| Mean-Square Convergence of Split-Step Theta Milstein Method for Stochastic Delay Differential Equations |
| کد مقاله : 1037-FEMATH5-FULL |
| نویسندگان |
|
امید فرخنده روز *1، داوود احمدیان2 1آذربایجان شرقی، تبریز، خیابان آخونی روبروی انتقال خون پلاک 156 کد پستی 5183854664 2دانشگاه تبریز دانشکده ریاضی |
| چکیده مقاله |
| In this paper we introduce a split-step theta Milstein (SSTM) method for stochastic delay differential equations (SDDEs). The basic idea is to reformulate the original problem eliminating the dependence on the differentiation of the solution in the past values. We need to solve SSDE equation by SSTM scheme with a sufficiently small mesh size ∆t = 2^ {−15} and identify their outcomes as the “exact solution” X(t) for the error-comparison. The error-analysis below is based on a comparison to this “exact solution” as a kind of “reference solution”. To illustrate the convergence of the SSTM scheme, 5000 sample trajectories are simulated for the stepsize ∆ = 16∆t, 32∆t, 64∆t, 128∆t, 256∆t. To construct the confidence Intervals for the absolute mean square errors. Consequently, this paper concludes that the SSTM scheme converge strongly to the exact solution with the order 1. It is worth mentioning that the SSTM scheme has never been employed or analyzed for the numerical approximation of SDDEs, at least to the very best of our knowledge. |
| کلیدواژه ها |
| Split-step theta Milstein method; Mean-square convergence; Stochastic delay differential equations. |
| وضعیت: پذیرفته شده برای ارائه شفاهی |