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Joint distribution of a Lévy process and its running supremum

Published online by Cambridge University Press:  26 July 2018

Laure Coutin*
Affiliation:
Université Paul Sabatier
Monique Pontier*
Affiliation:
Université Paul Sabatier
Waly Ngom*
Affiliation:
Université Paul Sabatier
*
* Postal address: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse cedex, France.
* Postal address: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse cedex, France.
* Postal address: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse cedex, France.

Abstract

Let X be a jump-diffusion process and X* its running supremum. In this paper we first show that for any t > 0, the law of the pair (X*t, Xt) has a density with respect to the Lebesgue measure. This allows us to show that for any t > 0, the law of the pair formed by the random variable Xt and the running supremum X*t of X at time t can be characterized as a weak solution of a partial differential equation concerning the distribution of the pair (X*t, Xt). Then we obtain an expression of the marginal density of X*t for all t > 0.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2018 

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