Stefan Geiss

On singular integral and martingale transforms

Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on L p X (R 2 ) with p ∈ (1, ∞). Moreover, replacing equality by a linear equivalence, this is found to be a typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given. As a corollary we obtain that the norm of the real part of the Beurling-Ahlfors operator equals p * — 1 with p * := max{p, (p/(p - 1))}, where the novelty is the lower bound.

Quantitative approximation of certain stochastic integrals

We approximate certain stochastic integrals, typically appearing in Stochastic Finance, by stochastic integrals over integrands, which are path-wise constant within deterministic, but not necessarily equidistant, time intervals. We ask for rates of convergence if the approximation error is considered in L 2 . In particular, we show that by using non-equidistant time nets, in contrast to equidistant time nets, approximation rates can be improved considerably.

On approximation of a class of stochastic integrals and interpolation

Given a diffusion we give different equivalent conditions so that a stochastic integral has an L 2-approximation rate of n −η, if one approximates by integrals over piece-wise constant integrands where equidistant time nets of cardinality are used. In particular, we obtain assertions in terms of smoothness properties of g(Y T ) in the sense of Malliavin calculus. After optimizing over non-equidistant time-nets of cardinality in case , it turns out that one always obtains a rate of which is optimal. This applies to all functions g obtained in an appropriate way by the real interpolation method between the weighted Sobolev space D 1,2(μ) and L 2(μ), where μ is related to the law of Y T . Finally, we obtain the result that if and only if the equidistant time nets attain the optimal rate of convergence

Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces

We consider weak convergence of the rescaled error processes arising from Riemann discretizations of certain stochastic integrals and relate the

First time to exit of a continuous Itô process: general moment estimates and L1-convergence rate for discrete time approximations

L_p

Antisymmetric tensor products of absolutely p -summing operators

-integrability of the weak limit to the fractional smoothness in the Malliavin sense of the stochastic integral.

A COUNTEREXAMPLE CONCERNING THE RELATION BETWEEN DECOUPLING CONSTANTS AND UMD-CONSTANTS

We establish general moment estimates for the discrete and continuous exit times of a general It\^{o} process in terms of the distance to the boundary. These estimates serve as intermediate steps to obtain strong convergence results for the approximation of a continuous exit time by a discrete counterpart, computed on a grid. In particular, we prove that the discrete exit time of the Euler scheme of a diffusion converges in the

Fractional smoothness and applications in finance

L_1

On fractional smoothness and Lp-approximation on the Gaussian space

norm with an order

Contraction Principles for Vector Valued Martingales with Respect to Random Variables Having Exponential Tail with Exponent 2<α<∞

1/2