First-order numerical schemes for stochastic differential equations using coupling
Alnafisah, Yousef Ali
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We study a new method for the strong approximate solution of stochastic differential equations using coupling and we prove order one error bounds for the new scheme in Lp space assuming the invertibility of the diffusion matrix. We introduce and implement two couplings called the exact and approximate coupling for this scheme obtaining good agreement with the theoretical bound. Also we describe a method for non-invertibility case (Combined method) and we investigate its convergence order which will give O(h3/4 √log(h)j) under some conditions. Moreover we compare the computational results for the combined method with its theoretical error bound and we have obtained a good agreement between them. In the last part of this thesis we work out the performance of the multilevel Monte Carlo method using the new scheme with the exact coupling and we compare the results with the trivial coupling for the same scheme.