Optimal stopping, ruin probabilities and prophet inequalities for Lévy processes

E. Mordecki. Prepublicaciones de Matemática de la Universidad de la República, 2000/38.


Abstract: Solution to the optimal stopping problem

V(x)=sup{E[max(x+X(T) , 0)]; T is a stopping time}

is given, where X is a Lévy process. Results are expressed in terms of the distribution of the random variable

M=sup{ X(t), t > 0}

under the hypothesis of finitness of E(M), and simple conditions for this hypothesis to hold are given. Based on this, the prophet inequality

V(x) <= E(x+M) <= e V(x)

is obtained. Closed form solutions of the distribution of M are given for a wide class of Lévy processes.


Keywords: Optimal stopping, prophet inequalities, Lévy process, ruin probability.Classification: MSC 60G40.


Download: dvi file (60 Kb), ps file (490 Kb), pdf file (280 Kb)


Click Here!


Return to Mordecki's articles