Anton Thalmaier

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.

On the differentiation of heat semigroups and poisson integrals

We give a version of integration by parts on the level of local martingales; combined with the optional sampling theorem, this method allows us to obtain differentiation formulae for Poisson integrals in the same way as for heat semigroups involving boundary conditions. In particular, our results yield Bismut type representations for the logarithmic derivative of the Poisson kernel on regular domains in Riemannian manifolds corresponding to elliptic PDOs of Hormander type. Such formulae provide a direct approach to gradient estimates for harmonic functions on Riemannian manifolds

COMPLETE LIFTS OF CONNECTIONS AND STOCHASTIC JACOBI FIELDS

Differentiable families of V-martingales on manifolds are investigated: their infinitesimal variation provides a notion of stochastic Jacobi fields. Such objects are known (2) to be martingales taking values in the tangent bundle when the latter is equipped with the complete lift of the connection V. We discuss various characterizations of TM-valued martingales. When applied to specific families of V-martingales which appear in connection with the heat flow for maps between Riemannian manifolds, our results allow to establish formulas giving a stochastic representation for the differential of solutions to the nonlinear heat equation. As an application, we prove local and global gradient estimates for harmonic maps of bounded dilatation. 0 Elsevier, Paris

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).

Brownian motion and the formation of singularities in the heat flow for harmonic maps

SummaryWe develop a general framework for a stochastic interpretation of certain nonlinear PDEs on manifolds. The linear operation of takin expectations is replaced by the concept of “martingale means”, namely the notion of deterministic starting points of martingales (with respect to the Levi-Civita connection) ending up at a prescribed state. We formulate a monotonicity condition for the Riemannian quadratic variation of such martingales that allows us to turn smallness of the quadratic variation into a priori gradient bounds for solutions of the nonlinear heat equation. Such estimates lead to simple criteria for blow-ups in the nonlinear heat flow for harmonic maps with small initial energy.

Horizontal martingales in vector bundles

1. Introduction and Notations 2. Complete Lifts to Tangent Bundles 3. Deformed Antidevelopment 4. Horizontal Lifts to Vector Bundles 5. A General Class of Lifts to Vector Bundles 6. Complete Lifts to Cotangent Bundles 7. Complete Lifts to Exterior Bundles 8. Complete Lifts to Dirac Bundles 9. Martingales in the Tangent Space Related to Harmonic Maps 10. Martingales in the Exterior Tangent Bundle Related to Harmonic Forms References

Horizontal Diffusion in C 1 Path Space

We define horizontal diffusion in C 1 path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing un- der the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction properties in the Monge- Kantorovich minimization problem for probability measures evolving along the heat flow. For constant rank diffusions, differentiating a family of coupled diffusions gives a derivative process with a covariant derivative of finite variation. This con- struction provides an alternative method to filtering out redundant noise.

A probabilistic approach to the Yang-Mills heat equation

We construct a parallel transport U in a vector bundle E, along the paths of a Brownian motion in the underlying manifold, with respect to a time dependent covariant derivative ∇ on E, and consider the covariant derivative ∇0U of the parallel transport with respect to perturbations of the Brownian motion. We show that the vertical part U −1 ∇0U of this covariant derivative has quadratic variation twice the Yang–Mills energy density (i.e., the square norm of the curvature 2-form) integrated along the Brownian motion, and that the drift of such processes vanishes if and only if ∇ solves the Yang–Mills heat equation. A monotonicity property for the quadratic variation of U −1 ∇0U is given, both in terms of change of time and in terms of scaling of U −1 ∇0U . This allows us to find a priori energy bounds for solutions to the Yang–Mills heat equation, as well as criteria for non-explosion given in terms of this quadratic variation.  2002 Editions scientifiques et medicales Elsevier SAS.

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.  2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS

Manifold-valued martingales, changes of probabilities, and smoothness of finely harmonic maps

This paper is concerned with regularity results for starting points of continuous manifold-valued martingales with fixed terminal value under a possibly singular change of probability. In particular, if the martingales live in a small neighbourhood of a point and if the stochastic logarithm M of the change of probability varies in some Hardy space Hr for sufficiently large r< 2, then the starting point is differentiable at MD 0. As an application, our results imply that continuous finely harmonic maps between manifolds are smooth, and the differentials have stochastic representations not involving derivatives. This gives a probabilistic alternative to the coupling technique used by Kendall (1994).