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Existence and uniqueness results for neutral SDEs in Hilbert spaces. (English) Zbl 1110.60063

Let A(t) be a generator of a strongly continuous semigroup of bounded linear operators in a Hilbert space H. The author considers a stochastic differential equation (SDE) of the form

d[X(t)+g(t,X(t))]=[AX(t)+f(t,X(t))]dt+σ(t,X(t))dW(t),X(0)=x 0 H,(1)

with non-Lipshitz coefficients f,σ satisfying the estimate of the form

Ef(t,X)-f(t,Y) p G(t,EX-Y p )

for X,YL p (Ω,H), p>2, and a scalar function G possessing some additional properties. By a Picard type approximation the existence and uniqueness of a mild solution to (1) under some additional conditions on g,A,f and σ is proved.

MSC:
60H15Stochastic partial differential equations
34F05ODE with randomness
34G20Nonlinear ODE in abstract spaces
35R60PDEs with randomness, stochastic PDE