Optimal Stopping and Maximal Inequalities for Poisson Processes
D.O. Kramkov and E. Mordecki,
Publicaciones Matemáticas del Uruguay. Vol. 8 (1999) pp. 153-178.
Abstract: Closed form solutions for some optimal stopping problems for stochastic processes driven by a Poisson processes N are given. First, cost functions and optimal stopping rules are described for the problems
s(x)=supTE(max[x,sup0<t<T(Nt-at)]-cT),
v(x)=supTE(max[x,sup0<t<T(bt-Nt)]-cT),
with a,b,c positive constants and T a stopping time. Based on the obtained results, maximal inequalities in the spirit of "On optimal stopping and maximal inequalities for Bessel processes", by Dubins L., Shepp L.A., Shiryaev A.N. (Theory of Probability and its Applications, 38 (1993) 226-261) are obtained. To complete the picture, solutions to the problems
c(x)=supTE(x+NT-aT)+,
p(x)=supTE(x+bT-NT)+
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