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Institut fiir A:r:igewandte Analysis und Stochastik im Forschungsverbund Berlin e.V.

Proceedings of the 1st Workshop on Stochastic Numerics

E. Platen 1• 2 (Editor) submitted: 24th November 1992

1 Australian National University Institute of Advanced Studies SRS, GPO Box 4, Canberra A.C.T: 2601 Australia

2 Institut fiir Angewandte Analysis und Stochastik Hausvogteiplatz 5 - 7 D - 0 1086 Berlin Germany

Preprint No. 21 Berlin 1992

1991 Mathematics Subject Classification. 60H10. Key words and phrases. stochastic differential equations, numerical solution, statistical methods, simulation, limit theorems, financial mathematics.

Herausgegeben vom Institut fiir Angewandte Analysis und Stochastik Hausvogteiplatz 5-7 D - 0 1086 Berlin

Fax: + 49 30 2004975 e-Mail (X.400): c=de;a=dbp;p=iaas-berlin;s=preprint e-Mail (Internet): preprint@iaas-berlin.dbp.de

PROCEEDINGS

of the 1st Workshop on

STOCHASTIC NUMERICS

September 7 - 12, 1992 in Gosen (near'Berlin) organized by the Institute of Applied Analysis and Stochastics Berlin

together with the Humboldt University Berlin

Chairman: E. Platen (Canberra/Berlin) Scientific Program Committee: L. Arnold (Bremen), S. Artemev (Novosibirsk), P. Deuflhard (Berlin), J. Gartner (Berlin), B. Grigelionis (Vilnius), J. Jacod (Paris), P. Kloeden (Geelong), U. Kuchler (Berlin), G.N. Milstein (Ekaterinburg), T. Mitsui (Nagoya), H. Neunzert (Kaiserslautern) P. Protter (Purdue), R. Rebolledo (Santiago), D. Talay (Sophia Antipolis),

Local Organizing Committee: Mrs. M. Teuchert, IAAS, Hausvogteiplatz 5-7, D-0-1086 Berlin, Fax: (030) 2004975, Tel: (030) 20377 594

Forward

One aim of the pt Workshop on Stochastic Numerics was to bring together for the first time leading experts on this field from the eastern countries with those from the west. The support by Deutscher Akademischer Austauschdienst and Deutsche Forschungsgemeinschaft allowed the participation of many colleagues between Torun and Novosibirsk. Another goal of this workshop was achieved by confronting specialists from stochastic analysis, stochastic numerics, statistics of stochastic processes and different fields of application with new results and trends in the other fields. Thus applied and inter- disciplinary aspects played an important role. Modelling of challenging applications, stochastic numerical methods and statistical inference were understood in the frame- work of their interrelations. Several young mathematicians took the opportunity to consult specialists to improve their profile.

The response to the conference was so overwhelming that many participants expres- sed their hope that it would be regarded as a kick-off meeting to a series of workshops which enable mathematicians working at universities and in industry to discuss their work, exchange ideas, or even join forces to approach some of the more difficult and pressing problems. Announced as the first workshop on stochastic numerics, the conference provided the ideal setting for identifying such problems and stimulating research activities. Subsequent workshops could possibly involve a section, where progress is discussed and solutions are presented.

Areas that appear to be of particular interest are mathematical finance and quan- tum physics. We therefore decided to choose problems encountered in these fields to apply our techniques and put the theory to the test. Many papers delivered at the conference dealt with particular stochastic differential equations. Whereas in phy- sics, where relations are often governed by mechanistic laws, problems may well be described by such systems, in finance or quantum physics the dynamic processes are usually not fully understood until now and need to be identified. Hence, the objective is simply to adequately model observable phenomena and discover as much structure as possible. Some of the extended and revised abstracts e.g. by Breckling & Dal Dosso, Rebolledo and Chiarella list a few of the problems encountered which may be tackled using stochastic differential equations, stochastic numerics and statistics of processes. Those with an interest in the kind of problems are encouraged to contact the authors directly for a more detailed discussi~:m. Deutsche Bank Research for in- stance may also be in a position to provide the data necessary for analysing specific problems in mathematical finance.

These proceedings contain all abstracts submitted to the workshop including several extended and revised after the conference. We included also a list of addresses and e-mail addresses to provide a basis for further communication.

The organisers of the conference would like to thank all those who contributed to this workshop and announce that attempts will be made to have a 2nd Workshop on Stochastic Numerics near Berlin from 5 until 10 September 1994.

