Bayesian Quickest Detection Problems for Some Diffusion Processes

Abstract

We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process with linear and exponential penalty costs for a detection delay. The optimal times of alarms are found as the first times at which the weighted likelihood ratios hit stochastic boundaries depending on the current observations. The proof is based on the reduction of the initial problems into appropriate three-dimensional optimal stopping problems and the analysis of the associated parabolic-type free-boundary problems. We provide closed form estimates for the value functions and the boundaries, under certain nontrivial relations between the coefficients of the observable diffusion.2010 Mathematics Subject Classification: Primary 60G40, 62M20, 34K10.Secondary 62C10, 62L15, 60J60.

On the sequential testing and quickest change-point detection problems for Gaussian processes

“…Another possible natural extension µ((t − θ) + ) of the original linear drift function µ(t) = ρt is not studied in the paper. The problem setting in the case of observable diffusion processes in which the local drift rate depends on the running value of the observations was considered in [20] and [21].…”

“…In order to simplify further notations, we define the process Φ = ( Φ s ) s≥0 by Φ s = Φ 0 e λγ(t,s) L γ(t,s) for s ≥ 0. Since the process L has the form of (4.19), by applying the time-change formula for Itô integrals from [30, Theorems 8.5.1 and 8.5.7], we obtain 21) where the process B = ( B s ) s≥0 defined in (4.6) is a standard Brownian motion. Therefore, by using the definition of τ in (4.20) and taking into consideration the time change from (4.5), the stopping time β(t, τ ) can be represented as…”

“…The method of solution of these problems was based on the reduction of the associated optimal stopping problems to the equivalent free boundary problems for ordinary (integro-)differential operators and a unique characterization of the Bayesian risks by means of the smooth and continuous fit conditions for the value functions at the optimal stopping boundaries. Further investigations of both problems for observable Wiener processes were studied in [18,19] in the finite horizon setting, and for certain more general time-homogeneous diffusions in [20,21] on infinite time intervals.…”

Bayesian Switching Multiple Disorder Problems

“…Kolmogorov in [23] (see also [18]- [19]). For this, let us define the processes = ( ) ≥0 and = ( ) ≥0 by:…”

Strategies on Initial Public Offering of Company Equity at Stock Exchanges in Imperfect Highly Volatile Global Capital Markets with Induced Nonlinearities

“…The multiple evidences of the fact that the diffusion processes have the considerable influences on the various econophysical and econometrical parameters of the diffusion-type financial systems have been described in Bachelier (1900), Volterra (1906), Slutsky (1910Slutsky ( , 1912Slutsky ( , 1913Slutsky ( , 1914Slutsky ( , 1915Slutsky ( , 1922aSlutsky ( , b, 1923aSlutsky ( , b c, 1925aSlutsky ( , b, 1926Slutsky ( , 1927aSlutsky ( , b, 1929Slutsky ( , 1935Slutsky ( , 1937aSlutsky ( , b, 1942, Osborne (1959), Alexander (1961), Shiryaev (1961Shiryaev ( , 1963Shiryaev ( , 1964Shiryaev ( , 1965Shiryaev ( , 1967Shiryaev ( , 1978Shiryaev ( , 1998aShiryaev ( , b, 2002Shiryaev ( , 2008aShiryaev ( , b, 2010, Grigelionis, Shiryaev (1966), Graversen, Peskir, Shiryaev (2001), Kallsen, Shiryaev (2001, Jacod, Shiryaev (2003), Peskir, Shiryaev (2006), Feinberg, Shiryaev (2006), du Toit, Peskir, Shiryaev (2007), Eberlein, Papapantoleon, Shiryaev (2008), Shiryaev, Zryumov (2009), Shiryaev, Novikov (2009), Gapeev, Shiryaev (2010), Karatzas, Shiryaev, Shkolnikov (2011), , , Feinberg, Mandava, Shiryaev (2013), Akerlof, Stiglitz (1966), …”