Zooming in on a Lévy process at its supremum

Abstract

Let M and τ be the supremum and its time of a Lévy process X on some finite time interval. It is shown that zooming in on X at its supremum, that is, considering ((X τ +tε − M )/a ε ) t∈R as ε ↓ 0, results in (ξ t ) t∈R constructed from two independent processes having the laws of some self-similar Lévy process X conditioned to stay positive and negative. This holds when X is in the domain of attraction of X under the zoomingin procedure as opposed to the classical zooming out of Lamperti (1962). As an application of this result we establish a limit theorem for the discretization errors in simulation of supremum and its time, which extends the result of Asmussen, Glynn, and Pitman (1995) for the Brownian motion. Additionally, complete characterization of the domains of attraction when zooming in on a Lévy process at 0 is provided.2010 Mathematics Subject Classification. Primary 60G51, 60F17; secondary 60G18, 60G52.

Discretization error for a two-sided reflected Lévy process

“…Thus there are only the following possibilities for the limit process X: (i) (driftless) Brownian motion with α = 2, (ii) linear drift with α = 1 or (iii) strictly α-stable Lévy process with α ∈ (0, 2). A complete characterization of the respective domains of attraction can be found in [11]. Remark 1.…”

“…Moreover, (2.4) is satisfied by processes of bounded variation on compacts with non-zero drift component, in which case X is a linear drift process. In all the other cases, one needs to look at the behaviour of the Lévy measure Π(dx) at 0, see [11] for details, and the limit may be any of (i) -(iii) mentioned above. A sufficient condition is that the functions Π(x, ∞), Π(−∞, −x) are regularly varying at 0 with index −α for α ∈ (0, 1) ∪ (1, 2] and their ratio has a limit in [0, ∞].…”

“…on the event that the supremum is not achieved at the endpoints of the interval [0, 1]; the case of a Brownian motion X with drift was analyzed long before in [6]. The limiting random variable V is defined using the laws of X 'conditioned' to be positive and 'conditioned' to be negative and an independent uniform time shift, see [11] for the exact definition. In particular, when X is standard Brownian motion we may take V = min i∈Z B Υ+i as in [6], where (B t ) t≥0 , (B −t ) t≥0 are two independent copies of a 3-dimensional Bessel process and Υ is an independent [0, 1]-uniform random variable.…”

“…Thus one may expect that processes with a non-zero Brownian component are the worst ones to discretize in terms of the error rate, which is indeed true as explained in Section 2. This paper heavily relies on the recent results in [11] concerning discretization error when computing the supremum of a Lévy process, which in its turn employs the machinery of weak convergence of stochastic processes [18]. Here we avoid using technical concepts from [11] and mainly focus on the consequences for the intricate two-sided reflection.…”

“…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.…”