Look-back stopping times and their applications to liquidation risk and exotic options

In addition to first passage times, many look-back stopping times play a significant role in modeling various risks in insurance and finance as well as in defining financial instruments. Motivated by many recently arisen problems in risk management and exotic options, we study some look-back stopping times including drawdown and drawup, Parisian time and inverse occupation time of some time-homogeneous Markov processes such as diffusion processes and jump-diffusion processes.

Since the structures of these look-back stopping times are much more complex than fundamental stopping times such as first passage times, we aim to develop some general approaches to study these stopping times such as approximation approach and perturbation approach. These approaches can be transformed to a wide class of stochastic processes. Many interesting and explicit formulas for these stopping times are derived and based on which we gain quantitative understandings of these problems in insurance and finance.

In our study, we mainly use the techniques of Laplace transforms and partial differential equations (PDEs). Due to the complex structures, the distributions of these look-back stopping times are usually not explicit even for the simplest linear Brownian motion. However, under Laplace transforms, many important formulas become explicit and it enables us to conduct further derivations and analysis. Besides, PDE methodology provides us an effective and efficient approach in both theoretical investigation and numerical study of these stopping times.