From pseudo-random walk to pseudo-Brownian motion: first exit time from a one-sided or a two-sided interval

Abstract

Let $N$ be a positive integer, $c$ be a positive constant and $(U_n)_{n\ge 1}$ be a sequence of independent identically distributed pseudo-random variables. We assume that the $U_n$'s take their values in the discrete set $\{-N,-N+1,...,N-1,N\}$ and that their common pseudo-distribution is characterized by the \textit{(positive or negative) real} numbers \[\mathbb{P}\{U_n=k\}=\delta_{k0}+(-1)^{k-1} c\binom{2N}{k+N}\] for any $k\in\{-N,-N+1,...,N-1,N\}$. Let us finally introduce $(S_n)_{n\ge 0}$ the associated pseudo-random walk defined on $\mathbb{Z}$ by $S_0=0$ and $S_n=\sum_{j=1}^n U_j$ for $n\ge 1$. In this paper, we exhibit some properties of $(S_n)_{n\ge 0}$. In particular, we explicitly determine the pseudo-distribution of the first overshooting time of a given threshold for $(S_n)_{n\ge 0}$ as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudo-random walk to the pseudo-Brownian motion driven by the high-order heat-type equation $\partial/\partial t=(-1)^{N-1} c\;\partial^{2N}/$ $\partial x^{2N}$. We retrieve the corresponding pseudo-distribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, A.: First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation $\frac{\partial}{\partial t}=\pm \frac{\partial^N}{\partial x^N}$. Electron. J. Probab. 12 (2007), 300--353 [MR2299920]). In the same way, we get the pseudo-distribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudo-process.