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 = (Pi; i ∈ I) of continuous semimartingales, imposing no assumptions on the index set I and the subspace of R I 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 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.