PDE for joint law of the pair of a continuous diffusion and its running maximum

Laure Coutin (1) , Monique Pontier (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Let X be a d-dimensional diffusion process and M the running supremum of the first component. In this paper, in case of dimension d, we first show that for any t > 0, the law of the pair (M t , X t) admits a density with respect to Lebesgue measure. In uni-dimensional case, we 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 M t of X at time t can be characterized as a solution of a weakly valued-measure partial differential equation.

Mots clés: Partial differential equation; running supremum process; joint law

Domaines: Mathématiques [math]; Probabilités [math.PR]

Laure Coutin, Monique Pontier. PDE for joint law of the pair of a continuous diffusion and its running maximum. 2017. ⟨hal-01591946v2⟩