Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions

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Abstract

The classical existence-and-uniqueness theorem of the solution to a stochastic differential delay equation (SDDE) requires the local Lipschitz condition and the linear growth condition (see e.g. [11], [12] and [20]). The numerical solutions under these conditions have also been discussed intensively (see e.g. [4], [10], [13], [16], [17], [18], [21], [22] and [24]). Recently, Mao and Rassias [14] and [15] established the generalized Khasminskii-type existence-and-uniqueness theorems for SDDEs, where the linear growth condition is no longer imposed. These generalized Khasminskii-type theorems cover a wide class of highly nonlinear SDDEs but these nonlinear SDDEs do not have explicit solutions, whence numerical solutions are required in practice. However, there is so far little numerical theory on SDDEs under these generalized Khasminskii-type conditions. The key aim of this paper is to close this gap.



Original languageEnglish
Pages (from-to)5512-5524
Number of pages13
JournalApplied Mathematics and Computation
Volume217
Issue number12
Early online date15 Dec 2010
DOIs
Publication statusPublished - 15 Feb 2011

Keywords

  • brownian motion
  • stochastic differential delay equation
  • itô’s formula
  • euler–maruyama

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