# Optimal Stopping of Linear Diffusions under a Random Horizon – Wiljami Sillanpää (Department of Mathematics and Statistics)

We develop a general method based on the classical theory of diffusions for solving optimal stopping problems of one-dimensional regular diffusions under a random horizon. The presence of non-exponential discounting renders the problem time-inconsistent. Contrary to most literature on time-inconsistent optimal stopping, our discount function is log-superadditive and non-stationary. Under relatively weak assumptions, we characterize the equilibrium stopping time as a hitting time to a moving boundary, which in turn is characterized as a solution to a certain ordinary differential equation. The payoff associated to the equilibrium strategy is characterized as a solution to a certain boundary value problem. We illustrate the general theory with a numerical example where the underlying process is a geometric Brownian motion, the payoff is an American call and the horizon follows a Weibull distribution with an increasing hazard rate. This can be seen as a model for pricing options under default risk.