{
  "id": "math/0601029",
  "version": "v2",
  "published": "2006-01-02T15:07:31.000Z",
  "updated": "2006-07-22T07:43:29.000Z",
  "title": "An Adaptive Euler-Maruyama Scheme For SDEs: Convergence and Stability",
  "authors": [
    "H. Lamba",
    "J. C. Mattingly",
    "A. M. Stuart"
  ],
  "comment": "Corrected version. Cleaned up a number of proofs and replaced the incorrect proof in the appendex with a corrected one",
  "categories": [
    "math.NA",
    "math.PR"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-07-22T07:43:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adaptive euler-maruyama scheme",
      "convergence",
      "primary issue",
      "exponential moment bound",
      "explicit fixed time-step approximations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1029L"
    }
  }
}