Exit probability levels of diffusion processes

Donchev, Doncho

doi:10.1090/proc/13392

Exit probability levels of diffusion processes
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by Doncho S. Donchev PDF

Proc. Amer. Math. Soc. 145 (2017), 2241-2253 Request permission

Abstract:

We are interested in the probability that a diffusion process exits a domain between two curved boundaries through the upper one. In case of given boundaries that problem has closed solutions only in some special cases. We study a modification of the problem in which not only the exit probabilities but also the boundaries are unknown. Introducing the notion of exit probability levels, we show that this new problem can be reduced to a single non-linear second order PDE. In case of some important diffusion processes we find large families of solutions to this equation.

References

T. W. Anderson, A modification of the sequential probability ratio test to reduce the sample size, Ann. Math. Statist. 31 (1960), 165–197. MR 116441, DOI 10.1214/aoms/1177705996
A. Buonocore, V. Giorno, A. G. Nobile, and L. M. Ricciardi, On the two-boundary first-crossing-time problem for diffusion processes, J. Appl. Probab. 27 (1990), no. 1, 102–114. MR 1039187, DOI 10.2307/3214598
Doncho S. Donchev, Brownian motion hitting probabilities for general two-sided square-root boundaries, Methodol. Comput. Appl. Probab. 12 (2010), no. 2, 237–245. MR 2609650, DOI 10.1007/s11009-009-9144-4
J. Durbin, Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test, J. Appl. Probability 8 (1971), 431–453. MR 292161, DOI 10.2307/3212169
Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. MR 637061
E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. Teil I: Gewöhnliche Differentialgleichungen. 6. Aufl.; Teil II: Partielle Differentialgleichungen erster Ordnung für eine gesuchte Funktion. 4. Aufl, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Band 18, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1959 (German). MR 0106302
Hans Rudolf Lerche, Boundary crossing of Brownian motion, Lecture Notes in Statistics, vol. 40, Springer-Verlag, Berlin, 1986. Its relation to the law of the iterated logarithm and to sequential analysis. MR 861122, DOI 10.1007/978-1-4615-6569-7
A. A. Novikov, The stopping times of a Wiener process, Teor. Verojatnost. i Primenen. 16 (1971), 458–465 (Russian, with English summary). MR 0315798
Alex Novikov, Volf Frishling, and Nino Kordzakhia, Approximations of boundary crossing probabilities for a Brownian motion, J. Appl. Probab. 36 (1999), no. 4, 1019–1030. MR 1742147, DOI 10.1017/s0021900200017836
Goran Peskir and Albert Shiryaev, Optimal stopping and free-boundary problems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2006. MR 2256030
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integraly i ryady, “Nauka”, Moscow, 1981 (Russian). Èlementarnye funktsii. [Elementary functions]. MR 635931
Paavo Salminen, On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary, Adv. in Appl. Probab. 20 (1988), no. 2, 411–426. MR 938153, DOI 10.2307/1427397
Paavo Salminen and Marc Yor, On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes, Period. Math. Hungar. 62 (2011), no. 1, 75–101. MR 2772384, DOI 10.1007/s10998-011-5075-2
L. A. Shepp, On the integral of the absolute value of the pinned Wiener process, Ann. Probab. 10 (1982), no. 1, 234–239. MR 637389