Stochastic Processes in Continuous Time

This chapter reviews the theory of continuous-time stochastic processes, covering the concepts of adaptation, Lévy processes, diffusions, martingales, and Markov processes. Brownian motion is studied as the most important case, with properties that include the reflection principle and the strong Markov property. The technique of Skorokhod embedding is introduced, providing novel proofs of the central limit theorem and the law of the iterated logarithm. The family of processes derived from Brownian motion is reviewed and in the final section it is shown that a continuous process having finite variance and independent increments is Brownian motion.

Some results on optimal stopping under phase-type distributed implementation delay

We study optimal stopping of strong Markov processes under random implementation delay. By random implementation delay we mean the following: the payoff is not realised immediately when the process is stopped but rather after a random waiting period. The distribution of the random waiting period is assumed to be phase-type. We prove first a general result on the solvability of the problem. Then we study the case of Coxian distribution both in general and with scalar diffusion dynamics in more detail. The study is concluded with two explicit examples.

The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps

As is well-known, the benefit of restricting Lévy processes without positive jumps is the “ W , Z scale functions paradigm”, by which the knowledge of the scale functions W , Z extends immediately to other risk control problems. The same is true largely for strong Markov processes X t , with the notable distinctions that (a) it is more convenient to use as “basis” differential exit functions ν , δ , and that (b) it is not yet known how to compute ν , δ or W , Z beyond the Lévy, diffusion, and a few other cases. The unifying framework outlined in this paper suggests, however, via an example that the spectrally negative Markov and Lévy cases are very similar (except for the level of work involved in computing the basic functions ν , δ ). We illustrate the potential of the unified framework by introducing a new objective () for the optimization of dividends, inspired by the de Finetti problem of maximizing expected discounted cumulative dividends until ruin, where we replace ruin with an optimally chosen Azema-Yor/generalized draw-down/regret/trailing stopping time. This is defined as a hitting time of the “draw-down” process Y t = sup 0 ≤ s ≤ t X s - X t obtained by reflecting X t at its maximum. This new variational problem has been solved in a parallel paper.