Abstract
The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space ℂ. Firstly, we establish a global existence-uniqueness lemma for the SFDEs under the global Lipschitz condition in ℂ without the linear growth condition. Then, under the local Lipschitz condition in ℂ, we show that the non-continuable solution of SFDEs still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the state variable equals zero. And the solution of the considered equation must either explode at the end of the maximum existing interval or exist globally. Furthermore, some more general sufficient conditions for the global existence-uniqueness are obtained. Our conditions obtained in this paper are much weaker than some existing results. For example, we need neither the linear growth condition nor the continuous condition on the time t. Two examples are provided to show the effectiveness of the theoretical results.
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References
Doob J L. Stochastic Process. New York: Wiley, 1967
Dynkin E B. Markov Processes. Moscow: Fizmatgiz, 1963
Friedman A. Stochastic Differential Equations and Applications. New York: Academic Press, 1975
Gikhman I I, Skorokhod A V. Stochastic Differential Equations. New York: Springer-Verlag, 1973
Halmos P. Measure Theory. Princeton: Van Nostrand, 1950
Itô K. On stochastic differential equations. Memoirs Amer Math Soc, 1951, 4: 1–51
Khasminskii R Z. Stochastic Stability of Differential Equations. Maryland: Sijthoff and Noordhoff, 1980
Li X Y, Mao X R. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete Contin Dyn Syst, 2009, 24: 523–593
Ning H W, Liu B. Local existence-uniqueness and continuation of solutions for delay stochastic evolution equations. Appl Anal, 2009, 88: 563–577
Mao X R. Stochastic Differential Equations and Applications, 2nd ed. Chichester: Horwood Publishing, 2007
Mohammed S-E A. Stochastic Functional Differential Equations. Boston: Pitman, 1984
øksendal B. Stochastic Differential Equations: An Introduction with Applications. New York: Springer-Verlag, 1995
Pardoux E, Peng S. Adapted solution of a backward stochastic differential equation. Syst Control Lett, 1990, 14: 55–61
Shen Y, Luo Q, Mao X R. The improved LaSalle-type theorems for stochastic functional differential equations. J Math Anal Appl, 2006, 318: 134–154
Taniguchi T. On sufficient conditions for nonexplosion of solutions to stochastic differential equations. J Math Anal Appl, 1990, 153: 549–561
Taniguchi T. Successive approxmations to solutions of stochastic differential equations. J Differential Equations, 1992, 96: 152–169
Taniguchi T, Liu K, Truman A. Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces. J Differential Equations, 2002, 181: 72–91
Wei F Y, Wang K. The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay. J Math Anal Appl, 2007, 331: 516–531
Xu D Y, Yang Z G, Huang Y M. Existence-uniqueness and continuation theorems for stochastic functional differential equations. J Differential Equations, 2008, 45: 1681–1703
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Xu, D., Wang, X. & Yang, Z. Further results on existence-uniqueness for stochastic functional differential equations. Sci. China Math. 56, 1169–1180 (2013). https://doi.org/10.1007/s11425-012-4553-1
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DOI: https://doi.org/10.1007/s11425-012-4553-1