A Stochastic Two-Point Boundary Value Problem
S. J. Luo, FinancialCAD Corp.
John B. Walsh, University of British Columbia
Abstract
We investigate the two-point stochastic boundary-value problem
on $[0,1]$:
| U'' | = | f(U) dot{W} + g(U,U')
|
| U(0) | = | xi | (0)
|
| U(1) | = | eta.
|
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 $eta$, (0) 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.
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