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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

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