Stochastic Perron's Method for the Probability of lifetime ruin problem under transaction costs
Stochastic Perron's Method for the Probability of lifetime ruin problem under transaction costs
This paper applies stochastic Perron's method to a singular control problem: minimizing the probability of lifetime ruin for an investor facing proportional transaction costs. The investor consumes at a constant rate and can trade a risky asset with proportional costs. The value function is characterized as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) variational inequality, which includes gradient constraints due to transaction costs. The method avoids the need for a dynamic programming principle by constructing families of stochastic supersolutions and subsolutions. The infimum of supersolutions is shown to be a viscosity subsolution, and the supremum of subsolutions is a viscosity supersolution via Perron-type arguments. A complete proof of the comparison principle is provided, relying on the existence of a strict classical subsolution. The paper also derives explicit upper and lower bounds for the value function using frictionless ruin probabilities and liquidation strategies. The results demonstrate the applicability of stochastic Perron's method to singular control problems and provide a rigorous foundation for the lifetime ruin problem under transaction costs.
Highlights
- 1First application of stochastic Perron's method to singular control problems, specifically the probability of lifetime ruin under transaction costs.
- 2Characterization of the value function as the unique viscosity solution of an HJB variational inequality without relying on the dynamic programming principle.
- 3Complete proof of the comparison principle for the HJB equation, which is essential for stochastic Perron's method.
- 4Construction of explicit stochastic supersolutions and subsolutions that satisfy boundary conditions, simplifying the boundary analysis.
Methods
- MStochastic Perron's method: defines families of stochastic supersolutions and subsolutions, then shows their envelopes are viscosity subsolution and supersolution respectively.
- MViscosity solution theory: uses comparison principle to prove uniqueness and identify the value function as the unique viscosity solution.
- MSingular control formulation: models trading with proportional transaction costs via cumulative buying and selling processes.
- MConstruction of explicit bounds: uses frictionless ruin probabilities and liquidation strategies to obtain upper and lower bounds for the value function.
Results
- RThe value function ψ is the unique continuous viscosity solution to the HJB variational inequality (2.4) with boundary conditions (2.5).
- RUpper bound: ψ(x,y) ≤ ((c - rL(x,y))/(c - rb))^{β/r}.
- RLower bound: ψ(x,y) ≥ ((c - rL(x,y))/(c - rb))^d, where d > 1 is defined in (2.8).
- RStochastic Perron's method successfully handles gradient constraints arising from transaction costs.
- RThe comparison principle holds for the HJB equation under the given assumptions.
Analyze Paper
Generate insights from "Stochastic Perron's Method for the Probability of lifetime r...".