Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise

In this paper, successive approximation method is applied to investigate the existence and uniqueness of solutions to stochastic differential equations (SDEs) driven by Levy noise under non-Lipschitz condition which is a much weaker condition than Lipschitz one. The stability of solutions to non-Lipschitz SDEs driven by Levy noise is also considered, and the stochastic stability is obtained in the sense of mean square.

[1]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[2]  Peter E. Kloeden,et al.  Synchronization of systems of Marcus canonical equations driven by α-stable noises☆ , 2010 .

[3]  David Applebaum,et al.  STOCHASTIC STABILIZATION OF DYNAMICAL SYSTEMS USING LÉVY NOISE , 2010 .

[4]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[5]  Marc Yor,et al.  Time Changes for Lévy Processes , 2001 .

[6]  E. Seneta,et al.  The Variance Gamma (V.G.) Model for Share Market Returns , 1990 .

[7]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[8]  T. Taniguchi.  Successive Approximations to Solutions of Stochastic Differential Equations , 1992 .

[9]  Jerzy Zabczyk,et al.  Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach , 2007 .

[10]  Tao Cheng,et al.  Successive approximation to solutions of stochastic differential equations with jumps in local non-Lipschitz conditions , 2013, Appl. Math. Comput..

[11]  Carlo Bianca,et al.  Onset of nonlinearity in thermostatted active particles models for complex systems , 2012 .

[12]  B. Xie.  Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces , 2008 .

[13]  J. Bouchaud,et al.  Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications , 1990 .

[14]  Yong Ren,et al.  Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay , 2009, Appl. Math. Comput..

[15]  I. Bihari.  A generalization of a lemma of bellman and its application to uniqueness problems of differential equations , 1956 .

[16]  L. Arnold.  Stochastic Differential Equations: Theory and Applications , 1992 .

[17]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[18]  G. Yin,et al.  Stability of nonlinear regime-switching jump diffusion , 2012, 1401.4471.

[19]  Jinqiao Duan,et al.  An averaging principle for stochastic dynamical systems with Lévy noise , 2011 .

[20]  George Yin,et al.  Two-time-scale Jump-Diffusion Models with Markovian Switching Regimes , 2004 .

[21]  Toshio Yamada,et al.  On the successive approximation of solutions of stochastic differential equations , 1981 .

[22]  A. Friedman.  Stochastic Differential Equations and Applications , 1975 .

[23]  Huiqing Zhang,et al.  Lévy noise induced switch in the gene transcriptional regulatory system. , 2012, Chaos.

[24]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[25]  Fumiko Yonezawa,et al.  Introduction to focused session on ‘anomalous relaxation’ , 1996 .

[26]  Xuerong Mao,et al.  On the approximations of solutions to neutral SDEs with Markovian switching and jumps under non-Lipschitz conditions , 2014, Appl. Math. Comput..

[27]  G. Zaslavsky,et al.  Lévy Flights and Related Topics in Physics , 2013 .

[28]  Huiqing Zhang,et al.  Stochastic stability for nonlinear systems driven by Lévy noise , 2012 .

[29]  Sergio Albeverio,et al.  Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients , 2010 .

[30]  T. Chan.  Pricing contingent claims on stocks driven by Lévy processes , 1999 .