# Discretization error for a two-sided reflected Lévy process

*Søren Asmussen* () and
*Jevgenijs Ivanovs* ()

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Søren Asmussen: Ny Munkegade

Jevgenijs Ivanovs: Ny Munkegade

*Queueing Systems: Theory and Applications*, 2018, vol. 89, issue 1, No 8, 199-212

**Abstract:**
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 $$X^{(n)}$$ 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 $$X_\varepsilon /a_\varepsilon $$ X ε / a ε has a non-trivial weak limit for some scaling function $$a_\varepsilon $$ a ε as $$\varepsilon \downarrow 0$$ ε ↓ 0 , it is proved in particular that $$(Y_1-Y^{(n)}_n)/a_{1/n}$$ ( Y 1 - Y n ( n ) ) / a 1 / n converges weakly to $$\pm \, V$$ ± 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.

**Keywords:** Brownian motion; Conditioning; Refraction; Regular variation; Regulator; Scaling limits; Self-similarity; Skorokhod problem; Stable process; 60G51; 60G16; 60G52; 65C05 (search for similar items in EconPapers)

**Date:** 2018

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