Numerical Integration of Stochastic Differential Equations

ISBN-13: 9789048144877 / Angielski / Miękka / 2010 / 172 str.

G. N. Milstein

Opis

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.

Spis treści

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.