Sequential testing problems for Bessel processes

Johnson, Peter J.; Peškir, Goran

doi:10.1090/tran/7068

Cited by 14 publications

(17 citation statements)

References 12 publications

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“…This is motivated by the fact that in recent years several specific problems of this kind have appeared in the literature (cf. [6], [8], [10], [12], [16], [17]) and further ones are on their way. Often these papers are motivated by real-world applications where dimension two (or higher) plays a crucial role (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Often these papers are motivated by real-world applications where dimension two (or higher) plays a crucial role (cf. [16], [17]). This necessitates in establishing general results implying continuity of the optimal stopping boundary that would be applicable in these and similar other problems.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the known uniqueness arguments [22,, originally established in [19] and further refined in [11], extend to this fully two-dimensional case as well (cf. [6], [8], [12], [16], [17]) and this enables one to characterise the optimal stopping boundary as the unique solution to a nonlinear integral equation in the class of continuous functions (with the value function being expressed in terms of the optimal stopping boundary itself). This characterisation of the optimal stopping boundary is known to be sufficient for practical purposes of optimal stopping (using Picard iteration for numerics for instance) and higher degrees of regularity can then be studied subsequently as/if needed.…”

Section: Introductionmentioning

confidence: 99%

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Continuity of the optimal stopping boundary for two-dimensional diffusions

Peškir¹

2019

Ann. Appl. Probab.

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We first show that a smooth fit between the value function and the gain function at the optimal stopping boundary for a two-dimensional diffusion process implies the absence of boundary's discontinuities of the first kind (the right-hand and left-hand limits exist but differ). We then show that the smooth fit itself is satisfied over the flat portion of the optimal stopping boundary arising from any of its hypothesised jumps. Combining the two facts we obtain that the optimal stopping boundary is continuous whenever it has no discontinuities of the second kind. The derived fact holds both in the parabolic and elliptic case under the sole hypothesis of Hölder continuous coefficients, thus improving upon all known results in the parabolic case, and establishing the fact for the first time in the elliptic case. The method of proof relies upon regularity results for the second order parabolic/elliptic PDEs and makes use of the local time-space calculus techniques.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Continuity of the optimal stopping boundary for two-dimensional diffusions

Peškir¹

2019

Ann. Appl. Probab.

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“…Non-constant SNR processes give rise to two-dimensional sufficient statistics, which depend on both the posterior process and the current position of the observed process, making their analysis more challenging. However, recent breakthroughs have been made for certain processes with non-constant SNR [35,36]. For a general discussion of this problem and the causes of increased dimensionality, see [35, §2] and [36, §3].…”

Section: (A) General Diffusion Processesmentioning

confidence: 99%

Detecting changes in real-time data: a user’s guide to optimal detection

Johnson

Moriarty

Peškir

2017

Phil. Trans. R. Soc. A.

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The real-time detection of changes in a noisily observed signal is an important problem in applied science and engineering. The study of parametric optimal detection theory began in the 1930s, motivated by applications in production and defence. Today this theory, which aims to minimize a given measure of detection delay under accuracy constraints, finds applications in domains including radar, sonar, seismic activity, global positioning, psychological testing, quality control, communications and power systems engineering. This paper reviews developments in optimal detection theory and sequential analysis, including sequential hypothesis testing and change-point detection, in both Bayesian and classical (non-Bayesian) settings. For clarity of exposition, we work in discrete time and provide a brief discussion of the continuous time setting, including recent developments using stochastic calculus. Different measures of detection delay are presented, together with the corresponding optimal solutions. We emphasize the important role of the signal-to-noise ratio and discuss both the underlying assumptions and some typical applications for each formulation.This article is part of the themed issue 'Energy management: flexibility, risk and optimization'.

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“…It is well known that optimal stopping problems for multi-dimensional continuous-time Markov processes are analytically more difficult than the corresponding problems for the onedimensional ones and their solutions are very rarely found explicitly. Some necessarily multidimensional optimal stopping problems arising mostly from the problems of quickest changepoint detection were studied by Bayraktar and Poor [3] and Bayraktar et al [4] for discontinuous Poisson processes, Dayanik et al [7] for mixed jump-diffusion processes with mean-reverting components, as well as in Gapeev and Shiryaev [13]- [14] and Johnson and Peskir [23]- [24] for purely continuous diffusion processes. Some analytical results for such optimal stopping problems were recently obtained by Assing et al [2].…”

Section: Introductionmentioning

confidence: 99%

On Some Functionals of the First Passage Times in Models With Switching Stochastic Volatility

Gapeev

Brockhaus

Dubois

2018

Int. J. Theor. Appl. Finan.

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We compute some functionals related to the joint generalized Laplace transforms of the first times at which two-dimensional diffusion-type Markov processes exit half strips. It is assumed that the state space components are driven by constantly correlated Brownian motions and the dynamics of the coefficients are described by a continuous-time Markov chain. The method of proof is based on the solutions of the equivalent boundary-value problems for systems of elliptic-type partial differential equations for the associated value functions. The results are illustrated on several two-dimensional continuous mean-reverting or diverting models of switching stochastic volatility.

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Sequential testing problems for Bessel processes

Cited by 14 publications

References 12 publications

Continuity of the optimal stopping boundary for two-dimensional diffusions

Continuity of the optimal stopping boundary for two-dimensional diffusions

Detecting changes in real-time data: a user’s guide to optimal detection

On Some Functionals of the First Passage Times in Models With Switching Stochastic Volatility

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