[Paper Review] Two switching multiple disorder problems for Brownian motions

Pavel V. Gapeev|arXiv (Cornell University)|Oct 31, 2010

Stochastic processes and financial applications33 references3 citations

TL;DR

This paper solves two formulations of the switching multiple disorder problem for a Brownian motion with switching drift rates, using optimal switching control and free-boundary problems. It derives closed-form solutions where optimal stopping times correspond to posterior probability processes hitting constant boundaries, with explicit expressions for risk functions and boundaries via special functions like Heun’s and Kummer’s functions.

ABSTRACT

The multiple disorder problem seeks to determine a sequence of stopping times which are as close as possible to the unknown times of disorders at which the observation process changes its probability characteristics. We derive closed form solutions in two formulations of the multiple disorder problem for an observable Brownian motion with switching constant drift rates. The method of proof is based on the reduction of the initial problems to appropriate optimal switching problems and the analysis of the associated coupled free-boundary problems. We also describe the sequential switching multiple disorder detection procedures resulting from these formulations.

Motivation & Objective

• To address the sequential detection of multiple disorder times in a Brownian motion with switching drift rates, where the true state is unobservable.
• To formulate and solve two distinct optimal switching problems based on different dynamics of the hidden Markov process governing the drift changes.
• To derive explicit solutions for optimal stopping times that minimize a combination of false alarm and delay penalties under exponential discounting.
• To characterize the optimal detection procedures as first-passage times of posterior probabilities across specific thresholds.
• To provide closed-form expressions for the minimal Bayesian risk functions and the associated optimal switching boundaries.

Proposed method

• Reduces the multiple disorder problem to an optimal switching control problem for the filtering estimate of the hidden state.
• Uses the method of solving coupled free-boundary problems to determine optimal stopping boundaries for the posterior probability process.
• Applies exponential discounting to connect the problem to single disorder detection with delay penalties, enabling analytical tractability.
• Employs martingale techniques and the Feynman-Kac formula to derive the value functions and verify optimality via verification theorems.
• Derives explicit expressions for the optimal boundaries and risk functions using special functions: Heun’s double confluent and Kummer’s confluent hypergeometric functions.
• Validates the optimality of the stopping rules using Dynkin’s formula and the Lebesgue dominated convergence theorem on approximating sequences.

Experimental results

Research questions

• RQ1 What is the optimal strategy for detecting multiple disorder times in a Brownian motion with switching drift rates when the hidden state evolves as a continuous-time Markov chain?
• RQ2 How does the optimal detection policy change when disorder occurrences are constrained to happen only after previous alarms are triggered?
• RQ3 Can closed-form solutions be derived for the optimal switching boundaries and risk functions in such multiple disorder problems?
• RQ4 What is the structure of the optimal sequential detection procedure based on posterior probabilities of the hidden state?
• RQ5 How do the solutions relate to known results in single disorder detection and optimal stopping under exponential discounting?

Key findings

• The optimal stopping times are the first hitting times of the posterior probability process to constant boundaries, specifically $ p_* $ and $ q_* $, or $ g_* $ and $ h_* $, depending on the formulation.
• The minimal Bayesian risk functions are given by $ V^*( heta) = \min\{V^*_0(\theta), V^*_1(\theta)\} $ and $ U^*(\theta) = \min\{U^*_0(\theta), U^*_1(\theta)\} $, with explicit expressions derived.
• The optimal boundaries $ p_* $, $ q_* $, $ g_* $, and $ h_* $ are solutions to a system of coupled free-boundary problems, which are solved using Heun’s and Kummer’s special functions.
• The value functions satisfy verification conditions via martingale representations, and the optimality is proven using Fatou’s lemma and the dominated convergence theorem.
• The sequential detection procedure is fully characterized: upon exiting a threshold region, the system switches mode and restarts the detection process from the new state.
• The results generalize Shiryaev’s single disorder problem to multiple, switching disorders, with explicit solutions under exponential discounting and Markovian dynamics.