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We study the windings of the planar Brownian motion around points, following the previous works of Wendelin Werner in particular. In the first chapter, we motivate this study by the one of smoother curves. We prove in particular a Green formula for Young integration, without simplicity assumption for the curve. In the second chapter, we study the area of the set of points around which the Brownian motion winds at least N times. We give an asymptotic estimation for this area, up to the second order, both in the almost sure sense and in the Lp spaces, when N goes to infinity.The third chapter is devoted to the proof of a result which shows that the points with large winding are distributed in a very balanced way along the trajectory. In the fourth chapter, we use the results from the two previous chapters to give a new Green formula for the Brownian motion. We also study the averaged winding around randomly distributed points in the plan. We show that, almost surely for the trajectory, this averaged winding converges in distribution, not toward a constant (which would be the Lévy area), but toward a Cauchy distribution centered at the Lévy area. In the last two chapters, we apply the ideas from the previous chapters to define and study the Lévy area of the Brownian motion, when the underlying area measure is not the Lebesgue measure anymore, but instead a random and highly irregular measure. We deal with the case of the Gaussian multiplicative chaos in particular, but the tools can be used in a much more general framework., On s'intéresse dans cette thèse à l'enlacement du mouvement Brownien plan autour des points, dans la succession des travaux de Wendelin Werner en particulier. Dans le premier chapitre, on motive cette étude par celle du cas des courbes plus lisses que le mouvement Brownien. On y démontre notamment une formule de Green pour l'intégrale de Young, sans hypothèse de simplicité de la courbe. Dans le chapitre 2, on étudie l'aire de l'ensemble des points autour desquels l'enlacement du mouvement brownien est plus grand que N. On donne, au sens presque sûr et dans les espaces Lp, une estimation asymptotique au second ordre de cette aire lorsque N tend vers l'infini. Le chapitre 3 est dévoué à la preuve d'un résultat qui montre que les points de grands enlacements se répartissent de manière très équilibrée le long de la trajectoire. Dans le chapitre 4, on utilise les résultats des deux précédents chapitres pour énoncer une formule de Green pour le mouvement brownien. On étudie aussi l'enlacement moyen de points répartis aléatoirement dans le plan. On montre que cet enlacement moyen converge en distribution (presque surement pour la trajectoire), non pas vers une constante (qui serait alors l’aire de Lévy) mais vers une variable de Cauchy centrée en l’aire de Lévy. Dans les deux derniers chapitres, on applique les idées des précédents chapitres pour définir et étudier l’aire de Lévy du mouvement Brownien lorsque la mesure d’aire sous-jacente n’est plus la mesure de Lebesgue mais une mesure aléatoire particulièrement irrégulière. On traite le cas du chaos multiplicatif gaussien en particulier, mais la méthode s’applique dans un cadre plus général.

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

In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for multi-dimensional SDEs driven by Brownian motions. Giles has previously shown that if we combine a numerical approximation with strong order of convergence $O(\Delta t)$ with MLMC we can reduce the computational complexity to estimate expected values of functionals of SDE solutions with a root-mean-square error of $\epsilon$ from $O(\epsilon^{-3})$ to $O(\epsilon^{-2})$. However, in general, to obtain a rate of strong convergence higher than $O(\Delta t^{1/2})$ requires simulation, or approximation, of L\'{e}vy areas. In this paper, through the construction of a suitable antithetic multilevel correction estimator, we are able to avoid the simulation of L\'{e}vy areas and still achieve an $O(\Delta t^2)$ multilevel correction variance for smooth payoffs, and almost an $O(\Delta t^{3/2})$ variance for piecewise smooth payoffs, even though there is only $O(\Delta t^{1/2})$ strong convergence. This results in an $O(\epsilon^{-2})$ complexity for estimating the value of European and Asian put and call options., Comment: Published in at http://dx.doi.org/10.1214/13-AAP957 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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.

We provide sharp exponential moment bounds for (Stratonovich) iterated stochastic integrals under conditioning by certain small balls, including balls in certain Hölder-like norms of exponent greater than 1 /3. The proof uses a control of the variation of the Lévy area,under conditioning. The results are applied to the computation of the Onsager–Machlup functional of diffusion processes with constant diffusion matrix.