Optimal consumption and investment for markets with random coefficients.

Berdjane Belkacem (1) , Serguei Pergamenchtchikov (2)

1 UMMTO - Université Mouloud Mammeri [Tizi Ouzou]
2 LMRS - Laboratoire de Mathématiques Raphaël Salem

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.

Mots clés: Kac formula; Feynman; Kac formula.; Black-Scholes market; Stochastic volatility; Optimal consumption and Investment; Hamilton-Jacobi-Bellman equation; Banach fixed point theorem

Domaines: Probabilités [math.PR]

Berdjane Belkacem, Serguei Pergamenchtchikov. Optimal consumption and investment for markets with random coefficients.. 2011. ⟨hal-00563577v2⟩