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DTSTAMP:20240304T171251Z
UID:1700496000@ist.ac.at
DTSTART:20231120T170000
DTEND:20231120T180000
DESCRIPTION:Speaker: Benjamin Robinson\nhosted by M. Beiglböck\, N. Berest
ycki\, L. Erdös\, J. Maas\, F. Toninelli\, E. Schertzer\nAbstract: Given
a family of probability measures increasing in convex order\, does there e
xist a Markov martingale with the corresponding marginal laws? This que
stion was answered positively by Strassen in 1965 in discrete time on a ge
neral space\, and by Kellerer in 1972 in continuous time in dimension one.
The multidimensional continuous-time case\, however\, remained open until
recently.In this talk I present a multidimensional extension of Kellerer'
s theorem. In particular I show that\, after applying some Gaussian regula
rization to the given measures\, there exists a strongly Markovian marting
ale Itô diffusion with these marginals. The proof of this result makes us
e of the Bass martingales that arise in martingale optimal transport\, as
well as a mimicking theorem for Itô processes\, and a novel compactness r
esult for martingale diffusions. In dimensions two and higher\, I show tha
t uniqueness does not hold in general. In particular\, there exists a two-
dimensional strong Markov martingale with continuous paths that has the sa
me marginal laws as a Brownian motion and yet is not itself a Brownian mot
ion. Moreover\, I present counterexamples showing that the existence of Ma
rkov martingales with given marginal distributions may also fail in higher
dimensions. Thus\, Kellerer's theorem cannot be extended to higher dimens
ions in full generality. Joint work with Gudmund Pammer (ETH Zürich) and
Walter Schachermayer (Universität Wien).
LOCATION:Mondi 2 (I01.01.008)\, Central Building\, ISTA
ORGANIZER:birgit.oosthuizen-noczil@ist.ac.at
SUMMARY:Benjamin Robinson: A regularized Kellerer theorem in arbitrary dime
nsion
URL:https://talks-calendar.ista.ac.at/events/4605
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