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Weak reflection principle for Lévy processes. (English) Zbl 1330.60064
Summary: In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the classical reflection principle to work. We call this method a weak reflection principle and show that it provides solutions to many problems for which the classical reflection principle is typically used. In addition, unlike the classical reflection principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak reflection principle for a large class of time-homogeneous diffusions on a real line and then proceed to extend this method to the Lévy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak reflection principle in financial mathematics, computational methods and inverse problems.

MSC:
60G51 Processes with independent increments; Lévy processes
60J75 Jump processes (MSC2010)
60J60 Diffusion processes
91G20 Derivative securities (option pricing, hedging, etc.)
91G80 Financial applications of other theories
45Q05 Inverse problems for integral equations
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