Aurélien Deya

A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Levy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Levy area terms.

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 Holder exponent γ > 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 Holder exponent 1/3 1/3.

Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise

We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter

On the rough-paths approach to non-commutative stochastic calculus

H\in (1/3,1)

On stochastic calculus with respect to q-Brownian motion

and multiplicative noise component

On L2 modulus of continuity of Brownian local times and Riesz potentials

\sigma

Integration with Respect to the Hermitian Fractional Brownian Motion

. When

Non-linear rough heat equations

\sigma

Convergence of Wigner integrals to the tetilla law

is constant and for every

Rough Volterra equations 2: Convolutional generalized integrals

H\in (0,1)