{
  "id": "2107.11896",
  "version": "v1",
  "published": "2021-07-25T21:15:54.000Z",
  "updated": "2021-07-25T21:15:54.000Z",
  "title": "Reflected backward stochastic differential equations under stopping with an arbitrary random time",
  "authors": [
    "Safa Alsheyab",
    "Tahir Choulli"
  ],
  "categories": [
    "math.PR",
    "q-fin.MF"
  ],
  "abstract": "This paper addresses reflected backward stochastic differential equations (RBSDE hereafter) that take the form of \\begin{eqnarray*} \\begin{cases} dY_t=f(t,Y_t, Z_t)d(t\\wedge\\tau)+Z_tdW_t^{\\tau}+dM_t-dK_t,\\quad Y_{\\tau}=\\xi, Y\\geq S\\quad\\mbox{on}\\quad \\Lbrack0,\\tau\\Lbrack,\\quad \\displaystyle\\int_0^{\\tau}(Y_{s-}-S_{s-})dK_s=0\\quad P\\mbox{-a.s..}\\end{cases} \\end{eqnarray*} Here $\\tau$ is an arbitrary random time that might not be a stopping time for the filtration $\\mathbb F$ generated by the Brownian motion $W$. We consider the filtration $\\mathbb G$ resulting from the progressive enlargement of $\\mathbb F$ with $\\tau$ where this becomes a stopping time, and study the RBSDE under $\\mathbb G$. Precisely, we focus on answering the following problems: a) What are the sufficient minimal conditions on the data $(f, \\xi, S, \\tau)$ that guarantee the existence of the solution of the $\\mathbb G$-RBSDE in $L^p$ ($p>1$)? b) How can we estimate the solution in norm using the triplet-data $(f, \\xi, S)$? c) Is there an RBSDE under $\\mathbb F$ that is intimately related to the current one and how their solutions are related to each other? We prove that for any random time, having a positive Az\\'ema supermartingale, there exists a positive discount factor ${\\widetilde{\\cal E}}$ that is vital in answering our questions without assuming any further assumption on $\\tau$, and determining the space for the triplet-data $(f,\\xi, S)$ and the space for the solution of the RBSDE as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-25T21:15:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "reflected backward stochastic differential equations",
      "arbitrary random time",
      "addresses reflected backward stochastic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}