Abstract
For a L\'evy process $X$ on a finite time interval consider the probability that it exceeds some fixed threshold $x>0$ while staying below $x$ at the points of a regular grid. We establish exact asymptotic behavior of this probability as the number of grid points tends to infinity. We assume that $X$ has a zooming-in limit, which necessarily is $1/\alpha$-self-similar L\'evy process with $\alpha\in(0,2]$, and restrict to $\alpha>1$. Moreover, the moments of the difference of the supremum and the maximum over the grid points are analyzed and their asymptotic behavior is derived. It is also shown that the zooming-in assumption implies certain regularity properties of the ladder process, and the decay rate of the left tail of the supremum distribution is determined.
Highlights
Consider a Lévy process X = (Xt, t ≥ 0) on the real line and letM = sup{Xt : t ∈ [0, 1]}, τ = inf{t ≥ 0 : Xt ∨ Xt− = M }be the supremum and its time, respectively, for the time interval [0, 1]
On the way towards this goal, we provide asymptotics of the moments E(M − M (n))p of the discretization error in the approximation of the supremum, markedly improving on the bounds in [33] and other works
The motivation comes from various applications, where it is vital to understand if the process X has exceeded a fixed threshold x > 0 or not
Summary
Be the supremum and its time, respectively, for the time interval [0, 1]. For any n ∈ N+ consider the maximum of X over the regular grid with step size 1/n:. The convergence in (1.2) readily suggests that E(M − M (n))p is of order b−n p and supplements it with exact asymptotics, but only when the underlying sequence of random variables (V (n))p is uniformly integrable Establishing the latter, is far from trivial and we could only do that thanks to [8] providing a representation of the pre- and post-supremum process using juxtaposition of the excursions in half-lines. Literature overview: The fundamental work in this area is [3], where the limit theorem for M − M (n) was established in the case of a linear Brownian motion X This sparkled research in various application areas including mathematical finance, see [15] for approximations of option prices in discrete-time models using continuoustime counterparts. There is a large body of literature concerned with the supremum of a Lévy process, see [16, 17, 38] among many others, and with the small-time behavior of Lévy processes, see [7, 22, 29, 32] and references therein
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