On Martingale Inequalities in Non-commutative Stochastic Analysis

We develop a non-commutativeLpstochastic calculus for the Clifford stochastic integral, anL2theory of which has been developed by Barnett, Streater, and Wilde. The main results are certain non-commutativeLpinequalities relating Clifford integrals and their integrands. These results are applied to extend the domain of the Clifford integral fromL2toL1integrands, and we give applications to optional stopping of Clifford martingales, proving an analog of a Theorem of Burkholder: The stopped Clifford processFThas zero expectation providedET<∞. In proving these results, we establish a number of results relating the Clifford integral to the differential calculus in the Clifford algebra. In particular, we show that the Clifford integral is given by the divergence operator, and we prove an explicit martingale representation theorem. Both of these results correspond closely to basic results for stochastic analysis on Wiener space, thus furthering the analogy between the Clifford process and Brownian motion.