The Annals of Probability

Integrated Brownian Motion, Conditioned to be Positive

Piet Groeneboom, Geurt Jongbloed, and Jon A. Wellner

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Abstract

We study the two-dimensional process of integrated Brownian motion and Brownian motion, where integrated Brownian motion is conditioned to be positive. The transition density of this process is derived from the asymptotic behavior of hitting times of the unconditioned process. Explicit expressions for the transition density in terms of confluent hypergeometric functions are derived, and it is shown how our results on the hitting time distributions imply previous results of Isozaki-Watanabe and Goldman. The conditioned process is characterized by a system of stochastic differential equations (SDEs) for which we prove an existence and unicity result. Some sample path properties are derived from the SDEs and it is shown that $t \to t^{9/10}$ is a “critical curve” for the conditioned process in the sense that the expected time that the integral part of the conditioned process spends below any curve $t \to t^{\alpha}$ is finite for $\alpha < 9 /10$ and infinite for $\alpha \geq 9/10$.

Article information

Source
Ann. Probab., Volume 27, Number 3 (1999), 1283-1303.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677447

Digital Object Identifier
doi:10.1214/aop/1022677447

Mathematical Reviews number (MathSciNet)
MR1733148

Zentralblatt MATH identifier
0983.60078

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J25: Continuous-time Markov processes on general state spaces

Keywords
Conditioning confluent hypergeometric functions hitting times integrated Brownian motion Kolmogorov diffusion stochastic differential equations

Citation

Groeneboom, Piet; Jongbloed, Geurt; Wellner, Jon A. Integrated Brownian Motion, Conditioned to be Positive. Ann. Probab. 27 (1999), no. 3, 1283--1303. doi:10.1214/aop/1022677447. https://projecteuclid.org/euclid.aop/1022677447


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References

  • 1 ABRAMOWITZ, M. and STEGUN, I. A. 1964. Handbook of Mathematical Functions. National Bureau of Standards 55, Washington, DC.
  • 2 GOLDMAN, M. 1971. On the first passage of the integrated Wiener process. Ann. Math. Statist. 42 2150 2155.
  • 3 IKEDA, N. and WATANABE, S. 1981. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
  • 4 ISOZAKI, Y. and WATANABE, S. 1994. An asymptotic formula for the Kolmogorov diffusion and a refinement of Sinai's estimates for the integral of Brownian motion. Proc. Japan Acad. Ser. A 70 271 276.
  • 5 ITO, K. and MCKEAN, H. P., JR. 1974. Diffusion Processes and Their Sample Paths. Springer, Berlin.
  • 6 JONGBLOED, G. 1995. Three statistical inverse problems. Ph.D. dissertation, Delft Univ.
  • 7 KOLMOGOROV, A. N. 1934. Zuffallige Bewegungen. Ann. Math. II 35 116 117. ¨
  • 8 LACHAL, A. 1991. Sur le premier instant de passage de l'integrale du mouvement brown´ ien. Ann. Inst. H. Poincare 27 385 405. ´
  • 9 LACHAL, A. 1992. Sur les excursions de l'integrale du mouvement brownien. C. R. Acad. ´ Sci. Paris Ser. I 314 1053 1056. ´
  • 10 LACHAL, A. 1996. Sur la distribution de certaines fonctionnelles de l'integrale du mouve´ ment Brownien avec derives parabolique et cubique. Comm. Pure Appl. Math. 49 ´ 1299 1338.
  • 11 MAMMEN, E. 1991. Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19 741 759.
  • 12 MCKEAN, H. P., JR. 1963. A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 227 235.
  • 13 OLVER, F. W. J. 1974. Asymptotics and Special Functions. Academic Press, New York.
  • 14 ROGERS, L. C. G. and WILLIAMS, D. 1987. Diffusions, Markov Processes and Martingales 2. Wiley, New York.
  • 15 ROGERS, L. C. G. and WILLIAMS, D. 1994. Diffusions, Markov Processes and Martingales 1, 2nd ed. Wiley, New York.
  • 16 SINAI, Y. G. 1992. Statistics of shocks in solutions of inviscid Burgers equation. Comm. Math. Phys. 148 601 621.
  • SEATTLE, WASHINGTON 98195-4322 E-MAIL: jaw@stat.washington.edu