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 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.

A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps / Cordoni, F.; Di Persio, L.; Oliva, I.. - In: NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 1021-9722. - 24:2(2017). [10.1007/s00030-017-0440-3]

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

Cordoni F.;Di Persio L.;
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 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 / Cordoni, F.; Di Persio, L.; Oliva, I.. - In: NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 1021-9722. - 24:2(2017). [10.1007/s00030-017-0440-3]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11572/322956
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