Stochastic integration with respect to arbitrary collections of continuous semimartingales and applications to mathematical finance

Stochastic integrals are defined with respect to a collection P = (P i;i ∈ I) of continuous semimartingales, imposing no assumptions on the index set I and the subspace of RI where P takes values. The integrals are constructed though finite-dimensional approximation, identifying the appropriate local geometry that allows extension to infinite dimensions. For local martingale integrators, the resulting space S(P) of stochastic integrals has an operational characterisation via a corresponding set of integrands R(C), constructed with only reference to the covariation structure C of P. This bijection between R(C) and the (closed in the semimartingale topology) set S(P) extends to families of continuous semimartingale integrators for which the drift process of P belongs to R(C). In the context of infinite-asset models in mathematical finance, the latter structural condition is equivalent to a certain natural form of market viability. The enriched class of wealth processes via extended stochastic integrals leads to exact analogues of optional decomposition and hedging duality as the finite-asset case. A corresponding characterisation of market completeness in this setting is provided.