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

Kardaras, Constantinos ORCID: 0000-0001-6903-4506 (2024) Stochastic integration with respect to arbitrary collections of continuous semimartingales and applications to mathematical finance. Annals of Applied Probability, 34 (3). 2566 - 2599. ISSN 1050-5164

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Abstract

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.

Item Type:	Article
Official URL:	https://projecteuclid.org/journals/annals-of-appli...
Additional Information:	© 2024 Institute of Mathematical Statistics
Divisions:	Statistics
Subjects:	H Social Sciences > HA Statistics
Date Deposited:	14 Dec 2023 16:09
Last Modified:	01 Nov 2025 07:33
URI:	http://eprints.lse.ac.uk/id/eprint/121057

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