The english translation

Lectures on stochastic differential equations and Malliavin calculus

We apply the techniques of stochastic integration with respect to the fractional Brownian motion and the Gaussian theory of regularity and supremum estimation to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by fractional Brownian motion with any level of H\"{o}lder-regularity (any \emph{Hurst} parameter) We prove existence and strong consistency of the MLE for linear and nonlinear equations We\ also prove that a basic discretized version of the MLE, is still a strongly consistent estimator

/pdf/statistical-aspects-of-the-fractional-stochastic-calculus-3r7pex50e6.pdf

Statistical aspects of the fractional stochastic calculus

https://bibos.math.uni-bielefeld.de/preprints/05-03-177.pdf

On bifractional Brownian motion

Bifractional Brownian motion

We give the Wiener–Itoˆ chaotic decomposition for the local time of the d-dimensional fractional Brownian motion with N-parameters and study its smoothness in the Sobolev–Watanabe spaces.

Regularity of the Local Time for the d-dimensional Fractional Brownian Motion with N-parameters

We establish a Krylov-type estimate and an Ito–Krylov change of variable formula for the solutions of one-dimensional quadratic backward stochastic differential equations (QBSDEs) with a measurable generator and an arbitrary terminal datum. This allows us to prove various existence and uniqueness results for some classes of QBSDEs with a square integrable terminal condition and sometimes a merely measurable generator. It turns out that neither the existence of exponential moments of the terminal datum nor the continuity of the generator are necessary to the existence and/or uniqueness of solutions. We also establish a comparison theorem for solutions of a particular class of QBSDEs with measurable generator. As a byproduct, we obtain the existence of viscosity solutions for a particular class of quadratic partial differential equations (QPDEs) with a square integrable terminal datum.

/pdf/quadratic-bsde-with-mathbb-l-2-terminal-data-krylov-s-p6qnnr2x6m.pdf

Quadratic BSDE with $\mathbb{L}^{2}$-terminal data: Krylov’s estimate, Itô–Krylov’s formula and existence results

This paper deals with some additive functionals of the fractional Brownian motion that arise as limits in law of some occupation times of this process. In concrete, this functionals are obtained via the Cauchy principal value and the Hadamard finite part. We derive some regularity properties of theses functionals in Sobolev-Watanabe sense.

/pdf/chaotic-expansion-and-smoothness-of-some-functionals-of-the-gm4kf6f6r2.pdf

Chaotic expansion and smoothness of some functionals of the fractional Brownian motion

We use the stochastic calculus of variations for the fractional Brownian motion to derive formulas for the replicating portfolios for a class of contingent claims in a Bachelier and a Black-Scholes markets modulated by fractional Brownian motion. An example of such a model is the Black-Scholes process whose volatility solves a stochastic differential equation driven by a fractional Brownian motion that may depend on the underlying Brownian motion.

Hedging options in market models modulated by the fractional Brownian motion

In a first step, we establish the existence (and sometimes the uniqueness) of solutions for a large class of quadratic backward stochastic differential equations (QBSDEs) with continuous generator and a merely square integrable terminal condition. Our approach is different from those existing in the literature. Although we are focused on QBSDEs, our existence result also covers the BSDEs with linear growth, keepingsquare integrable in both cases. As byproduct, the existence of viscosity solutions is established for a class of quadratic partial differential equations (QPDEs) with a square integrable terminal datum. In a second step, we consider QBSDEs with measurable generator for which we establish a Krylov's type a priori estimate for the solutions. We then deduce an Ito-Krylov's change of variable formula. This allows us to establish various existence and uniqueness results for classes of QBSDEs with square integrable terminal condition and sometimes a merely measurable generator. Our results show, in particular, that neither the existence of expo- nential moments of the terminal datum nor the continuity of the generator are necessary to the existence and/or uniqueness of solutions for quadratic BSDEs. Some comparison theorems are also established for solutions of a class of QBSDEs.

M’hamed Eddahbi

Papers

Regularity of the Local Time for the d-dimensional Fractional Brownian Motion with N-parameters

Quadratic BSDE with $\mathbb{L}^{2}$-terminal data: Krylov’s estimate, Itô–Krylov’s formula and existence results

Chaotic expansion and smoothness of some functionals of the fractional Brownian motion

Hedging options in market models modulated by the fractional Brownian motion

Quadratic BSDEs with L 2 -terminal data Existence results, Krylov's estimate and Itô-Krylov's formula ∗