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Stochastic Perron’s method for the probability of lifetime ruin problem under transaction costs. (English) Zbl 1343.93094

Summary: We apply the stochastic Perron method to a singular control problem, where an individual targets at a given consumption rate, invests in a risky financial market in which trading is subject to proportional transaction costs, and seeks to minimize her probability of lifetime ruin. Without relying on the dynamic programming principle, we characterize the value function as the unique viscosity solution of an associated Hamilton-Jacobi-Bellman variational inequality. We also provide a complete proof of the comparison principle, which is the main assumption of the stochastic Perron method.

MSC:

93E20 Optimal stochastic control
49J55 Existence of optimal solutions to problems involving randomness
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49L20 Dynamic programming in optimal control and differential games
49J40 Variational inequalities
60H30 Applications of stochastic analysis (to PDEs, etc.)
60G46 Martingales and classical analysis
91B30 Risk theory, insurance (MSC2010)
35Q93 PDEs in connection with control and optimization
35D40 Viscosity solutions to PDEs
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