Optimal pointwise approximation of SDEs based on brownian motion at discrete points

By Thomas Muller-Gronbach. 2004

58 citations recorded by the source corpus.

We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a measurable way) on a finite number of sequential observations of the driving Brownian motion. The resulting lower error bounds hold in particular for all methods that are implementable on a computer and use a random number generator to simulate the driving Brownian motion at finitely many points. Our analysis shows that approximation at a single point is strongly connected to an integration problem for the driving Brownian motion with a random weight. Exploiting general ideas from estimation of weighted integrals of stochastic processes, we introduce an adaptive scheme, which is easy to implement and performs asymptotically optimally.

Subfields: Finance, Management Science and Operations Research, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/2f60fd70ad08b3a5ebcaa154b56a44f3ec2b822e

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Citations in the corpus

On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient (2018, background): “ation of X1 the question is open up to now. For this problem it is so far only known that under the assumptions (A1) and (A2) it is impossible to obtain an Lp-error rate better than 1 in general, see [8, 21]. Whether or not there exists an algorithm that approximates X1 under the assumptions (A1) an”
Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift (2016, unknown): “the accuracy of numerical approximations, see [4, 11, 15, 16, 17, 18, 24, 28] and the references therein.”