Topological Solutions of Noncommutative Stochastic Differential Equations

Abstract

A general approach is introduced to studying the properties of solutions of an arbitrary noncommutative stochastic differential equation (NSDE) in several interesting locally convex operator topologies, grouped into two main sets comprising the strong/λ^⋆-topologies and the weak topologies. Results concerning the existence and uniqueness of solutions in these topologies are established. The approach is based on two reformulations of the NSDE, corresponding to the two sets of topologies, and is well-suited for characterizing, both analytically and numerically, various topological features of the solutions of an NSDE.

Keywords:

• Existence and uniqueness of solutions
• Itô calculus
• Lipschitzian maps
• Locally convex operator topologies
• Noncommutative stochastic differential equations
• Noncommutative stochastic integration
• Semimartingales
• Semimartingale inequalities

AMS Subject Classification 2000:

• 46L51
• 46L53
• 46L89
• 81R15

Acknowledgments

This work was done during my visit as a Senior Associate to the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy. The author is grateful to the Swedish International Development Cooperation Agency (SIDA) for proving the funds for my visit, and also to the Director of the ICTP, Professor K. R. Sreenivasan, the Head of the Mathematics Section of the ICTP, Professor Le Dung Trang, Professor Charles Chidume, and other staff of the ICTP, for their hospitality. Finally, the author thank the referee for his/her comments and suggestions.