On maximal inequalities for some jump processes
- Gapeev, Pavel
2010 Mathematics Subject Classification
- 60G40 34K10 60E15 60J60 60J75
- Jump process, stochastic differential equation, maximum process, optimal stopping problem, compound Poisson process, Ito's formula, integro-differential free-boundary problem, normal reflection, continuous and smooth fit, maximality principle, maximal inequalities
We present a solution to the considered in  and  optimal stopping problem for some jump processes. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem where the normal reflection and smooth fit may break down and the latter then be replaced by the continuous fit. The derived result is applied for determining the best constants in maximal inequalities for a compound Poisson process with linear drift and exponential jumps.