From Pseudorandom 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 a positive constant and (ξn)n≥1 be a sequence of independent identically distributed pseudorandom variables. We assume that the ξn’s take their values in the discrete set {-N,-N+1,…,N-1,N} and that their common pseudodistribution is characterized by the (positive or negative) real numbers ℙ{ξn=k}=δk0+(-1)k-1c(2Nk+N) for any k∈{-N,-N+1,…,N-1,N}. Let us finally introduce (Sn)n≥0 the associated pseudorandom walk defined on ℤ by S0=0 and Sn=∑j=1n‍ξj for n≥1. In this paper, we exhibit some properties of (Sn)n≥0. In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for (Sn)n≥0 as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation ∂/∂t=(-1)N-1c∂2N/∂x2N. We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess.