2014/08/26 by Goran Peskir
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #Function (biology) #Infinitesimal #Nonlinear system #Optimal stopping #Point processes and geometric inequalities #Process (computing) #Random variable #Range (aeronautics) #Stochastic processes and statistical mechanics #Stopping time #math.PR
paper · pdf · doi:10.1214/13-aap979
published as Annals of Applied Probability 2014, Vol. 24, No. 6, 2340-2370 · Published in at http://dx.doi.org/10.1214/13-AAP979 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2014/08/26 · arxiv created 2014/09/05 · arxiv updated 2014/09/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let Z=(Zt)t≥0 be a regular diffusion process started at 0, let ℓ be an independent random variable with a strictly increasing and continuous distribution function F, and let τℓ=inf\t≥0\vert Zt=ℓ\ be the first entry time of Z at the level ℓ. We show that the quickest detection problem infτ[P(τ<τℓ)+cE(τ -τℓ)+] is equivalent to the (three-dimensional) optimal stopping problem supτE[Rτ-∫0τc(Rt) dt], where R=S-I is the range process of X=2F(Z)-1 (i.e., the difference between the running maximum and the running minimum of X ) and c(r)=cr with c>0. Solving the latter problem we find that the following stopping time is optimal: τ*=inf \t≥0\vert f*(It,St)≤ Xt≤ g*(It,St)\, where the surfaces f* and g* can be characterised as extremal solutions to a couple of first-order nonlinear PDEs expressed in terms of the infinitesimal characteristics of X and c. This is done by extending the arguments associated with the maximality principle [Ann. Probab. 26 (1998) 1614–1640] to the three-dimensional setting of the present problem and disclosing the general structure of the solution that is valid in all particular cases. The key arguments developed in the proof should be applicable in similar multi-dimensional settings.