Numerical Methods in Computational Finance : A Partial Differential Equation (PDE/FDM) Approach

Cover -- Title Page -- Copyright -- Contents -- Preface -- Who Should Read this Book? -- PART A Mathematical Foundation for One‐Factor Problems -- CHAPTER 1 Real Analysis Foundations for this Book -- 1.1 Introduction and Objectives -- 1.2 Continuous Functions -- 1.2.1 Formal Definition of Continuity -- 1.2.2 An Example -- 1.2.3 Uniform Continuity -- 1.2.4 Classes of Discontinuous Functions -- 1.3 Differential Calculus -- 1.3.1 Taylor's Theorem -- 1.3.2 Big O and Little o Notation -- 1.4 Partial Derivatives -- 1.5 Functions and Implicit Forms -- 1.6 Metric Spaces and Cauchy Sequences -- 1.6.1 Metric Spaces -- 1.6.2 Cauchy Sequences -- 1.6.3 Lipschitz Continuous Functions -- 1.7 Summary and Conclusions -- CHAPTER 2 Ordinary Differential Equations (ODEs), Part 1 -- 2.1 Introduction and Objectives -- 2.2 Background and Problem Statement -- 2.2.1 Qualitative Properties of the Solution and Maximum Principle -- 2.2.2 Rationale and Generalisations -- 2.3 Discretisation of Initial Value Problems: Fundamentals -- 2.3.1 Common Schemes -- 2.3.2 Discrete Maximum Principle -- 2.4 Special Schemes -- 2.4.1 Exponential Fitting -- 2.4.2 Scalar Non‐Linear Problems and Predictor‐Corrector Method -- 2.4.3 Extrapolation -- 2.5 Foundations of Discrete Time Approximations -- 2.6 Stiff ODEs -- 2.7 Intermezzo: Explicit Solutions -- 2.8 Summary and Conclusions -- CHAPTER 3 Ordinary Differential Equations (ODEs), Part 2 -- 3.1 Introduction and Objectives -- 3.2 Existence and Uniqueness Results -- 3.2.1 An Example -- 3.3 Other Model Examples -- 3.3.1 Bernoulli ODE -- 3.3.2 Riccati ODE -- 3.3.3 Predator‐Prey Models -- 3.3.4 Logistic Function -- 3.4 Existence Theorems for Stochastic Differential Equations (SDEs) -- 3.4.1 Stochastic Differential Equations (SDEs) -- 3.5 Numerical Methods for ODEs -- 3.5.1 Code Samples in Python -- 3.6 The Riccati Equation.