D. F. Kuznetsov
Department of Mathematics
St.-Petersburg State Technical University
St.-Petersburg, Russia,
The book is devoted to the problem of numerical analysis of Ito stochastic differential equations. The book consists of seven chapters. Chapter 1 is an introduction and begins with an exposition of general facts from the elementary theory of probability. Some problems formulated in terms of stochastic differential equations are presented. Chapter 2 deals with the problem of integration order replacement for multiple stochastic Ito integrals. For one class of multiple stochastic Ito integrals we give proofs of the integration order replacement theorems. Stochastic (Taylor-Ito, Taylor-Stratonovich, and unified Taylor-Ito) expansions of Ito processes are considered in Chapter 3. The unified Taylor-Ito expansions are constructed via integration order replacement theorems for multiple stochastic Ito integrals obtained in Chapter 2. Examples of the unified Taylor-Ito expansions for solutions of certain scalar and vector stochastic differential Ito equations are given. Chapter 4 provides methods of expansion and approximation of multiple stochastic Stratonovich and Ito's integrals. We give a new method of multiple Stratonovich stochastic integral approximation based on multiple Fourier series on full orthonormal systems of functions. The comparison of this method with the Milstein method of expansion and approximation of multiple stochastic Stratonovich integrals is given. General formulas for expansion, approximation, and mean-square error of approximation of multiple stochastic Stratonovich integral of a multiplicity k are obtained. We suggest a new method of multiple Ito stochastic integral approximation based on multiple integral sums. Chapter 4 ends with a discussion of various expansions and approximations of multiple stochastic Stratonovich integrals on polynomial and trigonometric systems of functions. Chapter 5 contains an exposition of different explicit strong one-step numerical methods for stochastic differential Ito equations. Based on Taylor-Ito, Taylor-Stratonovich, and unified Taylor-Ito expansions these methods are of 1.0, 1.5, 2.0, 2.5, and r/2 (r=6,7, ...) strong orders. The methods of strong orders 2.0, 2.5 and r/2 (r = 6,7, ...) are new since these methods include the approximations of multiple Stratonovich and Ito stochastic integrals of the multiplicity k (k=4,5,...). For stochastic differential Ito equations with multidimensional noise, we suggest new finite-difference 1.5, 2.0 and 2.5 strong order methods based on the Taylor-Ito and unified Taylor-Ito expansions. Exact and approximate methods for numerical solution of linear stationary stochastic differential equations are considered in chapter 6. Based on the Cauchy formula, the exact method gives the exact representation for the stochastic component of a solution of a linear system of stationary stochastic differential equations. The approximate method is based on the approximate representation of a linear stationary stochastic differential equations system. The comparison of these two methods is given. The exact and approximate methods are generalized for linear stationary stochastic differential equations systems of order k. In Chapter 7 we give examples of the numerical simulation of linear and nonlinear stochastic differential equations: solutions of Lorenz and Rossler's equations in chaotic regime under the influence of stochastic perturbations, Chandler oscillations, solar activity, price dynamics of shares, population dynamics, chemical reactions oscillations, profitability of shares portfolio. CONTENTS PREFACE CHAPTER 1 STOCHASTIC DIFFERENTIAL EQUATIONS: DEFINITIONS, PROPERTIES, PROBLEMS, APPLICATIONS. 1.1 Numerical approaches for stochastic differential equations 1.2 Some facts from the probability theory 1.3 Mathematical models of dynamical systems under the action of stochastic perturbations 1.4 Stochastic models of physical and technical systems 1.4.1 Stochastic model of heat fluctuations of particles in substances, electric charges, and conductors Nyquist formula 1.4.2 Self-oscillatory electric system (lamp generator) 1.4.3 Chandler 's oscillations 1.4.4 Stochastic models of chemical kinetics and population dynamics 1.4.5 Models of finance mathematics 1.4.6 Solar activity 1.5 Stochastic differential equations 1.6 Ito formula 1.7 Relations between stochastic integrals and equations in the Ito and the Stratonovich forms 1.8 Impossibility of applications of numerical methods for ordinary differential equations to stochastic differential equations CHAPTER 2 INTEGRATION ORDER REPLACEMENT THEOREMS IN MULTIPLE STOCHASTIC ITO INTEGRALS 2.1 Multiple stochastic integrals 2.2 Integration order replacement problem in multiple stochastic Ito integrals 2.3 Integration order replacement in multiple stochastic Ito integrals of the second order 2.4 Integration order replacement in multiple stochastic Ito integrals of the order k CHAPTER 3 STOCHASTIC EXPANSIONS OF ITO