Joint distribution of a Lévy process and its running supremum

Laure Coutin (1) , Waly Ngom (1) , Monique Pontier (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Let X be a jump-diusion process and X * its running supremum. In this paper, we rst show that for any t > 0, the law of the pair (X * t , X t) has a density with respect to Lebesgue measure and compute this one. This allows us to show that for any t > 0, the pair formed by the random variable X t and the running supremum X * t of X at time t can be characterized as a solution of a weakly valued-measure partial dierential equation. Then we compute the marginal density of X * t for all t > 0.

Mots clés: Lévy process; partial dierential equation; running supremum process; First hitting time

Domaines: Mathématiques [math]; Statistiques [stat]

Laure Coutin, Waly Ngom, Monique Pontier. Joint distribution of a Lévy process and its running supremum. Journal of Applied Probability, 2018, 55 (2), pp.488-512. ⟨10.1017/jpr.2018.32⟩. ⟨hal-01527939⟩