Formules de dualité sur l'espace de Poisson
Gaussian approximations of multiple integrals.
Gaussian density estimates for the solution of singular stochastic Riccati equations
Stochastic Riccati equation is a backward stochastic differential equation with singular generator which arises naturally in the study of stochastic linear-quadratic optimal control problems. In this paper, we obtain Gaussian density estimates for the solutions to this equation.
Generalized functionals of Brownian motion.
Harmonic functions on loop groups
Heat kernel measure on central extension of current groups in any dimension.
Hedging in complete markets driven by normal martingales
This paper aims at a unified treatment of hedging in market models driven by martingales with deterministic bracket , including Brownian motion and the Poisson process as particular cases. Replicating hedging strategies for European, Asian and Lookback options are explicitly computed using either the Clark-Ocone formula or an extension of the delta hedging method, depending on which is most appropriate.
Hsu-Robbins and Spitzer's theorems for the variations of fractional Brownian motion.
Invariance of Poisson measures under random transformations
We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai (Probab. Theory Related Fields103 (1995) 409–429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples...
Invariant measure for some differential operators and unitarizing measure for the representation of a Lie group. Examples in finite dimension
Consider a Lie group with a unitary representation into a space of holomorphic functions defined on a domain 𝓓 of ℂ and in L²(μ), the measure μ being the unitarizing measure of the representation. On finite-dimensional examples, we show that this unitarizing measure is also the invariant measure for some differential operators on 𝓓. We calculate these operators and we develop the concepts of unitarizing measure and invariant measure for an OU operator (differential operator associated to...
Itô formula and local time for the fractional Brownian sheet.
Itô-Skorohod stochastic equations and applications to finance.
LAMN property for hidden processes : the case of integrated diffusions
In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process X. Our data are given by ∫01X(s+i)/n dμ(s) for i=0, …, n−1 and the unknown parameter appears in the diffusion coefficient of the process X only. Although the data are neither markovian nor gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic...
Large deviation principle for a stochastic heat equation with spatially correlated noise.
Local volatility in the Heston model: a Malliavin calculus approach.
Lp-theory for the stochastic heat equation with infinite-dimensional fractional noise
In this article, we consider the stochastic heat equation , with random coefficientsf and gk, driven by a sequence (βk)k of i.i.d. fractional Brownian motions of index H>1/2. Using the Malliavin calculus techniques and a p-th moment maximal inequality for the infinite sum of Skorohod integrals with respect to (βk)k, we prove that the equation has a unique solution (in a Banach space of summability exponent p ≥ 2), and this solution is Hölder continuous in both time and space.
Lp-theory for the stochastic heat equation with infinite-dimensional fractional noise*
In this article, we consider the stochastic heat equation , with random coefficients f and gk, driven by a sequence (βk)k of i.i.d. fractional Brownian motions of index H>1/2. Using the Malliavin calculus techniques and a p-th moment maximal inequality for the infinite sum of Skorohod integrals with respect to (βk)k, we prove that the equation has a unique solution (in a Banach space of summability exponent p ≥ 2), and this solution is Hölder continuous in both time and space.
Malliavin Calculus for a general manifold
Malliavin calculus for stable processes on homogeneous groups
Let be a symmetric semigroup of stable measures on a homogeneous group, with smooth Lévy measure. Applying Malliavin calculus for jump processes we prove that the measures have smooth densities.
Malliavin calculus in Lev́y spaces and applications to finance.