Optimal consumption and investment for markets with random coefficients

Belkacem Berdjane () and Sergey Pergamenshchikov ()

Finance and Stochastics, 2013, vol. 17, issue 2, 419-446

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 dynamic programming approach leads to an investigation of the Hamilton–Jacobi–Bellman (HJB) equation which is a highly nonlinear partial differential equation (PDE) of the second order. By using the Feynman–Kac representation, we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of iterative numerical schemes for both the value function and the optimal portfolio. We show that in this case, the optimal convergence rate is super-geometric, i.e., more rapid than any geometric one. We apply our results to a stochastic volatility financial market. Copyright Springer-Verlag 2013

Keywords: Black–Scholes market; Stochastic volatility; Optimal consumption and investment; Hamilton–Jacobi–Bellman equation; Feynman–Kac formula; Fixed-point solution; 91G10; 91G80; 93E20; G11 (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (13)

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DOI: 10.1007/s00780-012-0193-0

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