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

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 $\{-N,-N+1,\mathrm{.}\mathrm{.}\mathrm{.},N-1,N\}$ and that their common pseudo-distribution is characterized by the $\mathit{\text{(positive}}\mathit{\text{or}}\mathit{\text{negative)}}\mathit{\text{real}}$ numbers $$\mathbb{P}\{U_n=k\}={\delta }_{k0}+(-1{)}^{k-1}c\left(\genfrac{}{}{0px}{}{2N}{k+N}\right)$$ for any $k\in \{-N,-N+1,\mathrm{.}\mathrm{.}\mathrm{.},N-1,N\}$. 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 $\mathrm{\partial }\mathrm{/}\mathrm{\partial }t=(-1{)}^{N-1}c{\mathrm{\partial }}^{2N}\mathrm{/}$ $\mathrm{\partial }{x}^{2N}$. 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 $\frac{\mathrm{\partial }}{\mathrm{\partial }t}=\pm \frac{{\mathrm{\partial }}^N}{\mathrm{\partial }{x}^N}$. 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.