A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients

AbstractIn this note, we study the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations under some Carathéodory-type conditions on the coefficients by means of the successive approximation. In particular, we generalize and improve the results that appeared in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139–154] and Bao and Hou [J. Bao, Z. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl. 59 (2010) 207–214]

Mild solutions of non-Lipschitz stochastic integrodifferential evolution equations

In this work we study the existence and uniqueness of mild solutions for stochastic partial integrodifferential equations under local non-Lipschitz conditions on the coefficients. Our analysis makes use of the theory of resolvent operators as developed by R. Grimmer as well as a stopping time technique. Our results complement and improve several earlier related works. An example is provided to illustrate the theoretical results.Ministerio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Simons Foundatio

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

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

Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion

This paper deals with the existence and uniqueness of mild solutions to neutral stochastic delay functional integro-di erential equations perturbed by a fractional Brownian motion BH, with Hurst parameter H 2 ( 1 2; 1). We use the theory of resolvent operators developed in R.Grimmer [5] to show the existence of mild solutions. An example is provided to illustrate the results of this work

Asymptotic behavior of neutral stochastic partial functional integro-differential equations driven by a fractional Brownian motion

This paper deals with the existence, uniqueness and asymptotic behavior of mild solutions to neutral stochastic delay functional integro-di erential equations perturbed by a fractional Brownian motion BH, with Hurst parameter H 2 ( 1 2 ; 1). The main tools for the existence of solution is a xed point theorem and the theory of resolvent operators developed in Grimmer [R. Grimmer, Trans. Amer. Math. Soc., 273 (1982), 333{349.], while a Gronwall-type lemma plays the key role for the asymptotic behavior. An example is provided to illustrate the results of this work