Constructing functions which converge in measure to the solutions of certain nonlinear PDEs and integro-differential equations
by
Senem Alkan
Universita Bocconi
Coauthors: Rod Freed (California State University)

It is well known (see Bass (1998)) that the heat equation with convection, Poisson's equation, the Dirichlet problem, some PDEs involving Schrodinger operators, and some nonlinear PDEs (see Dynkin (2004)) can be solved by probabilistic methods involving conditional expectation. To do so, one utilizes the properties of the conditional probability measure on the space of sample functions of a stochastic process.

However, if the functional forms of the coefficients (which will generally be functions of the independent variables) are unknown, then the necessary conditional expectations cannot be computed. And since the forms of the coefficients are unknown, numerical methods cannot be used.

In an earlier paper, we show that if observations on the system of interest are available, then we can produce estimators of the solutions to any of the above-mentioned PDEs, despite the fact that the functional forms of the coefficients are unknown. These estimators converge in measure to the actual solutions of the PDEs.

In this paper, we extend the analysis to a larger collection of measures. This permits us to apply our procedure to a larger collection of PDEs (e.g., some nonlinear Poisson equations and some nonlinear Schrodinger equations) as well as some integro-differential equations. Once again, the functional forms of the coefficients need not be known. And once again, the estimators of the solutions converge in measure to the actual solutions.

We present applications

Date received: March 7, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # canu-14.