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Ioannis Karatzas
A note on Bayesian detection of change-points with an expected miss criterion
A process X is observed continuously in time; it behaves like Brownian motion with drift, which changes from zero to a known constant θ>0 at some time τ that is not directly observable. It is important to detect this change when it happens, and we attempt to do so by selecting a stopping rule T* that minimizes the “expected miss” E|T−τ| over all stopping rules T. Assuming that τ has an exponential distribution with known parameter λ>0 and is independent of the driving Brownian motion, we show that the optimal rule T* is to declare that the change has occurred, at the first time t for which
λ∫0teθ(Xt−Xs)+(λ−frac{θ2}{2})(t−s)ds≥frac{p*}{1−p*}.
Here, with Λ=2λ/θ2, the constant p* is uniquely determined in (½,1) by the equation
∫0½frac{(1−2π)e−Λ/π}{(1−π)2+Λπ2−Λ}dπ = ∫½p*frac{(2π−1)e−Λ/π}{(1−π)2+Λπ2−Λ}dπ.
Statistics & Decisions, Oldenbourg Wissenschaftsverlag
Print ISSN: 0721-2631
Volume: 21, 01/2003
Pages: 003