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The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay. (English) Zbl 1121.60064

Let \((\varOmega,\mathcal F, P)\) 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.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Mao, X. R., Stochastic Differential Equations and Applications (1997), Horwood Publication: Horwood Publication Chichester · Zbl 0892.60057
[2] Taniguchi, T., Successive approximations to solutions of stochastic differential equations, J. Differential Equations, 96, 152-169 (1992) · Zbl 0744.34052
[3] Arnold, L., Stochastic Differential Equations: Theory and Applications (1974), John Wiley and Sons: John Wiley and Sons New York · Zbl 0278.60039
[4] Gihman, I. I.; Skorohod, A. V., The Theory of Stochastic Processes (1975), Springer-Verlag: Springer-Verlag Berlin · Zbl 0305.60027
[5] Friedman, A., Stochastic Differential Equations and Their Applications (1976), Academic Press: Academic Press San Diego · Zbl 0323.60057
[6] Mohammed, S.-E. A., Stochastic Functional Differential Equations (1984), Pitman Advanced Publishing Program · Zbl 0584.60066
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