The Davis-Varaiya Multiplicity and Branching Numbers of a Filtration

We give a functional description of the space of stochastic integrals with respect to a given family of martingales, based on the notion of direct integral of Hilbert spaces. We define the multiplicity function of a filtration and show, using our Hilbert space construction, that the multiplicity function is equal to the rank of the quadratic covariance matrix of the generating martingales. In the discrete time case the multiplicity function coincides with the branching multiplicity of the event tree. We also show that the Davis and Varaiya (1974) martingale types can be realized through linear combinations of original martingales without stochastic integration