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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

$d\left[x\left(t\right)+g\left(t,x\left(t-r\right)\right)\right]=\left[Ax\left(t\right)+f\left(t,{x}_{t}\right)\right]\phantom{\rule{0.166667em}{0ex}}dt+\sigma \left(t,{x}_{t}\right)\phantom{\rule{0.166667em}{0ex}}dW\left(t\right)+{\int }_{𝒰}h\left(t,{x}_{t},u\right)\stackrel{˜}{N}\left(dt,du\right)$

in a Hilbert space $H$ with an initial condition $x\left(t\right)=\varphi \left(t\right)$ for $t\in \left[-r,0\right]$ is proved. Here ${x}_{t}\left(s\right)=x\left(t+s\right)$, $s\in \left[-r,0\right]$, $A$ generates a holomorphic semigroup of contractions on $H$, $W$ is a cylindrical Wiener process, $\stackrel{˜}{N}$ is a compensated Poisson martingale measure generated by a stationary Poisson point process in a $\sigma$-finite measure space $\left(𝒰,ℰ,\nu \right)$, 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 $\epsilon max\left\{1,|\rho \left(\epsilon \right)|\right\}$ where $\rho$ is of multiples of iterated logarithms growth near the origin.

##### MSC:
 60H15 Stochastic partial differential equations 34G20 Nonlinear ODE in abstract spaces 60J65 Brownian motion 60J75 Jump processes
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