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Crossing probabilities for diffusion processes with piecewise continuous boundaries. (English) Zbl 1122.60070
Summary: We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein-Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach, explicit formulas for boundary crossing probabilities for certain nonlinear boundaries are obtained, which are useful in evaluation and comparison of various computational algorithms. Moreover, numerical computation can be easily done by Monte Carlo integration and the approximation errors for general boundaries are automatically calculated. Some numerical examples are presented.
MSC:
60J60Diffusion processes
60J70Applications of Brownian motions and diffusion theory
60J25Continuous-time Markov processes on general state spaces
60G40Stopping times; optimal stopping problems; gambling theory
65C05Monte Carlo methods
References:
[1]T. W. Anderson,”A modification of the sequential probability ratio test to reduce the sample size,” Annals of Mathematical Statistics vol. 31 pp. 165–197, 1960. · Zbl 0089.35501 · doi:10.1214/aoms/1177705996
[2]W. Bischoff, E. Hashorva, J. Hüsler, and F. Miller,”Exact asymptotics for boundary crossings of the Brownian brisge with trend with application to the Kolmogorov test,” Annals of the Institute of Statististical Mathematics vol. 55, pp. 849–864, 2003. · Zbl 1049.62048 · doi:10.1007/BF02523397
[3]F. Black, and M. Scholes,”The pricing of options and corporate liabilities,” Journal of Political Economy vol.81, pp. 637–659, 1973. · doi:10.1086/260062
[4]G. Bluman, ”In the transformation of diffusion processes into the Wiener process,” SIAM Journal on Applied Mathematics vol. 39, pp. 238–247, 1980. · Zbl 0448.60056 · doi:10.1137/0139021
[5]K. Borovkov and A. Novikov,”Explicit bounds for approaximation rates of boundary crossing probabilities for the Wiener process,” Journal of Applied Probability vol. 42, pp. 82–92, 2005. · Zbl 1077.60057 · doi:10.1239/jap/1110381372
[6]I. D. Cherkasov,”On the transformation of the diffusion process to a Wiener process,” Theory of Probability and Its Applications vol. 2, pp. 373–377, 1957. · doi:10.1137/1102028
[7]J. C. Cox, J. E. Ingersoll, and S. A. Ross, ”A theory of the term structure of interest rates,” Econometrica vol. 53, pp. 385–407, 1985. · Zbl 1274.91447 · doi:10.2307/1911242
[8]H. E. Daniels, ”The minimum of stationary Markov process superimposed on a U-shaped trend,” Journal of Applied Probability vol.6, pp. 399–408, 1969. · Zbl 0209.19702 · doi:10.2307/3212009
[9]H. E. Daniels, ”Approximating the first crossing-time density for a curved boundary,” Bernoulli vol. 2, pp. 133–143, 1996. · Zbl 0865.60069 · doi:10.2307/3318547
[10]J. L. Doob, ”Heuristic approach to the Kolmogorov–Smirnov theorems,” Annals of Mathematical Statistics vol. 20, pp. 393–403, 1949. · Zbl 0035.08901 · doi:10.1214/aoms/1177729991
[11]J. Dupuis, and D. Siegmund,” Boundary crossing probabilities in linkage analysis,” Game theory, optimal stopping, probability and statistics, pp. 141–152, IMS Lecture Notes Monogr. Ser., 35, Inst. Math. Statist., Beachwood, Ohio, 2000.
