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

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  • Kardaras, Constantinos

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.

Suggested Citation

  • Kardaras, Constantinos, 2024. "Stochastic integration with respect to arbitrary collections of continuous semimartingales and applications to mathematical finance," LSE Research Online Documents on Economics 121057, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:121057
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    References listed on IDEAS

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    More about this item

    Keywords

    infinite-dimensional stochastic integration; continuous semimartingales; mathematical finance; fundamental theorem;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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