Stochastic Feynman–Kac Equations Associated to Lévy–Itô Diffusions

Abstract

We consider linear parabolic stochastic integro-PDEs of Feynman–Kac type associated to Lévy–Itô diffusions. The solution of such equations can be represented as certain Feynman–Kac functionals of the associated diffusion such that taking expectation yields the deterministic Feynamn–Kac formula. We interpret the problem in the framework of white noise analysis and consider differentiation in the sense of stochastic distributions. This concept allows for relaxed assumptions on the equation coefficients, identically to those required in problems of similar deterministic integro-PDEs.

Keywords:

• Backward diffusion equation
• Lévy processes
• Stochastic Feynman–Kac formula
• Stochastic integro partial differential equation
• White noise

2000 Mathematics Subject Classification:

• Primary 60H15
• Secondary 60H40, 60G51

ACKNOWLEDGMENTS

The author thanks Frank Proske and Kenneth Hvistendahl Karlsen for discussions and suggestions as well as an anonymous referee for valuable comments.