On the transition densities for reflected diffusions

Advances in Applied Probability - Tập 37 Số 2 - Trang 435-460 - 2005
Vadim Linetsky1
1Industrial Engineering and Management Sciences

Tóm tắt

Diffusion models in economics, finance, queueing, mathematical biology, and electrical engineering often involve reflecting barriers. In this paper, we study the analytical representation of transition densities for reflected one-dimensional diffusions in terms of their associated Sturm-Liouville spectral expansions. In particular, we provide explicit analytical expressions for transition densities of Brownian motion with drift, the Ornstein-Uhlenbeck process, and affine (square-root) diffusion with one or two reflecting barriers. The results are easily implementable on a personal computer and should prove useful in applications.

Từ khóa


Tài liệu tham khảo

10.1142/S0219024904002451

Abramowitz, 1972, Handbook of Mathematical Functions

10.1080/135048698334709

Itô, 1974, Diffusion Processes and their Sample Paths

Lebedev, 1972, Special Functions and Their Applications

10.1007/BF00736006

10.1016/0304-3932(91)90026-K

10.1090/S0002-9947-1957-0091566-1

10.1090/S0002-9947-1956-0087012-3

Erdelyi, 1953, Higher Transcendental Functions

10.1023/A:1021804515162

10.1023/A:1009803506170

10.1287/moor.1040.0119

10.3150/bj/1068128980

Bertolla, 1992, Target zones and realignments, Amer. Econom. Rev., 82, 520

10.1007/978-3-0348-7652-0

10.1007/978-3-642-88396-5

10.1111/1467-9965.00059

Slater, 1960, Confluent Hypergeometric Functions

10.1017/S0001867800016773

10.1017/S0001867800016785

10.1287/moor.23.2.257

10.1287/opre.51.2.185.12782

10.1002/jae.3950090104

10.1111/1540-6261.00548

10.2307/1969318

10.1006/jmaa.1994.1429

10.1007/BF00532802

Goldstein R. and Keirstead W. P. (1997). On the term structure of interest rates in the presence of reflecting and absorbing boundaries. Res. Rep. 97-1, Ohio State University.

10.1111/j.0960-1627.2004.00181.x

10.1017/S1074070800008762

Harrison, 1985, Brownian Motion and Stochastic Flow Systems

Karlin, 1981, A Second Course in Stochastic Processes

10.2307/2937922

10.1002/mana.3211480102

10.1215/S0012-7094-51-01806-6

Levitan, 1950, Expansion in Characteristic Functions of Differential Equations of the Second Order

10.21314/JCF.2004.120

10.1007/s00780-003-0120-5

10.1287/opre.1040.0113

10.1017/S0021900200014339

Ricciardi, 1987, On the probability densities of an Ornstein–Uhlenbeck process with a reflecting boundary, J. Appl. Prob., 24, 355, 10.2307/3214260

10.1145/229493.229496

10.2307/1911242

10.1023/B:CSEM.0000049491.13935.af

10.1023/A:1024403704190

10.1287/opre.1040.0136

Levitan, 1975, Introduction to Spectral Theory

Wong, 1964, Proc. Symp. Appl. Math., 264