Thomas Müller-Gronbach

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.

The Optimal Discretization of Stochastic Differential Equations

We study pathwise approximation of scalar stochastic differential equations. The mean squared L2-error and the expected number n of evaluations of the driving Brownian motion are used for the comparison of arbitrary methods. We introduce an adaptive discretization that reflects the local properties of every single trajectory. The corresponding error tends to zero like c·n?1/2, where c is the average of the diffusion coefficient in space and time. Our method is justified by the matching lower bound for arbitrary methods that are based on n evaluations on the average. Hence the adaptive discretization is asymptotically optimal. The new method is very easy to implement, and about 7 additional arithmetical operations are needed per evaluation of the Brownian motion. Hereby we can determine the complexity of pathwise approximation of stochastic differential equations. We illustrate the power of our method already for moderate accuracies by means of a simulation experiment.

Optimal designs for approximating the path of a stochastic process

Abstract We consider a centered stochastic process {X(t):t ∈ T} with known and continuous covariance function. On the basis of observations X(t1), …, X(tn) we approximate the whole path by orthogonal projection and measure the performance of the chosen design d = (t1, …, tn)′ by the corresponding mean squared L2-distance. For covariance functions on T2 = [0, 1]2, which satisfy a generalized Sacks-Ylvisaker regularity condition of order zero, we construct asymptotically optimal sequences of designs. Moreover, we characterize the achievement of a lower error bound, given by Micchelli and Wahba (1981), and study the question of whether this bound can be attained.

Multi-level Monte Carlo algorithms for infinite-dimensional integration on RN

We study randomized algorithms for numerical integration with respect to a product probability measure on the sequence space R^N. We consider integrands from reproducing kernel Hilbert spaces, whose kernels are superpositions of weighted tensor products. We combine tractability results for finite-dimensional integration with the multi-level technique to construct new algorithms for infinite-dimensional integration. These algorithms use variable subspace sampling, and we compare the power of variable and fixed subspace sampling by an analysis of minimal errors.

Optimal approximation of stochastic differential equations by adaptive step-size control

We study the pathwise (strong) approximation of scalar stochastic differential equations with respect to the global error in the L2-norm. For equations with additive noise we establish a sharp lower error bound in the class of arbitrary methods that use a fixed number of observations of the driving Brownian motion. As a consequence, higher order methods do not exist if the global error is analyzed. We introduce an adaptive step-size control for the Euler scheme which performs asymptotically optimally. In particular, the new method is more efficient than an equidistant discretization. This superiority is confirmed in simulation experiments for equations with additive noise, as well as for general scalar equations.

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0, 1[d. The error of an algorithm is defined in L2-sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W. For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of nonuniform time discretizations is crucial.

Deterministic multi-level algorithms for infinite-dimensional integration on RN

Pricing a path-dependent financial derivative, such as an Asian option, requires the computation of E(g(B)), the expectation of a payoff function g, that depends on a Brownian motion B. Employing a standard series expansion of B the latter problem is equivalent to the computation of the expectation of a function of the corresponding i.i.d. sequence of random coefficients. This motivates the construction and the analysis of algorithms for numerical integration with respect to a product probability measure on the sequence space R^N. The class of integrands studied in this paper is the unit ball in a reproducing kernel Hilbert space obtained by superposition of weighted tensor product spaces of functions of finitely many variables. Combining tractability results for high-dimensional integration with the multi-level technique we obtain new algorithms for infinite-dimensional integration. These deterministic multi-level algorithms use variable subspace sampling and they are superior to any deterministic algorithm based on fixed subspace sampling with respect to the respective worst case error.

Optimal pointwise approximation of SDEs based on Brownian motion at discrete points

We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a measurable way) on a finite number of sequential observations of the driving Brownian motion. The resulting lower error bounds hold in particular for all methods that are implementable on a computer and use a random number generator to simulate the driving Brownian motion at finitely many points. Our analysis shows that approximation at a single point is strongly connected to an integration problem for the driving Brownian motion with a random weight. Exploiting general ideas from estimation of weighted integrals of stochastic processes, we introduce an adaptive scheme, which is easy to implement and performs asymptotically optimally.We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a measurable way) on a finite number of sequential observations of the driving Brownian motion. The resulting lower error bounds hold in particular for all methods that are implementable on a computer and use a random number generator to simulate the driving Brownian motion at finitely many points. Our analysis shows that approximation at a single point is strongly connected to an integration problem for the driving Brownian motion with a random weight. Exploiting general ideas from estimation of weighted integrals of stochastic processes, we introduce an adaptive scheme, which is easy to implement and performs asymptotically optimally.

Variable Subspace Sampling and Multi-level Algorithms

We survey recent results on numerical integration with respect to measures μ on infinite-dimensional spaces, e.g., Gaussian measures on function spaces or distributions of diffusion processes on the path space. Emphasis is given to the class of multi-level Monte Carlo algorithms and, more generally, to variable subspace sampling and the associated cost model. In particular we investigate integration of Lipschitz functionals. Here we establish a close relation between quadrature by means of randomized algorithms and Kolmogorov widths and quantization numbers of μ. Suitable multi-level algorithms turn out to be almost optimal in the Gaussian case and in the diffusion case.

Optimal Pointwise Approximation of a Linear Stochastic Heat Equation with Additive Space-Time White Noise

We consider a linear stochastic heat equation on the spatial domain ]0, 1[ with additive space-time white noise, and we study approximation of the mild solution at a fixed time instance. We show that a drift-implicit Euler scheme with a non-equidistant time discretization achieves the order of convergence N -1/2, where N is the total number of evaluations of one-dimensional components of the driving Wiener process. This order is best possible and cannot be achieved with an equidistant time discretization.