{
  "id": "0905.3701",
  "version": "v3",
  "published": "2009-05-22T15:07:22.000Z",
  "updated": "2010-10-11T07:55:08.000Z",
  "title": "On the Martingale Property of Certain Local Martingales",
  "authors": [
    "Aleksandar Mijatovic",
    "Mikhail Urusov"
  ],
  "comment": "Appendix on local time of diffusions added; 27 pages, 1 figure; to appear in PTRF",
  "categories": [
    "math.PR",
    "q-fin.GN"
  ],
  "abstract": "The stochastic exponential $Z_t=\\exp\\{M_t-M_0-(1/2) <M,M>_t\\}$ of a continuous local martingale $M$ is itself a continuous local martingale. We give a necessary and sufficient condition for the process $Z$ to be a true martingale in the case where $M_t=\\int_0^t b(Y_u)\\,dW_u$ and $Y$ is a one-dimensional diffusion driven by a Brownian motion $W$. Furthermore, we provide a necessary and sufficient condition for $Z$ to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function $b$ and the drift and diffusion coefficients of $Y$. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-10-11T07:55:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G44",
      "60G48",
      "60H10"
    ],
    "keywords": [
      "martingale property",
      "continuous local martingale",
      "sufficient condition",
      "one-dimensional diffusion driven",
      "brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.3701M"
    }
  }
}