2005/09/10 by Erhan Bayraktar, H. Vincent Poor, Bayraktar, Erhan +1
Computer Science · Mathematics · #Computational Engineering #FOS: Computer and information sciences #Finance #G.3 #Information Theory (cs.IT) #and Science (cs.CE) #cs.CE #cs.IT #math.IT
paper · pdf · doi:10.48550/arxiv.cs/0509029
To appear in the SIAM Journal on Control and Optimization
arxiv created 2007/04/25 · arxiv updated 2009/12/01
A multi-source quickest detection problem is considered. Assume there are two independent Poisson processes X1 and X2 with disorder times θ1 and θ2, respectively; that is, the intensities of X1 and X2 change at random unobservable times θ1 and θ2, respectively. θ1 and θ2 are independent of each other and are exponentially distributed. Define θ\triangleq θ1 \wedge θ2=min\θ1,θ2\ . For any stopping time τ that is measurable with respect to the filtration generated by the observations define a penalty function of the form Rτ=ℙ(τ<θ)+c 𝔼[(τ-θ)+], where c>0 and (τ-θ)+ is the positive part of τ-θ. It is of interest to find a stopping time τ that minimizes the above performance index. Since both observations X1 and X2 reveal information about the disorder time θ, even this simple problem is more involved than solving the disorder problems for X1 and X2 separately. This problem is formulated in terms of a three dimensional sufficient statistic, and the corresponding optimal stopping problem is examined. A two dimensional optimal stopping problem whose optimal stopping time turns out to coincide with the optimal stopping time of the original problem for some range of parameters is also solved. The value function of this problem serves as a tight upper bound for the original problem's value function. The two solutions are characterized by iterating suitable functional operators.