On martingale inequalities in non-commutative stochastic analysis

We develop a non-commutative L-p stochastic calculus for the Clifford stochastic integral, an L-2 theory of which has been developed by Barnett, Streater, and Wilde. The main results are certain non-commutative L-p inequalities relating Clifford integrals and their integrands. These results are applied io extend the domain of the Clifford integral from L-2 to L-1 integrands, and we give applications to optional stopping of Clifford martingales, proving an analog of a Theorem of Burkholder: The stopped Clifford process F-T has zero expectation provided E root T < infinity. 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. (C) 1998 Academic Press.