Stochastic analysis on differential structures

Abstract

In most of the preceding work on stochastic integral and differential analysis, the study was based on the real line as its range space as well as its parameter set. However, modern developments demand that we consider the analogous theory on differential structures, more general than the real line. In this final chapter we first discuss the conformal properties of martingales with values in the complex plane and then devote Section 2 for a formulation and an extension of the work on certain (differential or smooth) manifolds, leading to Malliavin’s approach to the hypoellipticity problem. The study continues in Section 3 by discussing semimartingales with ℝⁿ, n ≥ 2, as the parameter set. There are substantial difficulties in this extension due to a lack of linear ordering. We observe that, even here, the L^2,2-boundedness principle relative to a (σ-finite) measure is at work and corresponding stochastic integrals can be defined. Finally we indicate in Section 4 how one considers stochastic partial differential equations in this context. Then some complements to the preceding work are given, as in earlier chapters, in the last section to conclude this monograph.