{
  "id": "math/0703805",
  "version": "v1",
  "published": "2007-03-27T13:54:25.000Z",
  "updated": "2007-03-27T13:54:25.000Z",
  "title": "The trap of complacency in predicting the maximum",
  "authors": [
    "J. du Toit",
    "G. Peskir"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117906000000638 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2007, Vol. 35, No. 1, 340-365",
  "doi": "10.1214/009117906000000638",
  "categories": [
    "math.PR"
  ],
  "abstract": "Given a standard Brownian motion $B^{\\mu}=(B_t^{\\mu})_{0\\le t\\le T}$ with drift $\\mu \\in \\mathbb{R}$ and letting $S_t^{\\mu}=\\max_{0\\le s\\le t}B_s^{\\mu}$ for $0\\le t\\le T$, we consider the optimal prediction problem: \\[V=\\inf_{0\\le \\tau \\le T}\\mathsf{E}(B_{\\tau}^{\\mu}-S_T^{\\mu})^2\\] where the infimum is taken over all stopping times $\\tau$ of $B^{\\mu}$. Reducing the optimal prediction problem to a parabolic free-boundary problem we show that the following stopping time is optimal: \\[\\tau_*=\\inf \\{t_*\\le t\\le T\\mid b_1(t)\\le S_t^{\\mu}-B_t^{\\mu}\\le b_2(t)\\}\\] where $t_*\\in [0,T)$ and the functions $t\\mapsto b_1(t)$ and $t\\mapsto b_2(t)$ are continuous on $[t_*,T]$ with $b_1(T)=0$ and $b_2(T)=1/2\\mu$. If $\\mu>0$, then $b_1$ is decreasing and $b_2$ is increasing on $[t_*,T]$ with $b_1(t_*)=b_2(t_*)$ when $t_*\\ne 0$. Using local time-space calculus we derive a coupled system of nonlinear Volterra integral equations of the second kind and show that the pair of optimal boundaries $b_1$ and $b_2$ can be characterized as the unique solution to this system. This also leads to an explicit formula for $V$ in terms of $b_1$ and $b_2$. If $\\mu \\le 0$, then $t_*=0$ and $b_2\\equiv +\\infty$ so that $\\tau_*$ is expressed in terms of $b_1$ only. In this case $b_1$ is decreasing on $[z_*,T]$ and increasing on $[0,z_*)$ for some $z_*\\in [0,T)$ with $z_*=0$ if $\\mu=0$, and the system of two Volterra equations reduces to one Volterra equation. If $\\mu=0$, then there is a closed form expression for $b_1$. This problem was solved in [Theory Probab. Appl. 45 (2001) 125--136] using the method of time change (i.e., change of variables). The method of time change cannot be extended to the case when $\\mu \\ne 0$ and the present paper settles the remaining cases using a different approach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-27T13:54:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G40",
      "35R35",
      "62M20",
      "60J65",
      "45G15",
      "60J60"
    ],
    "keywords": [
      "optimal prediction problem",
      "time change",
      "nonlinear volterra integral equations",
      "complacency",
      "standard brownian motion"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3805D"
    }
  }
}