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.
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