[12]J. Durbin,”Boundary crossing probabilities for the Brownian motion and the poisson processes and techniques for computing the power of the Kolmogorov–Smirnov test,” Journal of Applied Probability vol. 8, pp. 431–453, 1971. · Zbl 0225.60037 · doi:10.2307/3212169
[13]J. Durbin,”The first-passage density of the Brownian motion process to a curved boundary,” Journal of Applied Probability vol. 29, pp. 291–304, 1992. · Zbl 0806.60063 · doi:10.2307/3214567
[14]N. Ebrahimi,”System reliability bsed on diffusion models for fatigue crack frowth,” Naval Research Logistics vol. 52, pp. 46–57, 2005. · Zbl 1090.90047 · doi:10.1002/nav.20050
[15]B. Ferebee,”The tangent approximation to one-sided Brownian exit densities,” Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete vol. 61, pp. 309–326, 1982. · Zbl 0499.60085 · doi:10.1007/BF00539832
[16]B. Ferebee, ”An asymptotic expansion for one-sided Brownian exit densities,” Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete vol. 63, pp. 1–15, 1983. · Zbl 0488.60086 · doi:10.1007/BF00534172
[17]J. Garrido,”Stochastic differential equations for compounded risk reserves.,” Insurance. Mathematics & Economics vol. 8, pp. 165–173, 1989. · Zbl 0688.62056 · doi:10.1016/0167-6687(89)90054-1
[18]V. Giorno, A. G. Nobile, L. M. Ricciardi, and S. Sato,” On the evaluation of first-passage-time probability densities via non-singular integral equations,” Advances in Applied Probability vol. 21, pp. 20–36, 1989. · Zbl 0668.60068 · doi:10.2307/1427196
[19]M. T. Giraudo, and L. Sacerdote,”An improved technique for the simulation of first passage times for diffusion processes,” Communications in Statistics. Simulation and Computation vol. 28, pp. 1135–1163, 1999. · Zbl 0968.60504 · doi:10.1080/03610919908813596
[20]M. T. Giraudo, L. Sacerdote, and C. Zucca,”A Monte Carlo method for the simulation of first passage times of diffusion processes,” Methodology and Computing in Applied Probability vol. 3, pp. 215–231, 2001. · Zbl 1002.60066 · doi:10.1023/A:1012261328124
[21]I. Karatzas, and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn., Springer, Berlin Heidelberg New York, 1991.
[22]P. E. Kloeden and E. Platen, The Numerical Solution to Stochastic Differential Equations, Springer, Berlin Heidelberg New York, 1992.
[23]S. G. Kou, and H. Wang,”First passage times of a jump diffusion process,” Advances in Applied Probability vol. 35, pp. 504–531, 2003. · Zbl 1040.60057 · doi:10.1239/aap/1059486822
[24]A. Kolmogorov,”Über die analytischen methoden in der Wahrscheinlichkeitsrechnung,” Mathematische Annalen vol. 104, pp. 415–458, 1931. · doi:10.1007/BF01457949
[25]W. Krämer, W. Ploberger, and R. Alt, ”Testing for structural change in dynamic models,” Econometrica vol. 56, pp. 1355–1369, 1988. · Zbl 0655.62107 · doi:10.2307/1913102
[26]D. Lamberton, and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall, London, UK, 1996.
[27]H. R. Lerche, Boundary Crossing of Brownian Motion, In Lecture Notes in Statistics vol. 40. Springer, Berlin Heidelberg New York, 1986.
[28]X. S. Lin,”Double barrier hitting time distributions with applications to exotic options,” Insurance. Mathematics & Economics vol. 23, pp. 45–58, 1998. · Zbl 0942.60066 · doi:10.1016/S0167-6687(98)00021-3
[29]A. Martin-Löf,”The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier,” Journal of Applied Probability vol. 35, pp. 671–682, 1998. · Zbl 0919.92034 · doi:10.1239/jap/1032265215
[30]R. C. Merton,”Theory of rational option pricing,” Bell Journal of Economics and Management Science vol. 4, pp. 141–183, 1973. · doi:10.2307/3003143
[31]A. J. Michael,”Viscoelasticity, postseismic slip, fault interactions, and the recurrence of large earthquakes,” Bulletin of the Seismological Society of America vol. 95, pp. 1594–1603, 2005. · doi:10.1785/0120030208
[32]A. Novikov, V. Frishling, and N. Kordzakhia,”Approximations of boundary crossing probabilities for a Brownian motion,” Journal of Applied Probability vol. 99, pp. 1019–1030, 1999.
