Abstract
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.
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Cordoni, F., Di Persio, L. & Oliva, I. A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps. Nonlinear Differ. Equ. Appl. 24, 16 (2017). https://doi.org/10.1007/s00030-017-0440-3
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DOI: https://doi.org/10.1007/s00030-017-0440-3
Keywords
- Stochastic delay differential equations
- Quadratic variation
- Lévy processes
- Feynman–Kac formula
- Mild solution