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- Title
Discretization error for a two-sided reflected Lévy process.
- Authors
Asmussen, Søren; Ivanovs, Jevgenijs
- 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)<inline-graphic></inline-graphic>. 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 Xε/aε<inline-graphic></inline-graphic> has a non-trivial weak limit for some scaling function aε<inline-graphic></inline-graphic> as ε↓0<inline-graphic></inline-graphic>, it is proved in particular that (Y1-Yn(n))/a1/n<inline-graphic></inline-graphic> converges weakly to ±V<inline-graphic></inline-graphic>, 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, <xref>2018</xref>). Some further insight in the distribution of V is provided both theoretically and numerically.
- Subjects
LEVY processes; DISCRETIZATION methods; COMPUTER simulation; BROWNIAN motion; MATHEMATICAL functions; APPROXIMATION theory
- Publication
Queueing Systems, 2018, Vol 89, Issue 1/2, p199
- ISSN
0257-0130
- Publication type
Article
- DOI
10.1007/s11134-018-9576-z