It is well known that the quadratic Wasserstein distance W 2 (·, ·) is formally equivalent, for infinitesimally small perturbations, to some weighted H −1 homogeneous Sobolev norm. In this article I show that this equivalence can be integrated to get non-asymptotic comparison results between these distances. Then I give an application of these results to prove that the W 2 distance exhibits some localisation phenomenon: if µ and ν are measures on R n and ϕ : R n → R + is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between ϕ · µ and ϕ · ν by an explicit multiple of W 2 (µ, ν). ForewordThis article is divided into two sections, each of which having its own introduction.§ 1 deals with general results of comparison between Wasserstein distance and homogeneous Sobolev norm, while § 2 handles an application to localisation of W 2 distance.

We continue the analysis of our previous paper [13] pertaining to the existence of a shadow price process for portfolio optimisation under proportional transaction costs. There, we established a positive answer for a continuous price process S = (S t ) 0≤t≤T satisfying the condition (N U P BR) of "no unbounded profit with bounded risk". This condition requires that S is a semimartingale and therefore is too restrictive for applications to models driven by fractional Brownian motion. In the present paper, we derive the same conclusion under the weaker condition (T W C) of "two way crossing", which does not require S to be a semimartingale. Using a recent result of R. Peyre, this allows us to show the existence of a shadow price for exponential fractional Brownian motion and all utility functions defined on the positive half-line having reasonable asymptotic elasticity. Prime examples of such utilities are logarithmic or power utility. MSC 2010 Subject Classification: 91G10, 93E20, 60G48 JEL Classification Codes: G11, C61

For A and B two σ-algebras, the ρ-mixing coefficient ρ(A, B) between A and B is the supremum correlation between two real random variables X and Y being resp. A-and B-measurable; the τ (A, B) coefficient is defined similarly, but restricting to the case where X and Y are indicator functions. It has been known for long that the bound ρ Cτ (1 + |log τ |) holds for some constant C; in this article, I show that C = 1 fits and that that value cannot be improved. |Cov(X, Y)| Var(X) 1/2 Var(Y) 1/2 (where the supremum is taken only for non-constant X and Y). This coefficient is 0 if and only if A and B are independent; and we will say that A and B are as correlated (in the ρ-mixing sense) as ρ(A, B) is large. Note that one always has ρ(A, B) 1, because of the Cauchy-Schwarz inequality. There are other ways to measure dependence between A and B (see for instance the review paper [2]): in particular, rather than looking at correlation between A-and B-measurable random variables, we can look at correlation between events. The most classical measure of dependence in this category is the α-mixing coefficient: (2) α(A, B) := sup