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

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.