Imperial College of Science, Technology and Medicine Department of Computing Parallel Numerical methods for SDEs and Applications Nada Atallah Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Computing and the Diploma of Imperial College London, July 2016 Abstract Stochastic Differential Equations (SDEs) constitute an important mathematical tool with appli- cations in many areas of research such as finance, physics and computer science. The analytical study of these equations is problematic, especially in the multi-dimensional case, for this reason, numerical techniques prove to be necessary to solve such equations. In this project, parallel numerical techniques for SDEs are studied. Two kinds of parallelism will be explored: in space and in time. Implementation of these techniques applied to several systems of SDEs will be realised (using C++ and MPI) and performance measures like speedup and efficiency will be investigated on medium-scale computer clusters. In the second part of the thesis, a major application area in the field of computer and com- munication networks will be studied, that of second-order stochastic fluid networks. Recently, interest has been growing in networks with large numbers of components, with application in diverse fields, such as internet performance evaluation, the spread of computer viruses and biochemistry. Such models have such a large state space that discrete-state models are numer- ically infeasible due to the explosion in the size of the state space. Fluid approximations are therefore preferable and tend to be more accurate in large state spaces. In fluid models, an integer counter is replaced by a real number representing a volume and the solution method becomes based on differential equations rather than on difference equations. Some analytical solutions are possible in special cases but in general, numerical methods are required. In this project, parallel numerical studies of second order fluid networks will be conducted and re- sults in the context of the performance of computer and communication networks have been analysed, facilitating design improvement of their architecture. i ii Acknowledgements I would like to express my sincere gratitude to: • my supervisor Peter Harrison, • my second supervisor Tony Field, • AESOP group, • my brother, sister and all my friends who made this experience enjoyable and fruitful. iii iv Contents Abstract i Acknowledgements iii 1 Introduction 1 2 Mathematical background 5 2.1 Measure and probability . 5 2.1.1 Probability Spaces . 5 2.1.2 Random variables and stochastic processes . 6 2.1.3 Expected value and variance . 10 2.1.4 Distribution Functions and independence . 11 2.1.5 Limit Theorems . 12 2.1.6 Conditional expectation . 13 2.1.7 Martingales . 15 2.2 Brownian Motion . 17 2.2.1 Properties . 18 v vi CONTENTS 2.2.2 Brownian Martingales . 18 2.2.3 Reflection principle . 20 2.2.4 Path properties of Brownian motion . 22 2.3 Poisson Process . 24 2.3.1 Properties . 25 2.3.2 Poisson Martingales . 25 2.3.3 Path properties of Poisson processes . 27 2.4 Levy processes . 28 2.5 Stochastic Integrals and It^o'sFormula . 28 2.5.1 Stochastic integral for simple step integrand . 28 2.5.2 Stochastic integral for square integrable adapted integrands . 30 2.5.3 Further extension of the stochastic integral . 30 2.5.4 It^oformula (with respect to Brownian motion) . 31 2.5.5 It^o'sformula (with respect to It^oprocesses) . 33 2.6 Stochastic differential equations . 33 3 Numerical Methods for SDEs 36 3.1 Examples of explicitly solvable SDEs . 36 3.2 Existence and uniqueness of strong solutions . 38 3.2.1 Solutions for ODEs . 39 3.2.2 Solutions for SDEs . 40 3.3 Stochastic discrete time approximations . 43 CONTENTS vii 3.3.1 Types of approximation . 44 3.3.2 Wagner-Platen expansion . 44 3.3.3 Convergence . 46 3.4 Strong approximations . 48 3.4.1 Euler Scheme . 48 3.4.2 Milstein Scheme . 50 3.4.3 Higher order schemes . 53 3.4.4 Geometric Brownian motion simulation . 53 3.5 Weak approximations . 57 3.5.1 Weak error criterion . 59 3.5.2 Euler scheme . 59 3.5.3 Higher order schemes . 61 4 Parallel stochastic simulation 63 4.1 Phase space Parallelism . 63 4.1.1 Method . 