From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

Let be a positive integer, a positive constant and be a sequence of independent identically distributed pseudorandom variables. We assume that the ’s take their values in the discrete set and that their common pseudodistribution is characterized by the ( positive or negative ) real numbers for any . Let us finally introduce the associated pseudorandom walk defined on by and for . In this paper, we exhibit some properties of . In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for 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 . 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.