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Joint distribution of a Lévy process and its running supremum

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  26 July 2018

Laure Coutin ,
Monique Pontier  and
Waly Ngom
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.
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Abstract

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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.

Keywords

Lévy processpartial differential equationrunning supremum processfirst hitting time

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60H20: Stochastic integral equations 60H99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 488 - 512
Copyright
Copyright © Applied Probability Trust 2018 

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