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Existence-uniqueness and continuation theorems for stochastic functional differential equations. (English) Zbl 1161.34055
The authors investigate existence, uniqueness and continuation of solutions for stochastic functional differential equations driven by Brownian motion in which the coefficients map $\left[0,T\right)×{L}^{2}\left({\Omega },C\right)$ to ${L}^{2}\left({\Omega },C\right)$. Here, ${\Omega }$ is the underlying probability space and $C=C\left(\left[-r,0\right],{ℝ}^{n}\right)$, where $r>0$ is the maximal delay. Under suitable conditions like adaptedness and local Lipschitz conditions, they establish local existence and uniqueness of solutions. Due to the particular set-up (in ${L}^{2}$), maximal solutions are defined on a deterministic time interval. In addition, the authors provide sufficient conditions for global existence in terms of Lyapunov functions.
##### MSC:
 34K50 Stochastic functional-differential equations 34K05 General theory of functional-differential equations 60H10 Stochastic ordinary differential equations