[Paper Review] Asymptotic behaviour of first passage time distributions for Lévy processes

R. A. Doney, Víctor Rivero|arXiv (Cornell University)|Jul 22, 2011

Stochastic processes and statistical mechanics15 references4 citations

TL;DR

This paper establishes sharp asymptotic estimates for the first passage time distribution of Lévy processes in the domain of attraction of a stable law, distinguishing between continuous and discontinuous passage types. It derives local limit theorems for the density of first passage times under $$\mathbb{P}_x$$, showing that the asymptotic behavior depends on the stability index $\alpha$, positivity parameter $\rho$, and the presence of a positive drift in the ladder height process, with convergence rates tied to the scaling function $c(t)$ and the density of the limiting stable law at zero.

ABSTRACT

Let $X$ be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function $c.$ As an analogue of the random walk results in \cite{vw} and \cite{rad} we study the local behaviour of the distribution of the lifetime $ζ$ under the characteristic measure $\underline{n}$ of excursions away from 0 of the process $X$ reflected in its past infimum, and of the first passage time of $X$ below $0,$ $T_{0}=\inf \{t>0:X_{t}<0\},$ under $\mathbb{P}_{x}(\cdot),$ for $x>0,$ in two different regimes for $x,$ viz. $x=o(c(\cdot))$ and $x>D c(\cdot),$ for some $D>0.$ We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at $T_{0}$ and discontinuous passage. In the way to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to $X$ reflected in its past infimum.

Motivation & Objective

• To analyze the local asymptotic behavior of the first passage time distribution below zero for real-valued Lévy processes in the domain of attraction of a stable law.
• To distinguish between continuous and discontinuous passage at the first passage time, as their path behaviors lead to different distributional properties.
• To establish sharp local limit theorems for the density of the first passage time under $\mathbb{P}_x$, particularly in the regimes $x = o(c(t))$ and $x > Dc(t)$ for $D > 0$, and for the lifetime of excursions under the characteristic measure $\underline{n}$.
• To derive precise asymptotic expressions for the density of the first passage time, incorporating the density of the limiting stable law at zero and the scaling function $c(t)$.

Proposed method

• The authors use fluctuation theory and the theory of ladder processes, focusing on the downgoing ladder height process $H^*$ and its associated scale function $U^*$.
• They decompose the first passage time distribution into contributions from continuous passage ($C_0$) and discontinuous passage, using the event $C_0 = \{X(T_0-) = 0\}$ to separate path types.
• The analysis relies on sharp local estimates for the entrance law of the excursion process of the process reflected in its past infimum, using the scaling function $c(t)$ and the regularly varying behavior of $\underline{n}(\zeta > t)$.
• Key estimates involve the use of the density $g^*$ of the ladder height distribution and the function $\theta(s,y)$, which captures the local behavior of the excursion measure.
• The authors apply Tauberian and Abelian theorems to relate the asymptotic behavior of $\underline{n}(\zeta > t)$, which is regularly varying with index $-\overline{\rho}$, to the first passage time density.
• They use the functional limit theorem for the bivariate ladder process to derive the asymptotic behavior of the first passage time density, particularly in the case $\alpha\overline{\rho} < 1$ and $\alpha\overline{\rho} = 1$.

Experimental results

Research questions

• RQ1 How does the first passage time distribution of a Lévy process in the domain of attraction of a stable law behave asymptotically as time $t \to \infty$, particularly for $x = o(c(t))$ and $x > Dc(t)$?
• RQ2 What is the role of the path type—continuous versus discontinuous passage—at the first passage time in determining the existence and form of the density of $T_0$?
• RQ3 How do the asymptotics of the first passage time density depend on the density of the limiting stable law at zero, $f(0)$, and the scaling function $c(t)$?
• RQ4 What is the precise asymptotic behavior of the density of the first passage time when the ladder height process has a positive drift ($d^* > 0$), and how does this differ from the case $d^* = 0$?
• RQ5 In the case $\alpha\overline{\rho} = 1$, what is the exact asymptotic form of the first passage time density, and how does it relate to the constant $k_6 k_7$?

Key findings

• For $X \in D(\alpha, \rho)$ with $\alpha\overline{\rho} < 1$, the first passage time density satisfies $t \mathbb{P}_x(T \in (t, t+\Delta]) \sim \Delta k_6 \tilde{h}_{x_t}(1)$, where $\tilde{h}_{x_t}(1)$ is the density of the limiting stable law at time 1.
• When $\alpha\overline{\rho} = 1$, the asymptotic behavior is $t \mathbb{P}_x(T \in (t, t+\Delta]) \sim \Delta k_6 k_7 \tilde{h}_{x_t}(1)$, with $k_6 k_7 = p = d^*/(d^* + L(\infty))$ if $d^* > 0$, and $k_6 k_7 = 1$ if $d^* = 0$.
• The density of the first passage time exists and is asymptotically equivalent to $k_6 \tilde{h}_{x_t}(1)$ for discontinuous passage, while for continuous passage it may be singular when $d^* > 0$, necessitating separate treatment.
• The asymptotic behavior of the excursion lifetime $\zeta$ under $\underline{n}$ satisfies $\underline{n}(\zeta > t) \in RV(-\overline{\rho})$, and the local behavior is derived via sharp estimates on the entrance law of the reflected process.
• The constant $k_6$ is shown to satisfy $k_6 \sim C \cdot \frac{U^*(c(t))}{c(t) \underline{n}(\zeta > t)}$, with $U^*$ the scale function of the ladder height process.
• The paper confirms that $th_x(t) \to p \tilde{h}_{x_t}(1)$ as $t \to \infty$, uniformly for $x_t \in [D^{-1}, D]$, when $\Pi((-∞, 0)) > 0$, establishing a functional limit theorem for the first passage time density.