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Successive approximation of neutral functional stochastic differential equations with jumps. (English) Zbl 1196.60114
Existence and uniquenss of càdlàg mild solutions of a stochastic delay equation $d[x(t)+g(t,x(t-r))]=[Ax(t)+f(t,x_t)]\,dt+\sigma(t,x_t)\,dW(t)+\int_{\mathcal U}h(t,x_t,u)\tilde N(dt,du)$ in a Hilbert space $$H$$ with an initial condition $$x(t)=\varphi(t)$$ for $$t\in[-r,0]$$ is proved. Here $$x_t(s)=x(t+s)$$, $$s\in[-r,0]$$, $$A$$ generates a holomorphic semigroup of contractions on $$H$$, $$W$$ is a cylindrical Wiener process, $$\tilde N$$ is a compensated Poisson martingale measure generated by a stationary Poisson point process in a $$\sigma$$-finite measure space $$(\mathcal U,\mathcal E,\nu)$$, the nonlinearities $$g$$, $$f$$, $$\sigma$$ and $$h$$ are defined on suitable spaces and, roughly speaking, $$g$$ is Lipschitz of at most linear growth and the modulus of continuity of $$f$$, $$\sigma$$ and $$h$$ is at most $$\varepsilon\max\{1,|\rho(\varepsilon)|\}$$ where $$\rho$$ is of multiples of iterated logarithms growth near the origin.

MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 34G20 Nonlinear differential equations in abstract spaces 60J65 Brownian motion 60J75 Jump processes (MSC2010)
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References:
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