{
  "id": "math/0507193",
  "version": "v1",
  "published": "2005-07-10T03:34:00.000Z",
  "updated": "2005-07-10T03:34:00.000Z",
  "title": "Levy processes: Hitting time, overshoot and undershoot II - Asymptotic behaviour",
  "authors": [
    "Bernard Roynette",
    "Pierre Vallois",
    "Agnes Volpi"
  ],
  "comment": "Manuscript P661 submitted to SPA, October 2004. 28 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let (X_t, t>=0) be a Levy process started at 0, with Levy measure nu and T_x the first hitting time of level x>0: T_x:=inf{t>=0; X_t>x}. Let $F(theta, mu, rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F(theta,mu,rho,x) :=E(e^(-theta T_x - mu K_x \\rho L_x) 1_(T_x<+infinity)), where theta>=0, mu>=0, rho>=0, x>=0, K_x:=X_(T_x)-x and L_x:=x-X_(T_(x^-)). If we assume that nu has finite exponential moments we exhibit an asymptotic expansion for F(theta,mu,rho,x), as x -> +infinity. A limit theorem involving a normalization of the triplet (T_x,K_x,L_x) as x -> +infinity, may be deduced. At last, if nu_(|_R_+) has finite moment of fixed order, we prove that the ruin probability P(T_x<+infinity) has at most a polynomial decay.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-07-10T03:34:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "levy processes",
      "asymptotic behaviour",
      "undershoot",
      "joint laplace transform",
      "levy measure nu"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......7193R"
    }
  }
}