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 ε /a ε has a nontrivial weak limit for some scaling function a ε as ε ↓ 0, it is proved in particular that (Y 1 − 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.2010 Mathematics Subject Classification. Primary 60G51; secondary 60G16, 60G52, 65C05.

Optimal estimation of the supremum and occupation times of a self-similar Lévy process

“…It is noted that the proof of this result is far from trivial, since it requires precise understanding of the tail function 1 − F (x, y) for large x and the rate of growth of ξ (n) t as t → ∞ (uniformly in n) among other things. The identities (7) and (8) show that the statistics T mean n and T med n are first order equivalent to the standard estimator M n , and the knowledge of the distribution of X only enters through the n −1/α -order term. This fact will prove to be important in Section 4, where the parameters of the law of X will need to be estimated.…”

“…Instead of scaling X 1 , M n , Δ i with n 1/α we perform the simulation of the process X on the interval [0, n], which is allowed by self-similarity of X. Furthermore, X is simulated on the grid with step-size 1/m for m = 300, which yields an approximation of X n further corrected by the easily computable asymptotic mean error m −1/α EV , see [7]. Next, we take (at most) 30 terms in the product defining H n based on the observations closest to the maximum, that is we replace it by H n (•; 15) defined in §4.2.…”

Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement

“…We choose N = 24 and create a sample trajectory of the unrestricted Lévy walk starting from the value R(0) = 25, c.f. (24). Since X(1) = 20.5, we get Y (1) = −4.5.…”

“…We have mentioned before that the very concept of reflection from the barrier is still under dispute in the literature on α-stable Lévy processes. Nonetheless, there is a classic proposal, introduced by Skorokhod, [21,22] which (to our surprise) seems to have never been explicitly used in the physics-oriented research, [29,30,32,40,42], and has been rather seldom mentioned in the math-oriented papers (a notable exception is the series of publications by Asmussen and collaborators, [9,[23][24][25]. see also [26][27][28]).…”

“…To implement the reflecting walk we employ the recursion formula (24), where the regulators are visually absent. However, we can reconstruct them (this is the essence of the Skorokhod map) as follows.…”