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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các thiếu Ito-Clifford. IV: Một Tế-Nikodym định lý và khung Processes. | Copyright by INCREST 1984 J. OPERATOR THEORY 11 1984 255-271 THE IT0-CLIFFORD INTEGRAL. IV A RADON-NIKODYM THEOREM AND BRACKET PROCESSES c. BARNETT R. F. STREATER and I. F. WILDE 0. INTRODUCTION The construction and various properties of the Ito-Clifford stochastic integral have been discussed in 1 2 3 . In particular it was shown in 1 that any centred t L2-martingale is given as an Ito-Clifford stochastic integral xt Y s dlPs 0 where Ts 0 s is the Fermi-field. It was also shown in 1 that stochastic integrals of the form dx can be defined as elements of L2 7T the non-commu-tative L2-space associated with the Clifford probability gage space. We consider the relationship between the stochastic integral with respect to and that with respect to X. Specifically we prove a Radon-Nikodym theorem in the form ị dX Tdff . Using the Doob-Meyer decomposition of the submartingale x x given in 1 we define the pointed-bracket jU-process Xt Yty associated with z 2-martin-gales Xt and T . The stochastic integral àx is shown to be characterized as a process in terms of pointed-bracket processes. These results parallel those of standard i.e. commutative probability theory see for example 8 9 . In Sections 1 and 2 we review and generalize some of the results from 1 The stochastic integral dAf is defined in Section 3 this being a simplified version of that in 1 The Radon-Nikodym theorem is presented in Section 4 and in Section 5 an analogous result for stochastic integrals with respect to Wick martingales is proved. The pointed-bracket process is considered in Section 6 together with a characterization of the stochastic integral as a process. 256 c. BARNETT R. F. STREATER and 1. F. WILDE Finally in Section 7 we give a summary of the analogous results valid for left rather than right integrals. A Doob-Meyer decomposition and stochastic integration with respect to martingales over an arbitrary probability gage space is considered in 4 1. FOCK SPACE AND THE CLIFFORD ALGEBRA We recall .
