Steffen Dereich

An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process

We analyse the strong approximation of the Cox–Ingersoll–Ross (CIR) process in the regime where the process does not hit zero by a positivity preserving drift-implicit Euler-type method. As an error criterion, we use the pth mean of the maximum distance between the CIR process and its approximation on a finite time interval. We show that under mild assumptions on the parameters of the CIR process, the proposed method attains, up to a logarithmic term, the convergence of order 1/2. This agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations with Lipschitz coefficients, despite the fact that the CIR process has a non-Lipschitz diffusion coefficient.

On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces

Let μ be a centered Gaussian measure on a separable Banach space E and N a positive integer. We study the asymptotics as N→∞ of the quantization error, i.e., the infimum over all subsets ℰ of E of cardinality N of the average distance w.r.t. μ to the closest point in the set ℰ. We compare the quantization error with the average distance which is obtained when the set ℰ is chosen by taking N i.i.d. copies of random elements with law μ. Our approach is based on the study of the asymptotics of the measure of a small ball around 0. Under slight conditions on the regular variation of the small ball function, we get upper and lower bounds of the deterministic and random quantization error and are able to show that both are of the same order. Our conditions are typically satisfied in case the Banach space is infinite dimensional.

Infinite-Dimensional Quadrature and Approximation of Distributions

We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization and to the average Kolmogorov widths of the underlying probability measure. In addition to the general setting, we analyze, in particular, integration with respect to Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As auxiliary results, we determine the asymptotic behavior of quantization numbers and Kolmogorov widths for diffusion processes.

Constructive quantization: Approximation by empirical measures

In this article, we study the approximation of a probability measure

Random networks with sublinear preferential attachment: The giant component

\mu

Typical distances in ultrasmall random networks

on

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

\mathbb{R}^{d}

Small deviations of general Lévy processes.

by its empirical measure

Enlargement of Filtrations and Continuous Girsanov-Type Embeddings

\hat{\mu}_{N}

Real self-similar processes started from the origin

interpreted as a random quantization. As error criterion we consider an averaged