Discounted optimal stopping for maxima of some jump-diffusion processes
- Gapeev, Pavel
2010 Mathematics Subject Classification
- 60G40 34K10 91B70 60J60 60J75 91B28
- Discounted optimal stopping problem, Brownian motion, compound Poisson process, maximum process, integro-differential free-boundary problem, continuous and smooth fit, normal reflection, a change-of-variable formula with local time on surfaces, perpetual lookback American options
We present solutions to some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems where the normal reflection and smooth fit may break down and the latter then be replaced by the continuous fit. The results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.