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 Teor. Veroyatnost. i Primenen., 2018, Volume 63, Issue 3, Pages 565–583 (Mi tvp5154)

A note on the quasi-stationary distribution of the Shiryaev martingale on the positive half-line

A. S. Polunchenkoa, S. Martínezb, J. San Martínb

a Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton, NY, USA
b Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMI-CNRS 2807, Universidad de Chile, Santiago, Chile

Abstract: We obtain a closed-form formula for the quasi-stationary distribution of the classical Shiryaev martingale diffusion considered on the positive half-line $[A,+\infty)$ with $A>0$ fixed; the state space's left endpoint is assumed to be the killing boundary. The formula is obtained analytically as the solution of the appropriate singular Sturm–Liouville problem; the latter was first considered in section 7.8.2 of [P. Collet, S. Martínez, and J. San Martí­n, Quasi-Stationary Distributions. Markov Chains, Diffusions and Dynamical Systems, Springer, Heidelberg, 2013] but has heretofore remained unsolved.

Keywords: quasi-stationary distribution, martingale Shiryaev diffusion process, Whittaker function.

 Funding Agency Grant Number Simons Foundation 304574 The effort of A. S. Polunchenko was partially supported by the Simons Foundation via a Collaboration Grant in Mathematics under Award # 304574.

DOI: https://doi.org/10.4213/tvp5154

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English version:
Theory of Probability and its Applications, 2019, 63:3, 464–478

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Accepted:30.10.2017
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Citation: A. S. Polunchenko, S. Martínez, J. San Martín, “A note on the quasi-stationary distribution of the Shiryaev martingale on the positive half-line”, Teor. Veroyatnost. i Primenen., 63:3 (2018), 565–583; Theory Probab. Appl., 63:3 (2019), 464–478

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp5154
• https://doi.org/10.4213/tvp5154
• http://mi.mathnet.ru/eng/tvp/v63/i3/p565

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This publication is cited in the following articles:
1. Li K. Polunchenko A.S. Pepelyshev A., “Analytic Evaluation of the Fractional Moments For the Quasi-Stationary Distribution of the Shiryaev Martingale on An Interval”, Commun. Stat.-Simul. Comput.
2. A. S. Polunchenko, A. Pepelyshev, “Analytic moment and Laplace transform formulae for the quasi-stationary distribution of the Shiryaev diffusion on an interval”, Stat. Papers, 59:4 (2018), 1351–1377
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