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

DI PERSIO, Luca
;
CORDONI, Francesco Giuseppe;Oliva, Immacolata
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.
Feynman–Kac formula, Lévy processes, Mild solution , Quadratic variation , Stochastic delay differential equations
File in questo prodotto:
File Dimensione Formato  
SDDE_jumps_NoDEA.pdf

accesso aperto

Descrizione: Preprint version
Tipologia: Documento in Pre-print
Licenza: Dominio pubblico
Dimensione 538.14 kB
Formato Adobe PDF
538.14 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/959714
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 3
social impact