An Adaptive Euler-Maruyama Scheme For SDEs: Convergence and Stability

Details An Adaptive Euler-Maruyama Scheme For SDEs: Convergence and Stability An Adaptive Euler-Maruyama Scheme For SDEs: Convergence and Stability

2006-01-02

arxiv.org/abs/math/0601029v2

Mathematics

Numerical Analysis

Corrected version. Cleaned up a number of proofs and replaced the incorrect proof in the appendex with a corrected one

Scientific paper

The understanding of adaptive algorithms for SDEs is an open area where many issues related to both convergence and stability (long time behaviour) of algorithms are unresolved. This paper considers a very simple adaptive algorithm, based on controlling only the drift component of a time-step. Both convergence and stability are studied. The primary issue in the convergence analysis is that the adaptive method does not necessarily drive the time-steps to zero with the user-input tolerance. This possibility must be quantified and shown to have low probability. The primary issue in the stability analysis is ergodicity. It is assumed that the noise is non-degenerate, so that the diffusion process is elliptic, and the drift is assumed to satisfy a coercivity condition. The SDE is then geometrically ergodic (converges to statistical equilibrium exponentially quickly). If the drift is not linearly bounded then explicit fixed time-step approximations, such as the Euler-Maruyama scheme, may fail to be ergodic. In this work, it is shown that the simple adaptive time-stepping strategy cures this problem. In addition to proving ergodicity, an exponential moment bound is also proved, generalizing a result known to hold for the SDE itself.

Affiliated with

Also associated with

No associations

Rating

An Adaptive Euler-Maruyama Scheme For SDEs: Convergence and Stability 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 An Adaptive Euler-Maruyama Scheme For SDEs: Convergence and Stability, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and An Adaptive Euler-Maruyama Scheme For SDEs: Convergence and Stability will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-41582