Stochastic regression in terms of Brownian motion

Abstract

Given a continuous stochastic process (X _t ) _t∈[0,T], this article provides, in the first part, a stochastic process that is the best mean square approximation of the form , with W _t Brownian motion. The function coefficients a(t) and b(t) depend on the process X _t and are calculated in the case of several classical examples. In the second part, we extend the method for mean square approximations of the form . We also present simulations for each example, and show that replacing by the martingale is a more natural framework for the problem.

Keywords:

• stochastic process
• Brownian motion
• regression
• covariance
• best fit
• martingale

AMS Subject Classifications::

• 60G07
• 62M10

Acknowledgements

This research was partially supported by the NSF grant #0631541 and by the Hong Kong RGC grant #600607. Most of this material was finalized during the summer of 2009 when Ovidiu Calin visited the Hong Kong University of Science and Technology.