64 4.1.2 Numerical simulation . 65 4.2 Parallelism in time . 67 4.2.1 The Parareal algorithm . 68 4.2.2 Ordinary differential equations . 69 4.2.3 Stochastic differential equations . 73 5 Application: Second order stochastic fluid models 89 5.1 Queueing theory and fluid models . 89 5.2 Second order single fluid queue model . 90 5.2.1 Pathwise construction of the dynamics of a single server fluid queue . 91 5.2.2 Diffusion approximation and second order model . 93 5.2.3 Analytical study of second order fluid queues in a random environment . 96 5.2.4 Example of a single fluid queue and numerical results . 102 5.3 Second order stochastic queueing network model . 111 5.3.1 Traffic equations for single class Generalized Jackson network . 111 5.3.2 Pathwise construction of the dynamics of a single class open GJN . 113 5.3.3 Diffusion approximation and second order model . 115 5.3.4 Second order fluid network model analytically . 118 5.3.5 Second order fluid network model numerically . 123 6 Conclusion 134 viii List of Tables 4.1 Timing for a different number of processors. 66 ix x List of Figures 2.1 Histograms for different values of n . 14 2.2 Sample paths of Brownian Motion . 17 2.3 Sample path of Brownian Motion and its reflected path . 21 2.4 Sample path of a Poisson process . 24 2.5 Sample path of a Compound Poisson process of intensity 10 and marks uniformly distributed over 0-1 . 27 3.1 Sample path of the solution for N = 64 . 51 3.2 Sample path of the solution for N = 1024 . 51 3.3 Sample path of the solution for N = 16 . 54 3.4 Sample path of the solution for N = 64 . 54 3.5 GBM sample paths for different values of N . 56 3.6 Log-Log plot of the absolute error versus the time increment for the Euler scheme 58 3.7 Log-Log plot of the absolute error versus the time increment for the Milstein scheme . 58 3.8 Q-Q Plot comparing the simulation result of the Naive Euler algorithm with the exact solution of the GBM . 62 xi xii LIST OF FIGURES 4.1 Speedup versus the number of processors . 66 4.2 System efficiency versus number of processors . 67 4.3 Parareal algorithm [43] . 87 4.4 Parareal simulation of reflected SDE . 87 4.5 First Parareal simulation of GBM . 88 5.1 Single fluid queue . 91 5.2 Markov modulated process . 97 5.3 On-Off process . 103 5.4 Q-Q plot in the fluid flow case σ1 = σ2 = 0. 106 5.5 Q-Q plot when σ1 = 1; σ2 = 0. 107 5.6 Analytical and numerical solution of F(x) versus x when σ1 = 0; σ2 = 0. 107 5.7 Analytical and numerical solution of F(x) versus x when σ1 = 1; σ2 = 0. 108 5.8 Analytical and numerical solution of F(x) versus x when σ1 = 0; σ2 = 1. 108 5.9 Analytical and numerical solution of F(x) versus x when σ1 = σ2 = 1. 109 5.10 Stationary distribution of the buffer content. 110 5.11 Expectation of the buffer content as a function of the variance. 110 5.12 A generalized Jackson network (GJN) . 112 5.13 Tandem of fluid queues . 125 5.14 Mean queue length function of time with Q1(0) equal to 3 . 128 5.15 Mean queue length function of time with Q1(0) equal to 0 . 129 5.16 Variance of the queue length function of time with Q1(0) equal to 3 . 130 5.17 Variance of the queue length function of time with Q1(0) equal to 0 . 131 5.18 Stationary distribution with Q1(0) equal to 3 . 132 5.19 Stationary distribution with Q1(0) equal to 0 . 133 xiii xiv Chapter 1 Introduction Motivation, Objectives and Contribution Stochastic differential equations arise in many fields of study like physics, computing and biol- ogy. They extend the concept of differential equations by taking into consideration the noise factor. Few SDEs can be solved analytically, for this reason, the numerical approach constitute a very important method to solve especially high-dimensional SDEs. But in a lot of cases, they reveal to be highly demanding when the processing time or the memory utilisation is considered. Therefore, recurring to parallelize these methods seems to be a necessity for a lot of problems faced in this field of study. As for example, in the field of physics, a lot of problems related to nonequilibrium statistical mechanics, cosmology, ... seem to involve SDEs like the escape rate from a potential well problem.