Stochastic Perron's Method for the Probability of lifetime ruin problem under transaction costs

TL;DR

This paper employs stochastic Perron's method to analyze a singular control problem involving investment, consumption, and transaction costs, characterizing the value function without relying on the dynamic programming principle.

Contribution

It introduces a novel application of stochastic Perron's method to a lifetime ruin problem with transaction costs, establishing the value function as a unique viscosity solution.

Findings

Characterizes the value function as a viscosity solution of the HJB variational inequality.

Provides a complete proof of the comparison principle for the problem.

Demonstrates the effectiveness of stochastic Perron's method in singular control problems.

Abstract

We apply stochastic Perron's 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 (DPP), we characterize the value function as the unique viscosity solution of an associated Hamilton-Jacobi-Bellman (HJB) variational inequality. We also provide a complete proof of the comparison principle which is the main assumption of stochastic Perron's method.