Norbert Hofmann

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 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.

Stability of weak numerical schemes for stochastic differential equations

Abstract This paper considers numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise. For different explicit and implicit schemes, the regions of stability are also examined.

On quasi-Monte Carlo simulation of stochastic differential equations

In a number of problems of mathematical physics and other fields stochastic differential equations are used to model certain phenomena. Often the solution of those problems can be obtained as a functional of the solution of some specific stochastic differential equation. Then we may use the idea of weak approximation to carry out numerical simulation. We analyze some complexity issues for a class of linear stochastic differential equations (Langevin type), which can be given by dX t = -αX t dt + β(t)dW t , X0:= 0, where α > 0 and β: [0,T] → R. It turns out that for a class of input data which are not more than Lipschitz continuous the explicit Euler scheme gives rise to an optimal (by order) numerical method. Then we study numerical phenomena which occur when switching from (real) Monte Carlo simulation to quasi-Monte Carlo simulation, which is the case when we carry out the simulation on computers. It will easily be seen that completely uniformly distributed sequences yield good substitutes for random variates, while not all uniformly distributed (mod 1) sequences are suited. In fact we provide necessary conditions on a sequence in order to serve quasi-Monte Carlo purposes. This condition is expressed in terms of the measure of well-distributions. Numerical examples complement the theoretical analysis.

Linear vs Standard Information for Scalar Stochastic Differential Equations

We study pathwise approximation of scalar sdes with respect to the mean squared L2-error. We compare the power of linear and standard information about the driving Brownian motion. It turns out that asymptotically the corresponding minimal errors differ only by the factor 6/?.

Extrapolation methods for the weak approximation of Ito diffusions

Higher-order weak extrapolation methods for the approximation of functionals of Ito diffusions are considered. Under appropriate regularity conditions it is shown that extrapolations allow a considerable increase in the weak order of convergence of a discrete-time one-step approximation method. Numerical experiments indicate the efficiency of extrapolations based on higher-order weak schemes for stochastic differential equations with additive noise.

On the global error of Itô-Taylor schemes for strong approximation of scalar stochastic differential equations

We analyze the L 2 ([0,1])-error of general numerical methods based on multiple Ito-integrals for pathwise approximation of scalar stochastic differential equations on the interval [0,1]. We show that the minimal error that can be obtained is at most of order N-1/2, where N is the number of multiple Ito-integrals that are evaluated. As a consequence, there are no Ito-Taylor methods of higher order with respect to the global L 2 -error on [0,1], which is in sharp contrast to the well-known fact that arbitrary high orders can be achieved by these methods with respect to the error at the discretization points. In particular, it turns out that the asymptotic performance of piecewise linear interpolated Ito-Taylor schemes gets worse the more multiple Ito-integrals are involved.

Approximating Large Diversified Portfolios

This paper considers a financial market with asset price dynamics modeled by a system of lognormal stochastic differential equations. A one-dimensional stochastic differential equation for the approximate evolution of a large diversified portfolio formed by these assets is derived. This identifies the asymptotic dynamics of the portfolio as being a lognormal diffusion. Consequentially an efficient way for computing probabilities, derivative prices, and other quantities for the portfolio are obtained. Additionally, the asymptotic strong and weak orders of convergence with respect to the number of assets in the portfolio are determined. Copyright Blackwell Publishers, Inc. 2000.