The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay.

*(English)*Zbl 1121.60064Let \((\varOmega,\mathcal F, P)\) be a complete probability space with a filtration \(\{\mathcal F_t\}_{t\geqslant t_0}\) satisfying the usual conditions. Assume that \(B(t)\) is an \(m\)-dimensional Brownian motion defined on a complete probability space. Let BC\(((-\infty,0];\mathbb R^d)\) denote the family of bounded continuous \(\mathbb R^d\)-valued functions \(\varphi\) defined on \((-\infty, 0]\) with norm \(\|\varphi\|=\sup_{-\infty<\theta\leqslant0}| \varphi(\theta)| \). Consider the following \(d\)-dimensional stochastic functional differential equations with infinite delay (ISFDEs) at phase space BC\(((-\infty, 0];\mathbb R^d)\):

\[ dX(t) = f(X_t,t)\,dt+g(X_t,t)\,dB(t),\quad t_0\leqslant t\leqslant T,\tag{1} \] where \(X_t= \{X(t +\theta): -\infty <\theta\leqslant 0\}\) can be regarded as a BC\(((-\infty, 0];\mathbb R^d)\)-valued stochastic process, where \(f: \text{BC}((-\infty, 0];\mathbb R^d)\times [t_0, T]\to\mathbb R^d\) and \(g: \text{BC}((-\infty, 0];\mathbb R^d)\times [t_0, T]\to\mathbb R^{d+m}\) be Borel measurable.

This paper is devoted to build the existence-and-uniqueness theorem of solutions to equations (1). Under the uniform Lipschitz condition, the linear growth condition is weakened to obtain the moment estimate of the solution for ISFDEs. Furthermore, the existence-and-uniqueness theorem of the solution for ISFDEs is derived, and the estimate for the error between the approximate solution and the accurate solution is given. On the other hand, under the linear growth condition, the uniform Lipschitz condition is replaced by the local Lipschitz condition, the existence-and-uniqueness theorem is also valid for ISFDEs on \([t_0,T]\). Moreover, the existence-and-uniqueness theorem still holds on intervals \([t_0,\infty)\), where \(t_0\in\mathbb R\) is an arbitrary real number.

\[ dX(t) = f(X_t,t)\,dt+g(X_t,t)\,dB(t),\quad t_0\leqslant t\leqslant T,\tag{1} \] where \(X_t= \{X(t +\theta): -\infty <\theta\leqslant 0\}\) can be regarded as a BC\(((-\infty, 0];\mathbb R^d)\)-valued stochastic process, where \(f: \text{BC}((-\infty, 0];\mathbb R^d)\times [t_0, T]\to\mathbb R^d\) and \(g: \text{BC}((-\infty, 0];\mathbb R^d)\times [t_0, T]\to\mathbb R^{d+m}\) be Borel measurable.

This paper is devoted to build the existence-and-uniqueness theorem of solutions to equations (1). Under the uniform Lipschitz condition, the linear growth condition is weakened to obtain the moment estimate of the solution for ISFDEs. Furthermore, the existence-and-uniqueness theorem of the solution for ISFDEs is derived, and the estimate for the error between the approximate solution and the accurate solution is given. On the other hand, under the linear growth condition, the uniform Lipschitz condition is replaced by the local Lipschitz condition, the existence-and-uniqueness theorem is also valid for ISFDEs on \([t_0,T]\). Moreover, the existence-and-uniqueness theorem still holds on intervals \([t_0,\infty)\), where \(t_0\in\mathbb R\) is an arbitrary real number.

Reviewer: Alexandr B. Vasil’ev (Odessa)

##### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

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\textit{F. Wei} and \textit{K. Wang}, J. Math. Anal. Appl. 331, No. 1, 516--531 (2007; Zbl 1121.60064)

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##### References:

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