Berlin, September 1992

U. Kuchler E. Platen M. Teuchert

Contents

Arnold, L. : Stochastic Dynamical Systems 1

Artemev, S. : Certain Aspects of Application of Numerical Methods for Solving SDE Systems 4

Bally, V. : Approximation for the Solutions of SDE 6

Bossy, M. ,Talay,D.: A Stochastic Particle Method for Some One - Dimensional Nonlinear P.D.E. 7

Breckling, J. , Dal Dosso, L. : The Analysis of Term Structures 8

Chauvin, B. : A Stochastic Simulation for Solving Scalar Reaction - Diffusion Equations 12

Chiarella, C.: Stochastic Numeric Methods and Applications in Finance and Economics 15

Deuflhard, P.: Stochastic versus Deterministic Numerical ODE Integration 16

Dietz, H.M. , Kutoyants,T. A.: A Minimum - Distance Estimator for Diffusion

Processes with Ergodic Properties 20 Drummond, I. T.: Langevin Methods for Simulating Lattice

Field Theory 21 Fournie, E.: A Test of Type Kolmogorov-Smirnov for

Ergodic Diffusion Processes 26 Fritsch, H. : A Discretization of a Controlled Stochastic

Differential Inclusion 27 Gaines, J. G., Lyons, T. J. : Random Generation of Stochastic Area

Integrals 29 Gelbrich, M. : Approximation wrt. Time and Chance for Ito

SDE's with Rates 31 Hofmann, N. : An Application of Stochastic Numerical

Methods in Financing 32 Jacod, J. : Rate of Convergence for Some Functionals of

Brownian Semimartingales 34 Jakubowski, A. : An Eduction of the Langevin Equation 35

Kazimierczyk, P.: Correction Formulae for Maximum - Likelihood Estimator of Parameters of Stochastic Differential Equations 36

Kubilius, K. : On the Rate of Convergence of the Certain Diffusion Approximations 37

Lanska, V.: First - passage Time Problem for Simulated Diffusion. Application in Neural Modeling. 40

Lehn, J.: A Survey on Pseudorandom Number Generators 42

Lepingle, D. : Time Discretization of Reflected Stochastic Differential Equations 43

Lythe, G., Proctor, M. R. E.: The Effect of Noise on Solutions of Some ODE's. 45

MackeviCius, V.: Second Order Weak Runge - Kutta Approxi- mations for Stratonovich SDE's 47

Makroglou, A. : On the Numerical Solution of Stochastic Integro - Differential Equations 49

Martensson, B.: Gnans - A Program for Stochastic and Deter- ministic Dynamical Systems 51

Melgaard, H. , Madsen, H., Holst, J.: Methods for Parameter Estimation in

Stochastic Differential Equations 53 Melnikov, A. V.: On Martingale Approach to Stochastic

Approximation Algorithm 65 Milshtein, G. N.: The Solving of the Boundary Value Problem

for Parabolic Equation by the Numerical Integration of Stochastic Equations 67

Newton, N. J. : Variance Reduction for Simulated Diffusions 69

Ogawa, S.: Stochastic Numerical Treatment of Nonlinear Diffusions 70

Platen, E. : Numerical Solution of Stochastic Differen- tial Equations 72

MiculeviCius, R., Pragarauskas, H.: On Integro - Differential Bellman Equation 74

Rebolledo, R. : On the Stochastic Simulation of Quantum Systems 76

Richter, M. : Approximation of Random Elements 79 Schenk, K. R. : Computer Experiments for the Noisy Duffing -

van der Pol Equation 80 Schurz, H.: Approximate Markov Chain Filtering 83

Shkurko, I. : Arbitrary Order Approximations for Linear Systems of SDE's 85

Slominski, L.: On Approximation of Solutions of SDE's with · Reflecting Boundary Conditions 87

Kuchler, U. , S!Z!rensen, M. : Exponential Families of Stochastic Processes 88

Hern.andez, D. B., Spigler, R. : Convergence and Stability of Implicit Runge-

Kutta Methods for Systems with Multiplicative Noise 91

Bernard, P., Talay, D., Tubaro, L.: Analysis of a Stochastic Particle Method

for a Non - linear P. D. E. 92

Taraskin, A. F.:

Tudor, C., Tudor, M.:

Valkeila, E. :

Wefelmeyer, W. :

Xu, K.:

List of Addresses E-Mail-Addresses

Statistical Inferences about Semimartingales 94

Approximation Schemes for Stochastic Functi- onal Equations of Retarded Type 96 A Note on Poisson Process Generation 98

Efficient Empirical Estimators for Markov Step Processes 100 Simultaneous Normal Form and Center Manifold Reduction with MAPLE 101

103 109

FIRST \VORKSHOP ON STOCHASTIC NUMERICS BERLIN, SEPTEMBER 7-12, 1992

Stochastic1 Dynamical Systems by

Ludwig Arnold Institut fiir Dynamische Systeme

Universitat, Postfach D-2800 Bremen 33