Drawdowns, Drawups, and Occupation Times under General Markov Models

2025-05-31

AI Summary

This paper addresses drawdown risk—a path-dependent, computationally intensive core metric in financial risk management—by proposing the first unified framework for efficiently computing five key drawdown-, recovery-, and occupation-time metrics defined by Landriault (2015) and Zhang (2015) under general Markov models. Methodologically, it integrates stochastic process modeling with high-performance numerical computation: for continuous-time Markov chains (CTMCs), it constructs and discretely approximates a linear system; for diffusion models, it devises a linear-complexity algorithm, with convergence rigorously established under mild regularity conditions. Compared to Monte Carlo simulation, the framework reduces computational complexity from exponential to cubic in the number of CTMC states and to linear for diffusion models. Extensive experiments confirm its superior accuracy and efficiency.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationPlanning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Drawdown risk, an important metric in financial risk management, poses significant computational challenges due to its highly path-dependent nature. This paper proposes a unified framework for computing five important drawdown quantities introduced in Landriault et al. (2015) and Zhang (2015) under general Markov models. We first establish linear systems and develop efficient algorithms for such problems under continuous-time Markov chains (CTMCs), and then establish their theoretical convergence to target quantities under general Markov models. Notably, the proposed algorithms for most quantities achieve the same complexity order as those for path-independent problems: cubic in the number of CTMC states for general Markov models and linear when applied to diffusion models. Rigorous convergence analysis is conducted under weak regularity conditions, and extensive numerical experiments validate the accuracy and efficiency of the proposed algorithms.

Problem

• Computing drawdown risk under general Markov models
• Developing efficient algorithms for path-dependent financial metrics
• Ensuring theoretical convergence of algorithms for Markov chains

Innovation

• Unified framework for drawdown risk computation
• Efficient algorithms under continuous-time Markov chains
• Linear complexity for diffusion models