No CrossRef data available.
Article contents
Hitting-Time Densities of a Two-Dimensional Markov Process
Published online by Cambridge University Press: 27 July 2009
Abstract
This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x > 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.
- Type
- Articles
- Information
- Probability in the Engineering and Informational Sciences , Volume 6 , Issue 4 , October 1992 , pp. 561 - 580
- Copyright
- Copyright © Cambridge University Press 1992
References
Abrahams, J. (1984). A survey of recent progress on level-crossing problems for random processes. In Blake, I.F. & Poor, H.B. (eds.), Communications and networks. New York: Springer-Verlag.Google Scholar
Abramowitz, M. & Stegun, I.A. (1965). Handbook of mathematicalfunctions with formulas, graphs and mathemailcal tables. New York: Dover.Google Scholar
Bender, C.M. & Orszag, S.A. (1978). Advanced mathematical methods for scientists and engineers. New York: McGraw Hill.Google Scholar
Blake, I.F. & Lindsey, W.C. (1973). Level-crossing problems for random processes. IEEE Transactions on Information Theory lT-19: 295–315.CrossRefGoogle Scholar
Durbin, J. (1985). The first-passage density of a continuous Gaussian process to a general boundary. Journal of Applied Probability 22: 99–122.CrossRefGoogle Scholar
Franklin, J.N. & Rodemich, E. (1968). Numerical analysis of an elliptic-parabolic partial differential equation. SIAM Journal on Numerical Analysis 5: 680–716.CrossRefGoogle Scholar
Goldman, M. (1971). On the first passage of the integrated Wiener process. Annals of Mathematical Statisics 42: 2150–2155.CrossRefGoogle Scholar
Gor'kov, J.P. (1976). A formula for the solution of a certain boundary value problem for the stationary equation of Brownian motion. Soviet Mathematics Doklady 16: 904–908.Google Scholar
Hesse, C.H. (1990). Further refinements of a stochastic model for sedimentation. Technical Report, Department of Statistics, University of California at Berkeley.Google Scholar
Itô, K. & McKean, H.P. (1965). Diffusion processes and their sample paths. New York: Springer-Verlag.Google Scholar
Kendall, M., Stuart, A., & Ord, J.K. (1987). Kendall's advanced theory of statistics, Vol. 1, 5th ed.London: Charles Griffin Co.Google Scholar
Lefebvre, M. (1989). First-passage densities of a two-dimensional process. SIAM Journal of Applied Mathematics 5: 1514–1523.CrossRefGoogle Scholar
McKean, H.P. (1963). A winding problem for a resonator driven by a white noise. Journal of Mathematics: Kyoto University 2-2: 227–235.Google Scholar
Oppenheim, I., Shuler, K.E., & Weiss, G.H. (1977). Stochastic processes in chemical physics. Cambridge, MA: MIT Press.Google Scholar
Pickard, D.K. & Tory, E.M. (1982). Extensions of a Markov model for sedimentation. Journal of Mathematical Analysis and Applications 86: 442–470.Google Scholar
Pickard, D.K., Tory, E.M., & Tuckman, B. (1985). Simulation and analysis of steady-state transit times. Technical Report SS-P7, Harvard University.Google Scholar
van Kampen, N.G. (1981). Stochastic processes in physics and chemistry. Amsterdam: Elsevier North-Holland.Google Scholar