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PDF (9459 Kbyte) 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 2 Institut fiir Angewandte Analysis Institute of Advanced Studies und Stochastik SRS, GPO Box 4, Canberra Hausvogteiplatz 5 - 7 A.C.T: 2601 D - 0 1086 Berlin Australia 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): [email protected] 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.: Statistical Inferences about Semimartingales 94 Tudor, C., Tudor, M.: Approximation Schemes for Stochastic Functi- onal Equations of Retarded Type 96 Valkeila, E. : A Note on Poisson Process Generation 98 Wefelmeyer, W. : Efficient Empirical Estimators for Markov Step Processes 100 Xu, K.: Simultaneous Normal Form and Center Manifold Reduction with MAPLE 101 List of Addresses 103 E-Mail-Addresses 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 1 Basic concepts and results 1.1 Dynamical. systems The categories of metric, topological and smooth dynamical systems and their overlaps, time T = lK 1.2 The concept of a random dynamical system Cocycles on T = ID; and consequences, skew-product flows, example: random differential equations 1.3 Stochastic flows through stochastic differential equa- tions Two-sided stochastic calculus, two-parameter filtration :J'!, forward and back- ward integral, results of Kunita 1.4 Stochastic dynamical systems through stochastic dif- ferential equations White noise as a metric dynamical system, cocycle property, cocycles
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