Paul Malliavin

Fourier series method for measurement of multivariate volatilities

Abstract. We present a methodology based on Fourier series analysis to compute time series volatility when the data are observations of a semimartingale. The procedure is not based on the Wiener theorem for the quadratic variation, but on the computation of the Fourier coefficients of the process and therefore it relies on the integration of the time series rather than on its differentiation. The method is fully model free and nonparametric. These features make the method well suited for financial market applications, and in particular for the analysis of high frequency time series and for the computation of cross volatilities.

Stochastic calculus of variations in mathematical finance

Gaussian stochastic calculus of variations.- Pathwise propagation of Greeks in complete elliptic markets.- Market equilibrium and price-volatility feedback rate.- Multivariate conditioning and regularity of laws.- Non-elliptic markets and instability in HJM models.- Insider trading.- Rates of weak convergence and distribution theory on Gaussian spaces.-Fourier series method for the measurement of historical volatilities.

A Fourier transform method for nonparametric estimation of multivariate volatility

We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi-martingales, which is based on Fourier analysis. The co-volatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the prices process and the Fourier transform of the co-volatility process. A nonparametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes. The asymptotic properties of the random estimator are studied: namely, consistency in probability uniformly in time and convergence in law to a mixture of Gaussian distributions.

Non perturbative construction of invariant measure through confinement by curvature

Abstract A curvature criterium, computable by infinitesimal differential geometry, insures the existence of invariant probability measure for an a priori given elliptic operator; uniqueness, reversibility, formula of integration by part are discussed in this context; existence of invariant measure for some non linear OU operator on an Hilbert space.

Numerical error for SDE: Asymptotic expansion and hyperdistributions

The principal part of the error in the Euler scheme for an SDE with smooth coefficients can be expressed as a generalized Watanabe distribution on Wiener space. To cite this article: P. Malliavin, A. Thalmaier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Support of Virasoro unitarizing measures

A unitarizing measure is a probability measure such that the associated L 2 space contains a closed subspace of holomorphic functionals on which the Virasoro algebra acts unitarily. It has been shown that the unitarizing property is equivalent to an a priori given formula of integration by parts, which has been computed explicitly. We show in this Note that unitarizing measures must be supported by the quotient of the homeomorphism group of the circle by the subgroup of Mobius transformations. To cite this article: H. Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621-626. uf6d9 2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS

Invariant or quasi-invariant probability measures for infinite dimensional groups : Part II : Unitarizing measures or Berezinian measures

Some infinitesimal representations of Virasoro algebra are known to be unitarizable; is it possible to realize the underlying Hilbert space as an L2-space?

Energy Identities and Estimates for Anticipative Stochastic Integrals on a Riemannian Manifold

If x denotes an ℝd-valued P τ-adapted Brownian motion where P τ is the usual past filtration, the following energy identity n n

Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows

E{left| {int_0^1 {{u_tau }dx(tau )} } right|^2} = Eint_0^1 {{{left| {{u_tau }} right|}^2}dtau }