Discretization error for a two-sided reflected Lévy process

2017/08/13 by Asmussen, Søren, Ivanovs, Jevgenijs
#60G52 #65C05} #FOS: Mathematics #Primary 60G51 #Probability (math.PR) #secondary 60G16

paper · doi:10.48550/arxiv.1708.03948

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 resultant process Y and regulators L,U at the lower and upper barriers at some fixed time. Under the weak assumption that X_ε/a_ε has a non-trivial weak limit for some scaling function a_ε as ε\downarrow 0, it is proved in particular that (Y₁-Y⁽ⁿ⁾_n)/a_1/n converges weakly to ± 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.

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