Lifetime Ruin under High-watermark Fees and Drift Uncertainty

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.

Optimal Entry and Consumption under Habit Formation

“…For some related literature on optimal stopping using viscosity solution, we refer to [31] and [27]. See also some recent work on stochastic control problems using stochastic Perron's method among [2,3,4,5,23,33]. One important step to complete the argument of stochastic Perron's method is the comparison principle of the associated variational inequalities, which is also established in the present paper.…”