Theta schemes for SDDEs with non-globally Lipschitz continuous coefficients

• Xiaofeng Zong ^[1] ; Fuke Wu ^[2] ; Chengming Huang ^[2]
  1. [1] Chinese Academy of Sciences

     Chinese Academy of Sciences

     China

     
  2. [2] Huazhong University of Science and Technology

     Huazhong University of Science and Technology

     China

     
• Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 278, Nº 1 (15 April 2015), 2015, págs. 258-277
• Idioma: inglés
• DOI: 10.1016/j.cam.2014.10.014
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• Resumen

  • This paper establishes the boundedness, convergence and stability of the two classes of theta schemes, namely split-step theta (SST) scheme and stochastic linear theta (SLT) scheme, for stochastic differential delay equations (SDDEs) with non-globally Lipschitz continuous coefficients. When the drift f(x,y)f(x,y) satisfies one-sided Lipschitz condition with respect to the present state xx and the diffusion g(x,y)g(x,y) obeys the global Lipschitz condition with respect to the present term xx, but the delay terms yy in the drift and diffusion may be highly nonlinear, this paper first examines the strong convergence rates of the theta schemes for SDDEs. It is also proved that the two classes of theta schemes for θ∈(1/2,1]θ∈(1/2,1] converge strongly to the exact solution with the order 1/21/2 but for θ∈[0,1/2]θ∈[0,1/2] the linear growth condition on drift f(x,y)f(x,y) in xx is needed for the strong convergence rates. The exponential mean square stability of the theta schemes with θ∈(1/2,1]θ∈(1/2,1] is also investigated for highly nonlinear SDDEs.