PROCESSES 3.1 Introduction 3.2 Ito differentiability of stochastic processes 3.3 Unified Taylor-Ito expansions 3.3.1 The first form of the unified Taylor-Ito expansion 3.3.2 The second form of the unified Taylor-Ito expansion 3.4 Taylor-Ito expansion in the forms of W. Wagner and E. Platen 3.5 Stratonovich differentiability of stochastic processes 3.6 Stratonovich-Taylor expansion 3.7 Examples of Taylor-Ito expansions 3.7.1 Taylor-Ito expansions for solutions of some scalar stochastic differential Ito equations 3.7.2 Taylor-Ito expansions for solutions of some vector stochastic differential Ito equations CHAPTER 4 METHODS OF APPROXIMATION OF MULTIPLE STOCHASTIC STRATONOVICH INTEGRALS 4.1 Introduction 4.2 Relations between multiple stochastic Ito and Stratonovich integrals 4.3 A method of expansion and approximation of multiple stochastic Stratonovich integrals based on multiple Fourier series on full orthonormal systems 4.3.1 Expansion of multiple stochastic Stratonovich integrals into multiple series consisting of products of standard gaussian random functions 4.3.2 General relations for approximations of multiple stochastic Stratonovich integrals 4.3.3 Approximation of multiple stochastic Stratonovich integrals with the use of a trigonometric system 4.3.4 Approximation of multiple stochastic Stratonovich integrals with the use of a polynomial system 4.4 The Milstein method of expansion and approximation of multiple stochastic Stratonovich integrals 4.4.1 Introduction 4.4.2 Examples of expansions of some multiple stochastic Stratonovich integrals by the Milstein method 4.5 Comparison of the Milstein method and the method based on multiple Fourier series 4.6 Expansion of multiple stochastic integrals by using of Hermit polynomials 4.7 Method of multiple Ito stochastic integral approximation based on multiply integral sums CHAPTER 5 EXPLICIT STRONG ONESTEP NUMERICAL METHODS FOR STOCHASTIC DIFFRENTIAL ITO EQUATIONS 5.1 Introduction 5.2 Taylor-Ito expansion and numerical methods 5.3 Numerical methods based on Taylor-Ito expansion. 5.3.1 The Milstein method 5.3.2 The order 1.5 explicit strong one-step method 5.3.3 The order 2.0 explicit strong one-step method 5.3.2 The order 2.5 explicit strong one-step method 5.4 Numerical methods based on the Taylor-Ito expansion in Wagner and Platen's form 5.4.1 The order 1.5 explicit strong one-step method 5.4.2 The order 2.0 explicit strong one-step method 5.4.3 The order 2.5 explicit strong one-step method 5.4.4 Peculiarities of numerical methods based on the unified Taylor-Ito expansion and the Taylor-Ito expansion in Wagner and Platen's form 5.5 Numerical methods based on Stratonovich-Taylor expansion 5.5.1 The order r/2 explicit strong one-step method 5.5.2 The order 1.0 explicit strong one-step method 5.5.3 The order 1.5 explicit strong one-step method 5.5.4 The order 2.0 explicit strong one-step method 5.5.5 The order 2.5 explicit strong one-step method 5.6 Finite-difference methods based on the Taylor-Ito expansions 5.6.1 Taylor approximations for derivatives of nonrandom functions 5.6.2 The order 1.0 explicit strong one-step finite-difference method 5.6.3 The order 1.5 explicit strong one-step finite-difference method 5.6.4 The order 2.0 explicit strong one-step finite-difference method 5.6.5 The order 2.5 explicit strong one-step finite-difference method CHAPTER 6 NUMERICAL SIMULATION FOR SOLUTIONS OF STATIONARY LINEAR STOCHASTIC DIFFERENTIAL EQUATUIONS (LSDE) 6.1 Systems of LSDE: formulas and auxiliary results 6.1.1 Integral representation of LSDE solutions 6.1.2 Moments characteristics of LSDE solutions 6.1.3 Properties of discrete systems of stochastic equations in the stationary case 6.2 Exact method of simulation of LSDE solutions 6.2.1 A General approach to simulation of LSDE solutions 6.2.2 An algorithm for simulation of the dynamic component of LSDE solutions 6.2.3 An algorithm for simulation of systematic component of LSDE solutions 6.2.4 An algorithm for simulation of stochastic component of LSDE solutions 6.2.5 An algorithm for simulation of LSDE solutions 6.3 An approximate method for simulation of LSDE solutions 6.3.1 Introduction 6.3.2 An algorithm for simulation of LSDE solutions by an approximate method 6.3.3. Comparison of exact and approximate methods for simulation of LSDE solutions CHAPTER 7 EXAMPLES OF SIMULATION OF STOCHASTIC INTEGRALS AND SOLUTIONS OF STOCHASTIC DIFFERENTIAL ITO EQUATIONS 7.1 Simulation of multiple stochastic Stratonovich integrals 7.1.1 The case of trigonometric basis 7.1.2 The case of polynomial basis 7.2 Simulation of the price dynamics of shares 7.3 Lorenz equations in a chaotic regime under the influence of stochastic perturbations 7.4 Simulation of the population dynamics and oscillations in chemical reactions 7.5 Simulation of the profitability of shares portfolio 7.6 Simulation of Chandler's oscillations 7.7 Simulation of the solar activity 7.8 Rossler's equations in a chaotic regime under the influence of stochastic perturbations BIBLIOGRAPHY LIST OF SYMBOLS