{
  "id": "2003.03834",
  "version": "v1",
  "published": "2020-03-08T19:23:31.000Z",
  "updated": "2020-03-08T19:23:31.000Z",
  "title": "The shape of the value function under constrained optimal stopping",
  "authors": [
    "David Hobson"
  ],
  "comment": "16 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In a classical problem for the stopping of a diffusion process $(X_t)_{t \\geq 0}$, where the goal is to maximise the expected discounted value of a function of the stopped process ${\\mathbb E}^x[e^{-\\beta \\tau}g(X_\\tau)]$, maximisation takes place over all stopping times $\\tau$. In a constrained optimal stopping problem, stopping is restricted to event times of an independent Poisson process. In this article we consider whether the resulting value function $V_\\theta(x) = \\sup_{\\tau \\in {\\mathcal T}({\\mathbb T}^\\theta)}{\\mathbb E}^x[e^{-\\beta \\tau}g(X_\\tau)]$ (where the supremum is taken over stopping times taking values in the event times of an inhomogeneous Poisson process with rate $\\theta = (\\theta(X_t))_{t \\geq 0}$) inherits monotonicity and convexity properties from $g$. It turns out that monotonicity (respectively convexity) of $V_\\theta$ in $x$ depends on the monotonicity (respectively convexity) of the quantity $\\frac{\\theta(x) g(x)}{\\theta(x) + \\beta}$ rather than $g$. Our main technique is stochastic coupling.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-08T19:23:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G40",
      "90B50"
    ],
    "keywords": [
      "event times",
      "independent poisson process",
      "stopping times",
      "respectively convexity",
      "diffusion process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}