The Ito-Clifford Integral

by C. Barnett, R. F. Streater and I. F. Wilde

ABSTRACT

We consider a quantum analogue of a filtration; it is a family A(alpha) of hyperfinite type II(1) von Neumann algebras constructed using the Clifford algebra over the Hilbert space L^2(R^+). We define martingales, and stochastic integrals relative to martingales, and obtain a Doob-Meyer decomposition. We also prove an analogue of the Kunita-Watanabe theorem on square-integrable martingales, which here says that any square-integrable martingale can be written as the sum of terms, each being a polynomial martingale.

REMARKS

The Ito-Clifford process appears in the problem of quantisation of a classical system under constraints. The required analysis appears in A Feynman-Kac Formula for Anticommuting Brownian Motion, by Stephen Leppard and Alice Rogers, ArXiv:quant-ph/0008081.

Stopping times in quantum processes have been defined by Barnett and Wilde, Barnett Thakrar; see Attal for these references and further work on quantum stopping times.