Viscosity solution for optimal liquidation problems with randomly-terminated horizon

Abstract

In this paper, we study an optimal liquidation problem of a stressed asset, for which its value is modeled by a geometric Brownian motion. The default time of the stressed asset, therefore, becomes stochastic and predictable. Hence we deal with the optimal liquidation problem in a randomly-terminated horizon. We consider the liquidation of a large single-asset portfolio with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impacts. We prove that the value function is the unique viscosity solution to the Hamilton–Jacobi–Bellman (HJB) equation of the considered optimal liquidation problem.