Stochastic integrals in an arbitrary probability gauge space

by C. Barnett, R. F. Streater and I. F. Wilde; Math. Proc. Camb. Phil. Soc., 94, 541-551, 1983.

ABSTRACT

We consider an increasing family of finite von Neumann algebras {A(alpha)}, a non-commutative version of the idea of a filtration in measure theory. A quantum stochastic process is taken to be a family {X(alpha)} of operators, where X(alpha) lies in A(alpha), so that the family is adapted to the filtration. Let M(alpha) be the conditional expectation from L^p(A(infty)) onto L^p(A(alpha)). A process X(alpha) is called a martingale if

M(beta) X(alpha)=X(beta), when beta is less than alpha.

In a previous paper we showed if an L^2 martingale obeys "condition D" then we can define the stochastic integral of an adapted process with respect to the martingale. This definition made use of a Doob-Meyer decomposition. In this paper we extend these results and give a definition of integral that does not employ condition D.