Phasetype distributions, autoregressive processes and overshoot

TL;DR

This paper derives the joint distribution of overshoot and threshold-time for autoregressive processes with phasetype distributed innovations, enabling explicit solutions for optimal stopping problems.

Contribution

It provides a novel analytical framework for autoregressive processes with phasetype innovations, specifically addressing overshoot and threshold-time distributions.

Findings

Joint distribution formulas for overshoot and threshold-time

Explicit solutions for optimal stopping problems

Application of continuous fit principle

Abstract

Autoregressive processes are intensively studied in statistics and other fields of applied stochastics. For many applications the overshoot and the threshold-time are of special interest. When the upward innovations are in the class of phasetype distributions we determine the joint distribution of this two quantities and apply this result to problems of optimal stopping. Using a principle of continuous fit this leads to explicit solutions.