Abstract
The convergence of the split-step backward Euler (SSBE) method applied to stochastic differential equation with variable delay is proven inLp-sense. Almost sure convergence is derived from theLpconvergence by Chebyshev’s inequality and the Borel-Cantelli lemma; meanwhile, the convergence rate is obtained.
Highlights
In probability theory, there are several types of convergence of sequences of random variables such as convergence in pth mean (Lp sense), almost sure, in probability, and in distribution
Under assumptions (A1)–(A3), the proof of the three other cases follows in a similar manner that of Lemma 2.5 in [8]
Under assumptions (A1)–(A3), the approximate solution y(t) converges (a.s.) to the exact solution x(t) uniformly on [0, T] as hN → 0
Summary
There are several types of convergence of sequences of random variables such as convergence in pth mean (Lp sense), almost sure, in probability, and in distribution. The mean-square convergence analysis of numerical schemes for solving stochastic delay differential equation (SDDE) has gained considerable research attention, and we refer here to the papers of Baker and Buckwar [1], Buckwar [2], Hu et al [3], Liu et al [4], and Mao and Sabanis [5] just to mention a few of them. The almost sure and Lp convergence of a numerical method for an SDDE are rarely investigated in the literature. The aim of this paper is to obtain the almost sure convergence rate together with Lp convergence rate of the SSBE method for SDDE (1).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
