WEAK APPROXIMATIONS FOR STOCHASTIC DIFFERENTIAL

          EQUATIONS WITH BOUNDARY CONDITIONS

                Arturo Kohatsu-Higa

          Universitat Pompeu Fabra, Barcelona


Stochastic differential equations with boundary conditions
are straighforward generalizations of ordinary differential
equations with boundary conditions.
 
Nevertheless, their treatment is completely different due to
the anticipating nature of this type of stochastic equations.
In this talk we are interested in the generalization of the
shooting method to approximate solutions of such equations.
 
To tackle the problem we introduce a variation of the proof
for weak approximations that is suitable for stochastic
processes which are evaluations of the flow generated by
a stochastic differential equation on a random variable that
maybe anticipating.
 
Our main assumption is that the process and the initial
random variable have to be smooth in the Malliavin sense.
Furthermore if the inverse of the Malliavin covariance
matrix associated with the process under consideration
is integrable then approximations for densities and
distributions can also be achieved. We apply these ideas
to the case of stochastic differential equations with
boundary conditions and the composition of two diffusions.