Numerical Integration of Stochastic Differential Equations

G. N. Milstein

ISBN 10: 9048144876 / ISBN 13: 9789048144877

Editore: Springer Netherlands, Springer Netherlands, 2010

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Druck auf Anfrage Neuware - Printed after ordering - U sing stochastic differential equations we can successfully model systems that func tion in the presence of random perturbations. Such systems are among the basic objects of modern control theory. However, the very importance acquired by stochas tic differential equations lies, to a large extent, in the strong connections they have with the equations of mathematical physics. It is well known that problems in math ematical physics involve 'damned dimensions', of ten leading to severe difficulties in solving boundary value problems. A way out is provided by stochastic equations, the solutions of which of ten come about as characteristics. In its simplest form, the method of characteristics is as follows. Consider a system of n ordinary differential equations dX = a(X) dt. (O.l ) Let Xx(t) be the solution of this system satisfying the initial condition Xx(O) = x. For an arbitrary continuously differentiable function u(x) we then have: (0.2) u(Xx(t)) - u(x) = j (a(Xx(t)), ~~ (Xx(t))) dt.

Codice articolo 9789048144877

Dati bibliografici

• Titolo: Numerical Integration of Stochastic Differential Equations
• Editore: Springer Netherlands, Springer Netherlands
• Data di pubblicazione: 2010
• Lingua: Inglese
• ISBN 10: 9048144876
• ISBN 13: 9789048144877
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 235 L x 155 A x 11 P (millimetri) 235 millimetri di larghezza per 155 millimetri di altezza per 11 millimetri di profondità
• Peso: 289 grammi

Descrizione articolo

Riassunto

This book is devoted to mean-square and weak approximations of solutions of stochastic differential equations (Sde). These approximations represent two fundamental aspects in the contemporary theory of Sde. Firstly, the construction of numerical methods for such systems is important as the solutions provided serve as characteristics for a number of mathematical physics problems. Secondly, the employment of probability representations together with a Monte Carlo method allows us to reduce the solution of complex multidimensional problems of mathematical physics to the integration of stochastic equations.
Along with a general theory of numerical integrations of such systems, both in the mean-square and the weak sense, a number of concrete and sufficiently constructive numerical schemes are considered. Various applications and particularly the approximate calculation of Wiener integrals are also dealt with.
This book is of interest to graduate students in the mathematical, physical and engineering sciences, and to specialists whose work involves differential equations, mathematical physics, numerical mathematics, the theory of random processes, estimation and control theory.

Contenuti

Introduction. 1: Mean-square approximation of solutions of systems of stochastic differential equations. 1. Theorem on the order of convergence (theorem on the relation between approximation on a finite interval and one-step approximation). 2. Methods based on an analog of Taylor expansion of the solution. 3. Explicit and implicit methods of order 3/2 for systems with additive noises. 4. Optimal integration methods for linear systems with additive noises. 5. A strengthening of the main convergence theorem. 2: Modeling of Itô integrals. 6. Modeling Itô integrals depending on a single noise. 7. Modeling Itô integrals depending on several noises. 3: Weak approximation of solutions of systems of stochastic differential equations. 8. One-step approximation. 9. The main theorem on convergence of weak approximations and methods of order of accuracy two. 10. A method of order of accuracy three for systems with additive noises. 11. An implicit method. 12. Reducing the error of the Monte Carlo method. 4: Application of the numerical integration of stochastic equations for the Monte Carlo computation of Wiener integrals. 13. Methods of order of accuracy two for computing Wiener integrals of functionals of integral type. 14. Methods of order of accuracy four for computing Wiener integrals of functionals of exponential type. Bibliography. Index.