Convergent Numerical Scheme for Singular Stochastic Control with State Constraints in a Portfolio Selection Problem

We consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs. Numerical approximations for the value function using the Markov chain approximation method of Kushner and Dupuis are studied. The main resu...

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Veröffentlicht in:SIAM journal on control and optimization Jg. 45; H. 6; S. 2169 - 2206
Hauptverfasser: Budhiraja, Amarjit, Ross, Kevin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2007
Schlagworte:
Applied sciences
Approximation
Bank accounts
Costs
Decision theory. Utility theory
Exact sciences and technology
Investments
Investors
Markov analysis
Operational research and scientific management
Operational research. Management science
Portfolio theory
Queuing theory. Traffic theory
Scheduling, sequencing
Stochastic control theory
Utility functions
Viscosity
Solution uniqueness
Markov decision
Degenerate system
Optimal policy
State space
Function approximation
Boundary condition
Modeling
Uniqueness theorem
Convergence
Numerical approximation
Markov chain
Singular control
Control constraint
Portfolio selection
Continuous control
90B35
Cost function
Dirichlet problem
68M20
60K25
State constraint
Consumption
Neumann problem
Probabilistic approach
60J70
free boundary problem
Continuous function
State space method
Cost control
Skorohod problem
Selection problem
90B22
Value function
Transaction cost
Boundary value problem
Hamilton Jacobi equation
Approximation method
Particular solution
Optimal control
Stochastic control
Hamilton-Jacobi-Bellman equations
Investment cost
Portfolio management
ISSN:0363-0129, 1095-7138
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Zusammenfassung:We consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs. Numerical approximations for the value function using the Markov chain approximation method of Kushner and Dupuis are studied. The main result of the paper shows that the value function of the Markov decision problem (MDP) corresponding to the approximating controlled Markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits. All our convergence arguments are probabilistic; the main assumption that we make is that the value function be finite and continuous. In particular, uniqueness of the solutions of the associated HJB equations is neither needed nor available (in the generality under which the problem is considered). Specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function; degeneracies in the dynamics; mixed boundary (Dirichlet-Neumann) conditions; and presence of both singular and absolutely continuous controls in the dynamics. Finally, schemes for computing the value function and optimal control policies for the MDP are presented and illustrated with a numerical study.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0363-0129
1095-7138
DOI:10.1137/050640515