Optimal consumption and investment for markets with random coefficients |
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Authors: | Belkacem Berdjane Serguei Pergamenshchikov |
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Institution: | 1. Laboratoire L2CSP, Université Mouloud Mammeri, Tizi-Ouzou, 15000, Algeria 2. Laboratoire LMRS, Université de Rouen, Avenue de l’Université, BP. 12, 768001, Saint Etienne du Rouvray Cedex, France 3. Laboratoire de Mathématiques Raphael Salem, Université de Rouen, Avenue de l’Université, BP. 12, 76801, Saint Etienne du Rouvray Cedex, France 4. Department of Mathematics and Mechanics, Tomsk State University, Lenin str. 36, 634041, Tomsk, Russia
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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. |
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