Drawdowns of diffusions

TL;DR

This paper provides excursion theoretical proofs for formulas related to the joint distribution of drawdown times and maxima in diffusions, extending previous results and analyzing jump processes for varying drawdown sizes.

Contribution

It introduces a unified excursion theoretical approach to prove and extend formulas for drawdowns and maxima in diffusion processes, including cases with lower bounds.

Findings

Proofs of Lehoczky's and Malyutin's formulas using excursion theory

Extension of formulas to include lower bounds on diffusions

Analysis of jump processes for variable drawdown sizes

Abstract

In this paper we give excursion theoretical proofs of Lehoczky's formula (in an extended form allowing a lower bound for the underlying diffusion) for the joint distribution of the first drawdown time and the maximum before this time, and of Malyutin's formula for the joint distribution of the first hitting time and the maximum drawdown before this time. It is remarkable -- but there is a clean explanation -- that the excursion theoretical approach which we developed first for Lehoczky's formula provides also a proof for Malyutin's formula. Moreover, we discuss some generalizations and analyze the pure jump process describing the maximum before the first drawdown time when the size of the drawdown is varying