First Passage Time Distribution of multi-scale stationary Markovian processes

TL;DR

This paper explores how the correlation structure of stationary Markovian processes influences their First Passage Time distribution, revealing that tail behavior depends on correlation properties rather than stationary distribution, with implications for fields like finance.

Contribution

It demonstrates the dependence of FPTD tail behavior on correlation properties and provides a general relation between FPTDs of related stochastic processes.

Findings

Power-law correlated processes have power-law FPTD tails 1/t^(α+1)/2.

Infinite time-scale processes with bounded scales show exponential tail decay.

FPTD behavior is determined by the distribution of time-scale weights, not just the number of scales.

Abstract

The aim of this paper is to investigate how the correlation properties of a stationary Markovian stochastic processes affect the First Passage Time distribution. First Passage Time issues are a classical topic in stochastic processes research. They also have relevant applications, for example, in many fields of finance such as the assessment of the default risk for firms' assets. By using some explicit examples, in this paper we will show that the tail of the First Passage Time distribution crucially depends on the correlation properties of the process and it is independent from its stationary distribution. When the process includes an infinite set of time-scales bounded from above, the FPTD shows tails modulated by some exponential decay. In the case when the process is power-law correlated the FPTD shows power-law tails 1/t^(alfa+1)/2 and therefore the moments <t^n> of the FPTD are finite only when n< (alfa-1)/2. We will also show that such power-law behaviour is not merely due to the fact that the process includes an infinite and unbounded set of time-scales. Rather, the time-scale must enter the FPTD with weights that must be distributed according to a power-law for large time-scales values. Finally, we will give a general result connecting the FPTD of an additive stochastic processes x(t) to the FPTD of a generic process y(t) related by a coordinate transformation y=f(x) to the first one.