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Annealed functional CLT in the rough path topology is proved for the standard class of ballistic random walks in random environment. Moreover, the `area anomaly', i.e. a deterministic linear correction for the second level iterated integral of the rescaled path, is identified in terms of a stochastic area on a regeneration interval. The main theorem is formulated in more general settings, namely for any discrete process with uniformly bounded increments which admits a regeneration structure where the regeneration times have finite moments. Here the largest finite moment translates into the degree of regularity of the rough path topology. In particular, the convergence holds in the $\alpha$-H\"older rough path topology for all $\alpha, Comment: 15 pages, 2 figures

Consider a multidimensional diffusion process $X=\{X(t):t\in [0,1]\}$. Let $\varepsilon>0$ be a deterministic, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of $X$, we construct a probability space, supporting both $X$ and an explicit, piecewise constant, fully simulatable process $X_{\varepsilon}$ such that ¶ \[\sup_{0\leq t\leq1}\Vert X_{\varepsilon}(t)-X(t)\Vert_{\infty}

In this article, we give sharp bounds for the Euler- and trapezoidal discretization of the Levy area associated to a d-dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean-square convergence rate of the Euler scheme. For H3/4 the exact rate is n^{-1}. Moreover, the trapezoidal scheme has exact convergence rate n^{-2H+1/2} for H>1/2. Finally, we also derive the asymptotic error distribution of the Euler scheme. For H lesser than 3/4 one obtains a Gaussian limit, while for H>3/4 the limit distribution is of Rosenblatt type., 28 pages

We construct a Gibbs measure for the nonlinear Schrodinger equation (NLS) on the circle, conditioned on prescribed mass and momentum: d \mu_{a,b} = Z^{-1} 1_{\int_T |u|^2 = a} 1_{i \int_T u \bar{u}_x = b} exp (\pm1/p \int_T |u|^p - 1/2 \int_{\T} |u|^2) d P for a \in R^+ and b \in R, where P is the complex-valued Wiener measure on the circle. We also show that \mu_{a,b} is invariant under the flow of NLS. We note that i \int_\T u \bar{u}_x is the Levy stochastic area, and in particular that this is invariant under the flow of NLS., Comment: 17 pages. An error in Subsec. 2.1 is corrected (see Prop. 2.2.) Also, an argument in Subsec. 2.3 is simplified. To appear in J. Math. Soc. Japan

Let { X 1 ( t ) } 0 ≤ t ≤ 1 and { X 2 ( t ) } 0 ≤ t ≤ 1 be two independent continuous centered Gaussian processes with covariance functions R 1 and R 2 . We show that if the covariance functions are of finite p -variation and q -variation respectively and such that p − 1 + q − 1 > 1 , then the Levy area can be defined as a double Wiener–Ito integral with respect to an isonormal Gaussian process induced by X 1 and X 2 . Moreover, some properties of the characteristic function of that generalised Levy area are studied.

In this note we introduce the notion of Newton--C\^{o}tes functionals corrected by L\'{e}vy areas, which enables us to consider integrals of the type $\int f(y) \mathrm{d}x,$ where $f$ is a ${\mathscr{C}}^{2m}$ function and $x,y$ are real H\"{o}lderian functions with index $\alpha>1/(2m+1)$ for all $m\in {\mathbb{N}}^*.$ We show that this concept extends the Newton--C\^{o}tes functional introduced in Gradinaru et al., to a larger class of integrands. Then we give a theorem of existence and uniqueness for differential equations driven by $x$, interpreted using the symmetric Russo--Vallois integral., Comment: Published at http://dx.doi.org/10.3150/07-BEJ6015 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

http://www.math.washington.edu/~ejpecp/; International audience; We study the limit of functionals of stochastic processes for which an homogenization result holds. All these functionals involve stochastic integrals. Among them, we consider more particularly the Lévy area and those giving the solutions of some SDEs. The main question is to know whether or not the limit of the stochastic integrals is equal to the stochastic integral of the limit of each of its terms. In fact, the answer may be negative, especially in presence of a highly oscillating first-order differential term. This provides us some counterexamples to the theory of good sequence of semimartingales.