On the stochastic calculus method for spins systems

In this note we show how to generalize the stochastic calculus method introduced by Comets and Neveu [Comm. Math. Phys. 166 (1995) 549-564] for two models of spin glasses, namely, the SK model with external field and the perceptron model. This method allows to derive quite easily some fluctuation results for the free energy in those two cases.Comment: Published at http://dx.doi.org/10.1214/009117904000000919 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rough Volterra equations 1: the algebraic integration setting

We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with H\"older exponent greater than 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coefficient. In case of a driving signal with H\"older exponent in (1/3,1/2], we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coefficient H>1/3.Comment: 31 page

A construction of the rough path above fractional Brownian motion using Volterra's representation

This note is devoted to construct a rough path above a multidimensional fractional Brownian motion $B$ with any Hurst parameter $H\in(0,1)$, by means of its representation as a Volterra Gaussian process. This approach yields some algebraic and computational simplifications with respect to [Stochastic Process. Appl. 120 (2010) 1444--1472], where the construction of a rough path over $B$ was first introduced.Comment: Published in at http://dx.doi.org/10.1214/10-AOP578 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise

We study a least square-type estimator for an unknown parameter in the drift coefficient of a stochastic differential equation with additive fractional noise of Hurst parameter H>1/2. The estimator is based on discrete time observations of the stochastic differential equation, and using tools from ergodic theory and stochastic analysis we derive its strong consistency.Comment: 15 page

Some differential systems driven by a fBm with Hurst parameter greater than 1/4

This note is devoted to show how to push forward the algebraic integration setting in order to treat differential systems driven by a noisy input with H\"older regularity greater than 1/4. After recalling how to treat the case of ordinary stochastic differential equations, we mainly focus on the case of delay equations. A careful analysis is then performed in order to show that a fractional Brownian motion with Hurst parameter H>1/4 fulfills the assumptions of our abstract theorems.Comment: 32 page