The Euler scheme for Levy driven stochastic differential equations: limit theorems

Details The Euler scheme for Levy driven stochastic differential equations: limit theorems The Euler scheme for Levy driven stochastic differential equations: limit theorems

2004-10-05

arxiv.org/abs/math/0410118v1

Annals of Probability 2004, Vol. 32, No. 3A, 1830-1872

Mathematics

Probability

Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imsta

Scientific paper

10.1214/009117904000000667

We study the Euler scheme for a stochastic differential equation driven by a Levy process Y. More precisely, we look at the asymptotic behavior of the normalized error process u_n(X^n-X), where X is the true solution and X^n is its Euler approximation with stepsize 1/n, and u_n is an appropriate rate going to infinity: if the normalized error processes converge, or are at least tight, we say that the sequence (u_n) is a rate, which, in addition, is sharp when the limiting process (or processes) is not trivial. We suppose that Y has no Gaussian part (otherwise a rate is known to be u_n=\sqrt n). Then rates are given in terms of the concentration of the Levy measure of Y around 0 and, further, we prove the convergence of the sequence u_n(X^n-X) to a nontrivial limit under some further assumptions, which cover all stable processes and a lot of other Levy processes whose Levy measure behave like a stable Levy measure near the origin. For example, when Y is a symmetric stable process with index \alpha \in(0,2), a sharp rate is u_n=(n/\log n)^{1/\alpha}; when Y is stable but not symmetric, the rate is again u_n=(n/\log n)^{1/\alpha} when \alpha >1, but it becomes u_n=n/(\log n)^2 if \alpha =1 and u_n=n if \alpha <1.

Affiliated with

Also associated with

No associations

Rating

The Euler scheme for Levy driven stochastic differential equations: limit theorems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The Euler scheme for Levy driven stochastic differential equations: limit theorems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Euler scheme for Levy driven stochastic differential equations: limit theorems will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-103349