First exit time from a bounded interval for pseudo-processes driven by the equation $\partial/\partial t=(−1)^{N−1}\partial^{2N}/\partial x^{2N}$

Aimé Lachal (1)

1 PSPM - Probabilités, statistique, physique mathématique

Let $N$ be a positive integer. We consider pseudo-Brownian motion $X=(X(t))_{t\ge 0}$ driven by the high-order heat-type equation $\partial/\partial t=(−1)^{N−1}\partial^{2N}/\partial x^{2N}$. Let us introduce the first exit time τab from a bounded interval $(a,b)$ by $X$ ($a,b\in\mathbb{R}$). In this paper, we provide a representation of the joint pseudo-distribution of the vector $(\tau_{ab},X(\tau_{ab}))$ by means of Vandermonde-like determinants. The method we use is based on the Feynman-Kac functional related to pseudo-Brownian motion which leads to a boundary value problem. In particular, the pseudo-distribution of the location of $X$ at time $\tau_{ab}$, namely $X(\tau_{ab})$, admits a fine expression involving famous Hermite interpolating polynomials.

Mots clés: Pseudo-Brownian motion; First exit time; Laplace transform; Hermite interpolating polynomials

Domaines: Probabilités [math.PR]

Aimé Lachal. First exit time from a bounded interval for pseudo-processes driven by the equation $\partial/\partial t=(−1)^{N−1}\partial^{2N}/\partial x^{2N}$. Stochastic Processes and their Applications, 2014, 124 (2), pp.1084-1111. ⟨10.1016/j.spa.2013.09.016⟩. ⟨hal-00943582⟩