10,788 research outputs found
The maximum maximum of a martingale with given marginals
We obtain bounds on the distribution of the maximum of a martingale with
fixed marginals at finitely many intermediate times. The bounds are sharp and
attained by a solution to -marginal Skorokhod embedding problem in
Ob{\l}\'oj and Spoida [An iterated Az\'ema-Yor type embedding for finitely many
marginals (2013) Preprint]. It follows that their embedding maximizes the
maximum among all other embeddings. Our motivating problem is superhedging
lookback options under volatility uncertainty for an investor allowed to
dynamically trade the underlying asset and statically trade European call
options for all possible strikes and finitely-many maturities. We derive a
pathwise inequality which induces the cheapest superhedging value, which
extends the two-marginals pathwise inequality of Brown, Hobson and Rogers
[Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by
elementary arguments, is derived by following the stochastic control approach
of Galichon, Henry-Labord\`ere and Touzi [Ann. Appl. Probab. 24 (2014)
312-336].Comment: Published at http://dx.doi.org/10.1214/14-AAP1084 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Symmetry-breaking phase transition in a dynamical decision model
We consider a simple decision model in which a set of agents randomly choose
one of two competing shops selling the same perishable products (typically
food). The satisfaction of agents with respect to a given store is related to
the freshness of the previously bought products. Agents select with a higher
probability the store they are most satisfied with. Studying the model from a
statistical physics perspective, both through numerical simulations and
mean-field analytical methods, we find a rich behaviour with continuous and
discontinuous phase transitions between a symmetric phase where both stores
maintain the same level of activity, and a phase with broken symmetry where one
of the two shops attracts more customers than the other.Comment: 13 pages, 6 figures, submitted to JSTA
The quest for the ultimate anisotropic Banach space
We present a new scale (with and ) of
anisotropic Banach spaces, defined via Paley-Littlewood, on which the transfer
operator associated to a hyperbolic dynamical system has good spectral
properties. When and is an integer, the spaces are analogous to the
"geometric" spaces considered by Gou\"ezel and Liverani. When and
, the spaces are somewhat analogous to the geometric
spaces considered by Demers and Liverani. In addition, just like for the
"microlocal" spaces defined by Baladi-Tsujii, the spaces are
amenable to the kneading approach of Milnor-Thurson to study dynamical
determinants and zeta functions.
In v2, following referees' reports, typos have been corrected (in particular
(39) and (43)). Section 4 now includes a formal statement (Theorem 4.1) about
the essential spectral radius if (its proof includes the content of
Section 4.2 from v1). The Lasota-Yorke Lemma 4.2 (Lemma 4.1 in v1) includes the
claim that is compact.
Version v3 contains an additional text "Corrections and complements" showing
that s> t-(r-1) is needed in Section 4.Comment: 31 pages, revised version following referees' reports, with
Corrections and complement
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