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 L2-valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the cofficients. 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
A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps / Cordoni, Francesco; Di Persio, Luca; Oliva, Immacolata. - In: NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 1021-9722. - (2017). [10.1007/s00030-017-0440-3]
A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps
Immacolata Oliva
2017
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 L2-valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the cofficients. 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 differentiabilityFile | Dimensione | Formato | |
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