Markovian bridges and reversible diffusions with jumps

 

Nicolas Privault   Jean-Claude Zambrini
Département de Mathématiques   Grupo de Física Matemática
Université de la Rochelle   Universidade de Lisboa
17042 La Rochelle Cedex 1   16049-003 Lisboa
France   Portugal

 

Abstract:

Markovian bridges driven by Lévy processes are constructed from the data of an initial and a final distribution, as particular cases of a family of time reversible diffusions with jumps. The processes obtained in this way are essentially the only (not necessarily continuous) Markovian Bernstein processes. These processes are also characterized using the theory of stochastic control for jump processes. Our construction is motivated by Euclidean quantum mechanics in momentum representation, but the resulting class of processes is much bigger than the one needed for this purpose. A large collection of examples is included.

Key words: Lévy processes, bridges, time reversal, Euclidean quantum mechanics.
Mathematics Subject Classification (1991): 60J25, 81S20, 47D07.

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