Optimal real-time detection of a drifting Brownian coordinate

Ernst, Philip; Peškir, Goran; Zhou, Quan

doi:10.1214/19-aap1522

Cited by 11 publications

(9 citation statements)

References 19 publications

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“…The following results can be derived easily by writing its generating function in terms of the scale matrix via (19).…”

Section: And Hencementioning

confidence: 99%

“…To see this, note that differentiability of W z (x) in z follows from the identity (19) for the killed process, and then d dz…”

Section: And Hencementioning

confidence: 99%

“…Using this generalized scale matrix in Theorem 15, the identity (19) and Lemma 6 can be extended as follows. Below, it is understood that the first passage time τ a as in ( 18) is for the generalized X defined in (40).…”

Section: Extension To the Non-iid Case With A Change Pointmentioning

confidence: 99%

“…Exact computation of the performance measures, such as the average run length, average detection delay, and false alarm probability, is rarely achieved. Contrary to the continuous-time model, where analytical tools such as martingale methods and Itô calculus are available (see, e.g., [19,41,42]), exact computation of the first passage identities for discrete-time processes tends to be infeasible by traditional methods. For this reason, research on the CUSUM statistic, which is a discrete-time process, has focused on pursuing approximate results.…”

Section: Introductionmentioning

confidence: 99%

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A series expansion formula of the scale matrix with applications in change-point detection

Ivanovs¹,

Yamazaki²

2021

Preprint

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We introduce a new method analyzing the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. By using the theory of the so-called scale matrix and further developing it, we derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure. The proposed method is robust and applicable in a general setting with non-i.i.d. observations. Numerical results also are given.

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“…The following results can be derived easily by writing its generating function in terms of the scale matrix via (19).…”

Section: And Hencementioning

confidence: 99%

“…To see this, note that differentiability of W z (x) in z follows from the identity (19) for the killed process, and then d dz…”

Section: And Hencementioning

confidence: 99%

Section: Extension To the Non-iid Case With A Change Pointmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A series expansion formula of the scale matrix with applications in change-point detection

Ivanovs¹,

Yamazaki²

2021

Preprint

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“…Other optimal stopping problems for two-dimensional diffusion processes were studied by Assing et al [2], where the monotonicity and continuity of the value functions were proved by using time-change and coupling techniques. More recently, Ernst et al [16] solved the optimal stopping problem for a two-dimensional diffusion process related to the optimal real-time detection of a drifting Brownian coordinate which is is equivalent to a free-boundary problem the associated partial differential operator of elliptic type. The important recent results in the area comprise the continuity of the optimal stopping boundaries in optimal stopping problems for two-dimensional diffusions proved by Peskir [44] and the global C 1 -regularity of the value function in two-dimensional optimal stopping problems studied by De Angelis and Peskir [9].…”

Section: Introductionmentioning

confidence: 99%

Discounted optimal stopping problems in continuous hidden Markov models

Gapeev

2021

Stochastics

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We study a two-dimensional discounted optimal stopping problem related to the pricing of perpetual commodity equities in a model of financial markets in which the behaviour of the underlying asset price follows a generalized geometric Brownian motion and the dynamics of the convenience yield are described by an unobservable continuous-time Markov chain with two states. It is shown that the optimal time of exercise is the first time at which the commodity spot price paid in return to the fixed coupon rate hits a lower stochastic boundary being a monotone function of the running value of the filtering estimate of the state of the chain. We rigorously prove that the optimal stopping boundary is regular for the stopping region relative to the resulting two-dimensional diffusion process and the value function is continuously differentiable with respect to the both variables. It is verified by means of a change-of-variable formula with local time on surfaces that the value function and the boundary are determined as a unique solution of the associated parabolic-type free-boundary problem. We also give a closed-form solution to the optimal stopping problem for the case of an observable Markov chain.

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