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

Mahmudov, N. I.

Stochastic Anal. Appl. 24, No. 1, 79-95 (2006).

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 \[ E\| f(t,X)-f(t,Y)\| ^p\leq G(t,E\| X-Y\| ^p) \] 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.

Reviewer: Yana Belopolskaya (St. Petersburg)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
34F05	Ordinary differential equations and systems with randomness
34G20	Nonlinear differential equations in abstract spaces
35R60	PDEs with randomness, stochastic partial differential equations

Keywords:

neutral stochastic differential equations; non-Lipschitz condition

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