Optimal Consumption Until Ruin for an Endowment Described by an Autonomous ODE for an Infinite Time Horizon

We give an algorithmic solution of the optimal consumption problem sup c ∫ [ 0 , τ ] e − β t d C t , where C t denotes the accumulated consumption until time t , and τ denotes the time of ruin. Moreover, the endowment process X t is modeled by X t = x + ∫ 0 t μ ( X s ) d s − C t . We solve the probl...

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Bibliographic Details
Published in:Mathematics of operations research 2016-08, Vol.41 (3), p.953-968
Main Author: Grandits, Peter
Format: Article
Language:English
Subjects:
algorithmic solution
Analysis
Boundary value problems
Consumption
Differential equations
Endowment
Endowments
free boundary value problem
optimal consumption
Optimization algorithms
Ordinary differential equations
singular control problem
Studies
viscosity solutions
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Summary:We give an algorithmic solution of the optimal consumption problem sup c ∫ [ 0 , τ ] e − β t d C t , where C t denotes the accumulated consumption until time t , and τ denotes the time of ruin. Moreover, the endowment process X t is modeled by X t = x + ∫ 0 t μ ( X s ) d s − C t . We solve the problem by showing that the function provided by the algorithm solves the Hamilton-Jacobi (HJ) equation in a viscosity sense and that the same is true for the value function of the problem. The argument is finished by a uniqueness result. It turns out that one has to change the optimal strategy at a sequence of endowment values, described by a free boundary value problem. Finally we give an illustrative example.
ISSN:0364-765X
1526-5471
DOI:10.1287/moor.2015.0763