Optimal consumption and investment for markets with random coefficients

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...

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Bibliographic Details
Published in:Finance and stochastics 2013-04, Vol.17 (2), p.419-446
Main Authors: Berdjane, Belkacem, Pergamenshchikov, Serguei
Format: Article
Language:English
Subjects:
Capital market
Coefficients
Convergence
Dynamic programming
Economic theory
Economic Theory/Quantitative Economics/Mathematical Methods
Economics
Expected utility
Finance
Financial services
Institutional investments
Insurance
Investment decision
Investment policy
Management
Mathematics
Mathematics and Statistics
Partial differential equations
Probability Theory and Stochastic Processes
Quantitative Finance
Statistics for Business
Stochastic models
Studies
Utility functions
Volatility
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Summary: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.
ISSN:0949-2984
1432-1122
DOI:10.1007/s00780-012-0193-0