Beniamin Goldys

Transition Semigroups of Banach Space-Valued Ornstein–Uhlenbeck Processes

We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t)=AX(t) dt+dWH(t), t≥0, where A is the generator of a C0-semigroup S on a separable real Banach space E and WH(t)t≥0 is cylindrical white noise with values in a real Hilbert space H which is continuously embedded in E. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of S in H. In particular we investigate the interplay between analyticity of the transition semigroup, S-invariance of H, and analyticity of the restricted semigroup SH.

Lognormality of rates and term structure models

A term structure model with lognormal type volatility structure is proposed. The Heath, Jarrow and Morton (HJM) framework, coupled with the theory of stochastic evolution equations in infinite dimensions, is used to show that the resulting instantaneous rates are well defined (they do not explode) and remain positive, contrary to those derived in [2]. They are also bounded from below and above by lognormal processes. The model can be used to price and hedge caps, swaptions and other interest rate and currency derivatives including the Eurodollar futures contract, which requires integrability of one over zero coupon bond. This extends results obtained by Sandmann and Sondermann in [22] and [23] for Markovian lognormal short rates to (non-Markovian) lognormal forward rates. We show also existence of invariant measures for the proposed term structure dynamics

On regularity properties of nonsymmetric ornstein-uhlenbeck semigroup in LP spaces

We study properties of the transition semigroup Rt corresponding to the Hilbert space valued nonsymmctric Ornstein-Uhlenheck process possessing an invariant measure μ Necessary and sufficient condition is given for Rtφ to be infinitely smooth in the direction of the Reproducing Kernel of μ for every bounded Borel φ. Estimates on the derivatives of Rtφ are obtained and the same estimates are obtained for the adjoint semigroup. We give also necessary and sufficient conditions for Rt to be an integral operator on LP (H,μ) extending earlier results by Da Prato-Zabczyk and Fuhrman. It is shown also that the integral kernel possesses strong integrability properties. The transition semigroup is also investigated in the scale of Sobolev spaces generalizing those of Malliavin calculus. The transition semigroup turns out to be strongly continuous in those spaces and compact if it is integral in LP (H,μ) Finally, an application to some parabolic PDEs on Hilbert space is given.

LOWER ESTIMATES OF TRANSITION DENSITIES AND BOUNDS ON EXPONENTIAL ERGODICITY FOR STOCHASTIC PDE'S

A formula for the transition density of a Markov process defined by an infinite-dimensional stochastic equation is given in terms of the Ornstein–Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and V ergodicity are proved for a large class of equations. We also provide computable bounds on the convergence rates and the spectral gap for the Markov semigroups defined by the equations. The bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients. Examples of finite-dimensional stochastic equations and semilinear parabolic equations are given.

On Invariant Measures for Diffusions on Banach Spaces

We consider a Banach space valued diffusion process corresponding to a stochastic evolution equation with strongly nonlinear drift. Sufficient conditions are given for the existence of a unique martingale solution and existence of an invariant measure. The resulting diffusion process is shown to be strongly Feller and irreducible. These properties yield uniqueness of invariant measure and ergodicity of the process. We also show that the invariant measure is equivalent to the invariant measure of the diffusion without drift. The main tool to show these results is the Girsanov Transformation.

Ornstein-Uhlenbeck semigroups in open sets of Hilbert spaces

Abstract We consider Ornstein-Uhlenbeck semigroups restricted to open subsets of a separable Hubert space H . We study their regularity and their behaviour at the boundary.

On Closability of Directional Gradients

Let μ be a centred Gaussian measure on a separable real Banach space E, and let H be a Hilbert subspace of E. We provide necessary and sufficient conditions for closability in Lp(E,μ) of the gradient DH in the direction of H. These conditions are further elaborated in case when the gradient DH corresponds to a bilinear form associated with a certain nonsymmetric Ornstein–Uhlenbeck operator. Some natural examples of closability and nonclosability are presented.

Uniform Exponential Ergodicity of Stochastic Dissipative Systems

We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in ℝd with d⩽3.

An LQ Problem for the Heat Equation on the Halfline with Dirichlet Boundary Control and Noise

We study a linear quadratic problem for a system governed by the heat equation on a halfline with boundary control and Dirichlet boundary noise. We show that this problem can be reformulated as a stochastic evolution equation in a certain weighted

Ergodicity of Stochastically Forced Large Scale Geophysical Flows

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