Lifetime Ruin under High-watermark Fees and Drift Uncertainty

TL;DR

This paper investigates the lifetime ruin problem considering high-watermark hedge fund fees and drift uncertainty, employing stochastic Perron's method to characterize the value function as a viscosity solution to the HJB equation.

Contribution

It introduces a novel approach using stochastic Perron's method to analyze a multi-dimensional ruin problem with complex performance fees and drift uncertainty, bypassing traditional dynamic programming.

Findings

Value function characterized as unique viscosity solution.

Established comparison principle for the HJB equation.

Demonstrated the applicability of stochastic Perron's method in complex ruin problems.

Abstract

This paper aims to make a new contribution to the study of lifetime ruin problem by considering investment in two hedge funds with high-watermark fees and drift uncertainty. Due to multi-dimensional performance fees that are charged whenever each fund profit exceeds its historical maximum, the value function is expected to be multi-dimensional. New mathematical challenges arise as the standard dimension reduction cannot be applied, and the convexity of the value function and Isaacs condition may not hold in our ruin probability minimization problem with drift uncertainty. We propose to employ the stochastic Perron's method to characterize the value function as the unique viscosity solution to the associated Hamilton Jacobi Bellman (HJB) equation without resorting to the proof of dynamic programming principle. The required comparison principle is also established in our setting to close the loop of stochastic Perron's method.