{
  "id": "0912.4768",
  "version": "v1",
  "published": "2009-12-24T02:00:32.000Z",
  "updated": "2009-12-24T02:00:32.000Z",
  "title": "A new construction of the $σ$-finite measures associated with submartingales of class $(Σ)$",
  "authors": [
    "Joseph Najnudel",
    "Ashkan Nikeghbali"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In a previous paper, we proved that for any submartingale $(X_t)_{t \\geq 0}$ of class $(\\Sigma)$, defined on a filtered probability space $(\\Omega, \\mathcal{F}, \\mathbb{P}, (\\mathcal{F}_t)_{t \\geq 0})$, which satisfies some technical conditions, one can construct a $\\sigma$-finite measure $\\mathcal{Q}$ on $(\\Omega, \\mathcal{F})$, such that for all $t \\geq 0$, and for all events $\\Lambda_t \\in \\mathcal{F}_t$: $$ \\mathcal{Q} [\\Lambda_t, g\\leq t] = \\mathbb{E}_{\\mathbb{P}} [\\mathds{1}_{\\Lambda_t} X_t]$$ where $g$ is the last hitting time of zero of the process $X$. Some particular cases of this construction are related with Brownian penalisation or mathematical finance. In this note, we give a simpler construction of $\\mathcal{Q}$, and we show that an analog of this measure can also be defined for discrete-time submartingales.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-24T02:00:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite measures",
      "discrete-time submartingales",
      "filtered probability space",
      "brownian penalisation",
      "simpler construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.4768N"
    }
  }
}