# A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes

## TL;DR

This paper develops a unified mathematical framework to analyze the joint distribution of drawdown and drawup events in time-homogeneous Markov processes, with explicit solutions for specific jump processes, aiding risk assessment.

## Contribution

It introduces a novel integral equation approach to characterize the joint law of drawdown-related quantities for general Markov processes, including explicit solutions for Lévy processes.

## Key findings

- Explicit joint law formulas for Lévy processes.
- Unified integral equation framework for Markov processes.
- Application potential in financial risk management.

## Abstract

Drawdown (resp. drawup) of a stochastic process, also referred as the reflected process at its supremum (resp. infimum), has wide applications in many areas including financial risk management, actuarial mathematics and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (resp. drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a L\'{e}vy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to study various drawdown quantities for the general class of time-homogeneous Markov processes.