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

Belkacem Berdjane (1) , Sergei Pergamenshchikov (1, 2)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 Department of Mathematics and Mechanics

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.

Mots clés: Sequential analysis; Truncate sequential estimate; Black-Scholes market model; Stochastic volatility; Optimal Consumption and Investment; Hamilton-Jacobi-Bellman equation

Domaines: Gestion de portefeuilles [q-fin.PM]; Applications [stat.AP]

Belkacem Berdjane, Sergei Pergamenshchikov. Sequential $\delta$-optimal consumption and investment for stochastic volatility markets with unknown parameters. 2012. ⟨hal-00743164v2⟩