The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps. (English) Zbl 1167.60336

Luo, Jiaowan; Taniguchi, Takeshi

Stoch. Dyn. 9, No. 1, 135-152 (2009).

Summary: In this paper, we consider the existence and uniqueness of mild solutions to non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jump processes:
\[ \begin{cases} d[X(t)+f(t, X_t)]=[AX(t)+g(t,X_t)]\,dt +\int_U k(t,X(t-),y) q(dydt),\quad & t\geq 0,\\ X(s)=\varphi(s), & s (= [-r, 0],\;r>0\end{cases} \]
with an initial function \(X(s) = \varphi(s)\), \(-r\leq s\leq 0\), where \(\varphi : [-r,0]\to H\) is a cadlag function with \(E[\sup_{r\leq \leq 0}|\varphi(s)|^2_H] <\infty\).

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60H20	Stochastic integral equations

Keywords:

non-Lipschitz condition; Poisson jump processes; stochastic neutral delay evolution equations

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