参考文献:1. Akahori, J (1995) Some formulae for a new type of path-dependent option. Ann. Appl. Probab. 5: pp. 383-388 CrossRef 2. Barndorff-Nielsen, OE, Mikosch, T, Resnick, SI (2001) L茅vy processes: theory and applications. Birkh盲user, New York CrossRef 3. Baurdoux, E. and van Schaik, K. (2012) Predicting the time at which a L茅vy process attains its supremum. / Acta Applicandae Mathematicae (in press). arXiv:1207.476. 4. Bertoin, J (1996) L茅vy processes. Cambridge University Press, Cambridge 5. Bondesson, L (1992) Generalized gamma convolutions and related classes of distributions and densities, lectures notes in statistics. Springer-Verlag, New York CrossRef 6. Borodin, AN, Salminen, P (1996) Handbook of brownian motion-facts and formulae. Birkh盲user Verlag, Boston Berlin CrossRef 7. Buffet, E (2003) On the maximum of Brownian motion with drift. J. Appl. Math. Stoch. Anal., 16: pp. 201-207 CrossRef 8. Cifarelli, DM, Melilli, E (2000) Some new results for Dirichlet priors. Ann. Stat., 28: pp. 1390-1413 CrossRef 9. Cifarelli, DM, Regazzini, E (1990) Distribution functions of means of a Dirichlet process. Ann. Stat., 18: pp. 429-442 CrossRef 10. Dassios, A (2005) On the quantiles of Brownian motion and their hitting times. Bernoulli, 11: pp. 29-36 CrossRef 11. Du Toit, J. and Peskir, G. (2008) Predicting the Time of the Ultimate Maximum for Brownian Motion with Drift. In / Proceedings of the Mathematical Control Theory and Finance. Springer, pp. 95鈥?12. 12. Eberlein, E. (2001) Application of Hyperbolic L茅vy motions to finance. In / L茅vy Processes: Theory and Applications (O.E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick, eds.). Birkh盲user, pp. 319鈥?36. 13. Gradshteyn, IS, Ryzhik, IM (2000) Tables of Integrals, Series and Products. Academic Press, New York 14. Graversen, SE, Peskir, G, Shiryaev, AN (2001) Stopping Brownian motion without anticipation as close as possible to its ultimate maximum. Theory Prob. Appl., 45: pp. 125-136 CrossRef 15. James, LF, Roynette, B, Yor, M (2008) Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv., 5: pp. 346-415 CrossRef 16. James, LF, Ljioi, A, Pr眉nster, I (2008) Distributions of linear functionals of two parameter Poisson-Dirichlet random measures. Ann. Probab., 18: pp. 521-551 CrossRef 17. Karatzas, I, Shreve, SE (1988) Brownian motion and stochastic calculus. Springer-Verlag, New York CrossRef 18. Kyprianou, A.E. (2006) / Introductory Lectures on Fluctuation of L茅vy Processes with Applications. Springer-Verlag. 19. Lijoi, A, Pr眉nster, I (2009) Distributional properties of means of random probability measures. Stat. Surv., 3: pp. 47-95 CrossRef 20. M枚rters, P, Peres, Y (2010) Brownian motion. Cambridge University Press, Cambridge CrossRef 21. Pecherskii, EA, Rogozin, BA (1969) On joint distributions of random variables associated with fluctuations of a process with independent increments. Theory Probab. Appl., 14: pp. 410-423 CrossRef 22. Samorodnitsky, G, Taqqu, MS (1994) Stable Non-gaussian random processes. Chapman & Hall, New York, London 23. Sato, K-I (1999) L茅vy processes and infinitely divisible distributions. University Press, Cambridge 24. Yano, K., Yano, Y. and Yor, M. (2009) / On the laws of first hitting times of points for one-dimensional symmetric stable Levy processes. Seminaire de Probabilites XLII, Lecture Notes in Math. Springer, Berlin, pp. 187鈥?27.
刊物类别:Mathematics and Statistics
刊物主题:Statistics Statistical Theory and Methods Statistics Computing and Software Statistics
出版者:Springer India
ISSN:0976-8378
文摘
The article develops a theory for the last time the subordinated Brownian motion (SBM with negative drift reaches its supremum. The study includes obtaining expressions for the Laplace transform of the last time that the SBM reaches its supremum and also for its density. In the process, we establish that the last time that the SBM reaches its supremum is a member of the generalized gamma convolution (GGC) family. The theoretical results for the general case have been explicitly derived for some well-known subordinators. Numerical investigations show close agreement between the theoretical derivations and empirical computations.