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First-Passage Distributions of Bidimensional Processes

Published online by Cambridge University Press:  27 July 2009

Mario Lefebvre
Mario Lefebvre
Affiliation:
École Polytechnique de Montréal, Département de Mathématiques et de Génie Industriel, Ecole Polytechnique, C.P. 6079, Succ. Centre-ville, Montréal, Québec, Canada, H3C 3A7
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Abstract

Bidimensional processes defined by dx(t) = ρ(x, y) dt and dy(t) = f(y) dt + σ(y) dW(t), where W(t) is a Wiener process, are considered. Let T(x, y, ξ) = inf[t ≥ 0: x(t] = ξ| x(0) = x, y(0) = y). Explicit expressions for the moment generating function of T(x, y, 0) and for the characteristic function of y(T(x, y, ξ)) are obtained in two special cases. The method of similarity solutions is used. Applications to optimal control problems are presented.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 3 , July 1996 , pp. 325 - 339
Copyright
Copyright © Cambridge University Press 1996

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References

1.Abramowitz, M. & Stegun, I.A. (1965). Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York: Dover.Google Scholar
2.Arnold, L. (1974). Stochastic differential equations: Theory and applications. New York: Wiley.Google Scholar
3.Crandall, M.G. & Lions, P.L. (1983). Viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society 277: 1–42.Google Scholar
4.Fleming, W.H. & Soner, H.M. (1993). Controlled Markov processes and viscosity solutions. New York: Springer-Verlag.Google Scholar
5.Hesse, C.H. (1991). The one-sided barrier problem for an integrated Ornstein-Uhlenbeck process. Communications in Statistics—Stochastic Models 7: 447–480.CrossRefGoogle Scholar
6.Kannan, D. (1979). An introduction to stochastic processes. New York: North Holland.Google Scholar
7.Karlin, S. & Taylor, H. (1981). A second course in stochastic processes. New York: Academic Press.Google Scholar
8.Lachal, A. (1990). Sur l'intégrale du mouvement brownien. Comptes Rendus de l'Académie des Sciences de Paris, Série I: Mathématiques 311: 461–464.Google Scholar
9.Lefebvre, M. (1989). First-passage densities of a two-dimensional process. SIAM Journal on Applied Mathematics 49: 1514–1523.CrossRefGoogle Scholar
10.Lefebvre, M. (1989). Moment generating function of a first hitting place for the integrated Ornstein-Uhlenbeck process. Stochastic Processes and Their Applications 32: 281–287.CrossRefGoogle Scholar
11.Lefebvre, M. (1994). First-passage problems involving processes with lognormal density functions (to appear in Rendiconti dell'Istituto Lombardo (Scienze—Serie A)).Google Scholar
12.Lefebvre, M. (1995). On the inverse of the first hitting time problem for bidimensional processes (to appear in Journal of Applied Probability).Google Scholar
13.Lions, P.L. (1983). Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations II: Viscosity solutions and uniqueness. Communications in Partial Differential Equations 8: 1229–1276.CrossRefGoogle Scholar
14.Rishel, R. (1991). Controlled wear process: Modeling optimal control. IEEE Transactions on Automatic Control 36: 1100–1102.CrossRefGoogle Scholar
15.Whittle, P. (1982). Optimization over time. Vol. I. Chichester: Wiley.Google Scholar