A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps

作者：Cordoni F^*; Di Persio L; Oliva I

来源：Nodea-Nonlinear Differential Equations and Applications, 2017, 24(2): 16.

DOI：10.1007/s00030-017-0440-3

• Stochastic delay differential equations
• Quadratic variation
• Levy processes
• Feynman-Kac formula
• Mild solution

We consider a stochastic functional delay differential equation, namely an equation whose evolution depends on its past history as well as on its present state, driven by a pure diffusive component plus a pure jump Poisson compensated measure. We lift the problem in the infinite dimensional space of square integrable Lebesgue functions in order to show that its solution is an L-2-valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the coefficients. Coupling the aforementioned equation with a standard backward differential equation, and deriving some ad hoc results concerning the Malliavin derivative for systems with memory, we are able to derive a non-linear Feynman-Kac representation theorem under mild assumptions of differentiability.

• 出版日期2017-4