Discretization error for a two-sided reflected Lévy process

Abstract

An obvious way to simulate a Lévy 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 resulting 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 (Ann Appl Probab, 2018). Some further insight in the distribution of V is provided both theoretically and numerically.