Titel
Stochastic integration and differential equations for typical paths
Autor*in
Michael Kupper
Department of Mathematics, University of Konstanz
Autor*in
Ariel Neufeld
Division of Mathematical Sciences, NTU Singapore
Abstract
The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure. In the spirit of [33, 37] and motivated by the pricing duality result obtained in [4] we introduce an outer measure as a variant of the pathwise minimal superhedging price where agents are allowed to trade not only in ω but also in ∫ωdω:=ω2−⟨ω⟩ and where they are allowed to include beliefs in future paths of the price process expressed by a prediction set. We then call a property to hold true on typical paths if the set of paths where the property fails is null with respect to our outer measure. It turns out that adding the second term ω2−⟨ω⟩ in the definition of the outer measure enables to directly construct stochastic integrals which are continuous, even for typical paths taking values in an infinite dimensional separable Hilbert space. Moreover, when restricting to continuous paths whose quadratic variation is absolutely continuous with uniformly bounded derivative, a second construction of model-free stochastic integrals for typical paths is presented, which then allows to solve in a model-free way stochastic differential equations for typical paths.
Stichwort
Föllmer integrationpathwise stochastic integralpathwise SDEinfinite dimensional stochastic calculusVovk’s outer measure
Objekt-Typ
Sprache
Englisch [eng]
Persistent identifier
https://phaidra.univie.ac.at/o:1144050
Erschienen in
Titel
Electronic Journal of Probability
Band
24
ISSN
1083-6489
Erscheinungsdatum
2019
Verlag
Institute of Mathematical Statistics
Projektnummer
P28861 – Austrian Science Fund (FWF)
Erscheinungsdatum
2019
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