Discretization error for a two-sided reflected L\'evy process

TL;DR

This paper analyzes the discretization error in simulating two-sided reflected Lévy processes, revealing weak convergence of the scaled error to a limit related to the process's supremum, with theoretical and numerical insights.

Contribution

It establishes the weak convergence of the scaled discretization error for reflected Lévy processes under minimal assumptions, linking it to the supremum approximation problem.

Findings

The scaled error converges weakly to a limit V.

The limit V depends on the last barrier visited.

Numerical analysis provides insights into the distribution of V.

Abstract

An obvious way to simulate a L\'evy process $X$ is to sample its increments over time $1/n$, thus constructing an approximating random walk $X^{(n)}$. This paper considers the error of such approximation after the two-sided reflection map is applied, with focus on the value of the resultant process $Y$ and regulators $L,U$ at the lower and upper barriers at some fixed time. Under the weak assumption that $X_\varepsilon/a_\varepsilon$ has a non-trivial weak limit for some scaling function $a_\varepsilon$ as $\varepsilon\downarrow 0$, it is proved in particular that $(Y_1-Y^{(n)}_n)/a_{1/n}$ converges weakly to $\pm V$, where the sign depends on the last barrier visited. Here the limit $V$ is the same as in the problem concerning approximation of the supremum as recently described by Ivanovs (2017). Some further insight in the distribution of $V$ is provided both theoretically and numerically.