Pathwise convergence rate for numerical solutions of stochastic differential equations

Devoted to numerical solutions of stochastic differential equations (SDEs), this work constructs a sequences of re-embedded numerical solutions having the same distribution as those of the original SDE in a new probability space. It is shown that the re-embedded numerical solutions converge strongly to the solution of the SDE. Moreover, the rate of convergence is ascertained. The main theorem is obtained by deriving a number of technical lemmas, that are interesting in their own right. Different from the well-known results in numerical solutions of SDEs, in lieu of the usual Brownian motion increments in the algorithm, an easily implementable sequence of independent and identically distributed (i.i.d.) random variables is used. Being easier to implement compared to the construction of Brownian increments, such an i.i.d. sequence is preferable in the actual computation. As far as the convergence and uniform mean square error estimates are concerned, the use of the i.i.d. sequence does not introduce essential difficulties compared with that of the Brownian increments. Nevertheless, the analysis becomes much more difficult in the study of rates of convergence because one has to deal with the difference of the Brownian increments and the i.i.d. sequence in the almost sure sense. This paper presents a new approach to establishing rates of convergence.