We develop a non-commutative L-p stochastic calculus for the Clifford
stochastic integral, an L-2 theory of which has been developed by Barn
ett, Streater, and Wilde. The main results are certain non-commutative
L-p inequalities relating Clifford integrals and their integrands. Th
ese results are applied io extend the domain of the Clifford integral
from L-2 to L-1 integrands, and we give applications to optional stopp
ing of Clifford martingales, proving an analog of a Theorem of Burkhol
der: 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 integr
al is given by the divergence operator, and we prove an explicit marti
ngale representation theorem. Both of these results correspond closely
to basic results for stochastic analysis on Wiener space, thus furthe
ring the analogy between the Clifford process and Brownian motion. (C)
1998 Academic Press.