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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+σ(t,x t )dW(t)+ 𝒰 h(t,x t ,u)N ˜(dt,du)

in a Hilbert space H with an initial condition x(t)=ϕ(t) for t[-r,0] is proved. Here x t (s)=x(t+s), s[-r,0], A generates a holomorphic semigroup of contractions on H, W is a cylindrical Wiener process, N ˜ is a compensated Poisson martingale measure generated by a stationary Poisson point process in a σ-finite measure space (𝒰,,ν), the nonlinearities g, f, σ 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, σ and h is at most εmax{1,|ρ(ε)|} where ρ is of multiples of iterated logarithms growth near the origin.

MSC:
60H15Stochastic partial differential equations
34G20Nonlinear ODE in abstract spaces
60J65Brownian motion
60J75Jump processes
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