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Local and global existence for mild solutions of stochastic differential equations. (English) Zbl 0931.60053
Let $dX(t)= (AX(t)+ F(t,X(t))) dt= B(t, X(t)) dW(t),\quad X(0)= \xi,\tag{1}$ be a stochastic differential equation of Itô type in the separable Hilbert space. The author considers the problem of the local and global existence and uniqueness of a mild solution of (1) under more general conditions on the coefficients than Lipschitz and linear growth. The proofs are based on some results for measures of noncompactness and condensing operators.

##### MSC:
 60H20 Stochastic integral equations
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