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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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