On the quasi-stationary distribution of the Shiryaev–Roberts diffusion

ABSTRACT

We consider the diffusion generated by the equation with fixed, and where μ≠0 is given, and is standard Brownian motion. We assume that is stopped at with A>0 preset, and obtain a closed-from formula for the quasi-stationary distribution of , i.e., the limit , x∈[0,A]. Further, we also prove Q_A(x) to be unimodal for any A>0, and obtain its entire moment series. More importantly, the pair with r≥0 and A>0 is the well-known Generalized Shiryaev–Roberts change-point detection procedure, and its characteristics for r∼Q_A(x) are of particular interest, especially when A>0 is large. In view of this circumstance, we offer an order-three large-A asymptotic approximation of Q_A(x) valid for all x∈[0,A]. The approximation is rather accurate even if A is lower than what would be considered “large” in practice.

KEYWORDS:

• Quasi-stationary distribution
• quickest change-point detection
• Shiryaev–Roberts procedure
• Whittaker functions

MATHEMATICS SUBJECT CLASSIFICATION:

• 62L10
• 60G40
• 60J60

Acknowledgment

The author is grateful to the Editor-in-Chief, Nitis Mukhopadhyay (University of Connecticut–Storrs), and to the anonymous referee for the constructive feedback provided on the first draft of the article that helped improve the quality of the article and shape its current form.