{
  "id": "0912.0615",
  "version": "v3",
  "published": "2009-12-03T10:06:19.000Z",
  "updated": "2012-03-20T01:44:45.000Z",
  "title": "Predicting the supremum: optimality of \"stop at once or not at all\"",
  "authors": [
    "Pieter C. Allaart"
  ],
  "comment": "20 pages; added a few specific examples and additional references",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let X_t, 0<=t<=T be a one-dimensional stochastic process with independent and stationary increments. This paper considers the problem of stopping the process X_t \"as close as possible\" to its eventual supremum M_T:=sup{X_t: 0<=t<=T}, when the reward for stopping with a stopping time tau<=T is a nonincreasing convex function of M_T-X_tau. Under fairly general conditions on the process X_t, it is shown that the optimal stopping time tau is of \"bang-bang\" form: it is either optimal to stop at time 0 or at time T. For the case of random walk, the rule tau=T is optimal if the steps of the walk stochastically dominate their opposites, and the rule tau=0 is optimal if the reverse relationship holds. For Le'vy processes X_t with finite Le'vy measure, an analogous result is proved assuming that the jumps of X_t satisfy the above condition, and the drift of X_t has the same sign as the mean jump. Finally, conditions are given under which the result can be extended to the case of nonfinite Le'vy measure.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-03-20T01:44:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G40",
      "60G50",
      "60G51",
      "60G25"
    ],
    "keywords": [
      "optimality",
      "nonfinite levy measure",
      "one-dimensional stochastic process",
      "reverse relationship holds",
      "optimal stopping time tau"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.0615A"
    }
  }
}