Approximations of boundary crossing probabilities for a Brownian motion

Journal of Applied Probability

Volumn 36, Issue 4, 1999, Pages 1019-1030

  Novikov, Alex ^a   Frishling, Volf ^b   Kordzakhia, Nino ^a  

^a UNIVERSITY OF NEWCASTLE   (Australia)

^b Portfolio and Market Risk Management Department   (Australia)

Author keywords

First passage times; Girsanov transformation; Numerical integration; Werner process

Indexed keywords

EID: 0033426760     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0021900200017836     Document Type: Article

References (30)

1. ANDERSON, T. W. (1960). A modification of the sequential probability ratio test to reduce the sample size. Ann. Math. Statist. 31, 165-197.
2. BOROVKOV, K. A. (1982). Rate of convergence in a boundary problem. Theory. Prob. Appl. 27, 148-149.
3. BREIMAN, L. (1966). First exit time from a square root boundary. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2, 9-16.
4. DANIELS, H. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli 2, 133-143.
5. DELONG, D. (1981). Crossing probabilities for a square root boundary by a Bessel process. Commun. Statist.-Theory Meth. A 10, 2197-2213.
6. DURBIN, J. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291-304.
7. FEREBEE, B. (1983). An asymptotic expansion for one-sided Brownian exit densities. Z. Wahreinscheinlichkeitsch. 61, 309-326.
8. FRISHLING, V., ANTIC, A., KUCHERA, A. AND RIDER, P. (1997). Pricing barrier options with time-dependent drift, volatility and barriers. Working paper, April 1997, Commonwealth Bank of Australia.
9. GRADSHTEYN, I. AND RYZHIK, I. (1980). Table of Integrals, Series, and Products. Academic Press, New York.
10. HALL, W. J. (1997). The distribution of Brownian motion on linear stopping boundaries. Sequential Analysis 4, 345-352.
11. KHMALADZE, E. AND SHINJIKASHVILI, E. (1998). Calculations of non-crossing probabilities for Poisson processes and its corollaries. Preprint, The University of NSW, Australia.
12. LERCHE, H. R. (1986). Boundary Crossing of Brownian Motion (Lecture Notes in Statist. 40). Springer, Berlin.
13. LIPTSER, R. S. AND SHIRYAEV, A. N. (1977). Statistics of Random Processes, Vol. 1. Springer, New York.
14. NAGAEV, S. V. (1970). The rate of convergence in a certain boundary value problem, part I. Theory Prob. Appl. 15, 179-199; part II, 15, 419-444.
15. NAGAEV, S. V. (1970). The rate of convergence in a certain boundary value problem, part I. Theory Prob. Appl. 15, 179-199; part II, 15, 419-444.
16. NOVIKOV, A. A. (1971). On stopping times for a Wiener process. Theory Prob. Appl. 16, 458-465.
17. NOVIKOV, A. A. (1979). On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary. Mat. Sb. 110, 539-550. (English translation in Math. USSR Sb. 38.)
18. NOVIKOV, A. A. (1980). Estimates and asymptotic behavior of the probability of not crossing moving boundaries by sums of independent random variables. Izv. Akad. Nauk SSSR Ser. Mat. 44, 868-885.
19. NOVIKOV, A. A., KORDZAKHIA, N. AND WRIGHT, I. (1998). Time-dependent barrier options and boundary crossing probabilities. Working paper, Department of Statistics, the University of Newcastle, Australia.
20. RICCIARDI, L. M. (1977). Diffusion Processes and Related Topics in Biology (Lecture Notes in Biomath. 14). Springer, Berlin.
21. ROBERTS, G. O. AND SHORTLAND, C. F. (1997). Pricing barriers options with time-dependent coefficients. Math. Finance 7, 83-93.
22. SACERDOTE, L. AND TOMASSETI, F. (1996). On evaluations and asymptotics approximations of the first-passage-time probabilities. Adv. Appl. Prob. 28, 270-284.
23. SAHANENKO, A. I. (1974). On the rate of convergence in a boundary value problem. Theory. Prob. Appl. 19, 416-421.
24. SALMINEN, P. (1988). On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary. J. Appl. Prob. 20, 411-426.
25. SATO, S. (1977). Evaluation of the first passage probability to a square root boundary for the Wiener process. J. Appl. Prob. 14, 53-70.
26. SEN, P. K. (1981). Sequential Nonparametrics: Invariance Principles and Statistical Inference. Wiley, New York.
27. SHEPP, L. A. (1967). A first passage problem for the Wiener process. Ann. Math. Statist. 38, 1912-1914.
28. SIEGMUND, D. (1985). Sequential Analysis: Tests and Confidence Intervals. Springer, New York.
29. WANG, L. AND PÖTZELBERGER, K. (1997). Boundary crossing probability for Brownian motion and general boundaries. J. Appl. Prob. 34, 54-65.
30. WOLFRAM, ST. (1996). The Mathematica Book. Cambridge University Press.