Lifetime Ruin Under High-Water Mark Fees and Drift Uncertainty

Abstract

This paper aims to study lifetime ruin minimization 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 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

“…with μτ = μ τ , P-a.s. Moreover, the conditional variance ˆ (t) = E μ t − μt 2 F S t satisfies the deterministic Riccati ordinary differential equation (ODE) (see Equation (19) of [8] or Equation (23) of [25])…”

“…For some related works on optimal stopping using viscosity solutions, we refer to [31] and [27]. Recent works on stochastic control problems using the stochastic Perron method include [2, 3, 4, 5, 33, 23]. One important step in completing the argument for the stochastic Perron method is the comparison principle for the associated variational inequalities, which is established in the present paper.…”