[33]A. Novikov, V. Frishling, and N. Kordzakhia, ”Time-dependent barrier options and boundary crossing probabilities,” Georgian Mathematical Journal vol. 10, pp. 325–334, 2003.
[34]K. Pötzelberger, and L. Wang,”Boundary crossing probability for Brownian Motion,” Journal of Applied Probability vol. 38, pp. 152-164, 2001. · Zbl 0986.60079 · doi:10.1239/jap/996986650
[35]L. M. Ricciardi,”On the transformation of diffusion processes into the Wiener process,” Journal of Mathematical Analysis and Applications vol. 54, pp. 185-199, 1976. · Zbl 0361.60043 · doi:10.1016/0022-247X(76)90244-4
[36]L. M. Ricciardi, A. Di Crescenzo, V. Giorno, and A.G. Nobile,”An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling.,”Mathematica Japonica vol. 50, pp. 247–322, 1999.
[37]L. M. Ricciardi, L. Sacerdote, and S. Sato,”On an integral equation for first-passage-time probability densities,” Journal of Applied Probility vol. 21, pp. 302-314, 1984. · Zbl 0551.60081 · doi:10.2307/3213641
[38]G. O. Roberts, and C. F. Shortland,”Pricing barrier options with time-dependent coefficients,” Mathematical Finance vol. 7, pp. 83–93, 1997. · Zbl 0884.90044 · doi:10.1111/1467-9965.00024
[39]L. Sacerdote, and F. Tomassetti, ”On evaluations and asymptotic approximations of first-passage-time probabilities,” Advanced of Applied Probability vol. 28, pp. 270–284, 1996. · Zbl 0869.60034 · doi:10.2307/1427921
[40]P. K. Sen, Sequential Nonparametrics: Invariance Principles and Statistical Inference, Wiley, New York, 1981.
[41]D. Siegmund,”Boundary crossing probabilities and statistical applications,” Annals of Statistics vol. 14, pp. 361–404, 1986. · Zbl 0632.62077 · doi:10.1214/aos/1176349928
[42]A. N. Startsev,”Asymptotic analysis of the general stochastic epidemic with variable infectious periods,” Journal of Applied Probility vol. 38, pp. 18–35, 2001. · Zbl 1004.92032 · doi:10.1239/jap/996986640
[43]V. Strassen,”Almost sure behaviour of sums of independent random variables and martingales,” In L. M. Le Cam and J. Neyman (eds.),Proc. 5th Berkley Symp. Math. Statist. Prob., Vol. 11, Contributions to Probability Theory, pp. 315–343, Part 1, University of California Press, Berkley, California, 1967.
[44]H. C. Tuckwell, and F. Y. M. Wan,”First passage time to detection in stochastic population dynamical models for HIV-1,” Applied Mathematics Letters vol. 13, pp. 79–83, 2000. · Zbl 0962.92024 · doi:10.1016/S0893-9659(00)00037-9
[45]O. A. Vasicek,”An equilibrium characterization of the term structure,” Journal of Financial Economics vol. 5, pp. 177–188, 1977. · doi:10.1016/0304-405X(77)90016-2
[46]L. Wang, and K. Pötzelberger,”Boundary crossing probability for Brownian Motion and general boundaries,” Journal of Applied Probability vol. 34, pp. 54–65, 1997. · Zbl 0874.60034 · doi:10.2307/3215174
[47]T. Yamada, and S. Watanabe,”On the uniqueness of the solutions of stochastic differential equations,” Journal of Mathematics of Kyoto University vol. 11, pp. 155–167, 1971.
[48]A. Zeileis,”Alternative boundaries for CUSUM tests,” Statistical Papers vol. 45, pp. 123–131, 2004. · Zbl 1050.62075 · doi:10.1007/BF02778274