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seminars:stat:160505

Statistics Seminar
Department of Mathematical Sciences

 DATE: Thursday, May 05, 2016 1:15pm to 2:15pm WH 100E Aleksey Polunchenko, Binghamton University On a Diffusion Process that Arises in Quickest Change-Point Detection

Abstract

We consider the diffusion $(R_t)_{t\ge0}$ generated by the stochastic differential equation $dR_t=dt+\mu R_t dB_t$ with $R_0=0$, where $\mu\neq0$ is given and $(B_t)_{t\ge0}$ is standard Brownian motion. We obtain a closed-from expression for the quasi-stationary distribution of $(R_t)_{t\ge0}$, i.e., the limit $Q_A(x)=\lim_{t\to+\infty}\Pr(R_t\le x|T_A>t)$, $x\in[0,A]$, where $T_A=\inf\{t>0:R_t=A\}$ with $A>0$ fixed. The process $(R_t)_{t\ge0}$, its quasi-stationary distribution $Q_A(x)$, $x\in[0,A]$, and the stopping time $T_A$ are of importance in the theory of quickest change-point detection, especially the case when $A$ is large. We study the asymptotic behavior of $Q_A(x)$ for large $A$'s, and provide an order-three asymptotic approximation.