumption problem for a Black--Scholes financial market with stochastic volatility and unknown stock price 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.