A Stochastic Two-Point Boundary Value Problem

S. J. Luo (FinancialCAD Corp.)
John B. Walsh (University of British Columbia)


We investigate the two-point stochastic boundary-value problem on $[0,1]$: \begin{equation}\label{1} \begin{split} U'' &= f(U)\dot W + g(U,U')\\ U(0) &= \xi\\ U(1)&= \eta. \end{split} \tag{1} \end{equation} where $\dot W$ is a white noise on $[0,1]$, $\xi$ and $\eta$ are random variables, and $f$ and $g$ are continuous real-valued functions. This is the stochastic analogue of the deterministic two point boundary-value problem, which is a classical example of bifurcation. We find that if $f$ and $g$ are affine, there is no bifurcation: for any r.v. $\xi$ and $\eta$, (1) has a unique solution a.s. However, as soon as $f$ is non-linear, bifurcation appears. We investigate the question of when there is either no solution whatsoever, a unique solution, or multiple solutions. We give examples to show that all these possibilities can arise. While our results involve conditions on $f$ and $g$, we conjecture that the only case in which there is no bifurcation is when $f$ is affine.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-32

Publication Date: September 14, 2001

DOI: 10.1214/EJP.v7-111


  1. Dembo, A. and Zeitouni O., A change of variables formula for Stratonovich integrals and existence of solutions for two-point stochastic boundary value problems, Prob. Th. Rel Fields 84 (1990), 411-425.
    MR 91h:60061
  2. Krener, A., Reciprocal processes and the stochastic realization problem of a causal system, in Byrnes, C.Z., Lindquist, E. (Eds) Modelling, Identification, and Robust Control. Elsevier/Noble, Holland, 1986.
    MR 89c:93065
  3. Kunita, H., Stochastic differential equations and stochastic flow of diffeomorphisms, Lecture Notes in Math., 1047, Springer-Verlag 1984, 144-300.
    MR 87m:60127
  4. Nualart, D. and Pardoux, E., Second order stochastic differential equations with Dirichet boundary conditions, Stochastic Proc. Appl., 34 (1991) 1-24.
    MR 93b:60123
  5. Nualart, D. and Pardoux, E., Stochastic calculus with anticipative integrals, Prob. Th. Rel. Fields, 78 (1988) 535-581.
    MR 89h:60089
  6. Nualart, D. and Zakai, M., Generalized stochastic integrals and the Malliavin calculus, Prob. Th. Rel. Fields 73, (1986) 255-280.
    MR 88h:60110
  7. Ocone, D. and Pardoux, E., (1989) Linear stochastic differential equations with boundary conditions, Prob. Th. Rel. Fields 82 (1989) 489-526.
    MR 91a:60154
  8. Schaaf, R., Global solution branches of two-point boundary value problems, Lecture Notes in Math. 1458, 1990.
    MR 92a:34003
  9. Watanabe, S., Lectures on stochastic differential equations and Malliavin calculus, Springer Verlag, Berlin/Heidelberg/New York/Tokyo, 1984.
    MR 86b:60113

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.