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The disorder problem for diffusion processes with the ϵ-linear and expected total miss criteria

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  • Buonaguidi, B.

Abstract

We study the disorder problem for a time-homogeneous diffusion process. The aim is to determine an efficient detection strategy of the disorder time θ, at which the process changes its drift. We focus on the ϵ-linear and the expected total miss criteria, where, unlike the well known linear penalty criterion, the expected penalty for an early/wrong detection of θ is expressed as the frequency of false alarms launched at least ϵ units of time before θ and as the expected advance in the detection of θ, respectively. We show that the original optimal stopping problems can be reduced to a unifying optimal stopping problem; then, we derive the associated free-boundary problem and we provide sufficient conditions for the existence and uniqueness of its solution.

Suggested Citation

  • Buonaguidi, B., 2022. "The disorder problem for diffusion processes with the ϵ-linear and expected total miss criteria," Statistics & Probability Letters, Elsevier, vol. 189(C).
    • Handle: RePEc:eee:stapro:v:189:y:2022:i:c:s0167715222001092
    DOI: 10.1016/j.spl.2022.109548
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    References listed on IDEAS

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    1. Gapeev, Pavel V., 2020. "On the problems of sequential statistical inference for Wiener processes with delayed observations," LSE Research Online Documents on Economics 104072, London School of Economics and Political Science, LSE Library.
    2. Pavel V. Gapeev, 2020. "On the problems of sequential statistical inference for Wiener processes with delayed observations," Statistical Papers, Springer, vol. 61(4), pages 1529-1544, August.
    3. Bayraktar, Erhan & Dayanik, Savas & Karatzas, Ioannis, 2005. "The standard Poisson disorder problem revisited," Stochastic Processes and their Applications, Elsevier, vol. 115(9), pages 1437-1450, September.
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    5. Manuel Klein, 2009. "Comment on “Investment Timing Under Incomplete Information”," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 249-254, February.
    6. Erhan Bayraktar & Savas Dayanik, 2006. "Poisson Disorder Problem with Exponential Penalty for Delay," Mathematics of Operations Research, INFORMS, vol. 31(2), pages 217-233, May.
    7. Gapeev, P.V. & Peskir, G., 2006. "The Wiener disorder problem with finite horizon," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1770-1791, December.
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