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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 $$d[X(t)+g(t,X(t))]=[AX(t)+f(t,X(t))] dt+\sigma(t,X(t))dW(t), \quad X(0)=x_0\in H,\tag 1$$ with non-Lipshitz coefficients $f,\sigma$ satisfying the estimate of the form $$E\Vert f(t,X)-f(t,Y)\Vert ^p\le G(t,E\Vert X-Y\Vert ^p)$$ for $X,Y\in L^p(\Omega, H)$, $p>2$, 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 $\sigma$ is proved.

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