Pathwise construction of stochastic integrals

By Marcel Nutz. 2011

111 citations recorded by the source corpus.

We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Path-by-path, and without referring to a probability measure, we construct a sequence of Lebesgue-Stieltjes integrals whose medial limit coincides with the usual stochastic integral under essentially any probability measure such that the integrator is a semimartingale. This method applies to any predictable integrand.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/888ec6cf5a89829b467c9cf685004da60d7709c3

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In Management Science and Operations Research

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Citations in the corpus

The maximum maximum of a martingale with given $\mathbf{n}$ marginals (2012, unknown): “Consider a model P ∈ P which is calibrated to the market, i.e.”
Superreplication under Volatility Uncertainty for Measurable Claims (2012, background): “ts. 1 Here R HdB is, with some abuse of notation, the usual Itô integral under the ﬁxed measure P. We remark that we could also deﬁne the integral simultaneously under all P ∈P by the construction of [10]. This would yield a cosmetically nicer result, but we shall avoid the additional set-theoretic ”
Stochastic control for a class of nonlinear kernels and applications (2015, unknown): “(ii) Finally, using Proposition 5.1 of Neufeld and Nutz [60], we can then construct a process H P t ( ω ) satisfying the required conditions.”
ROBUST FUNDAMENTAL THEOREM FOR CONTINUOUS PROCESSES (2014, background): “Given H ∈ L(P) and P ∈ P, we can construct the usual stochastic integral H • S under P (the dependence on P is suppressed in the notation—but see also [33]).”
Pathwise stochastic integrals for model free finance (2013, result): “A more general pathwise construction of the Itô integral was given in [34], but it suffers from the same drawbacks with respect to applications in model free finance.”
Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix (2016, unknown): “2 in [27] under the Zermelo Fraenkel set theory with axiom of choice (ZFC) plus the Continuum Hypothesis.”