Stochastic Calculus and Stochastic Models

PrefaceAcknowledgmentsI. Introduction 0 Motivation, and a Forward Look 1 Random Variables 2 Conditional Expectations 3 Stochastic ProcessesII. Stochastic Integrals 1 Stochastic Models and Properties They Should Possess 2 Definition of the Integral 3 The Canonical Form 4 Elementary Properties of the Integral 5 The Itô-Belated IntegralIII. Existence of Stochastic Integrals 1 Fundamental Lemma 2 Existence of the Stochastic Integral: First Theorem 3 Second Existence Theorem 4 Third and Fourth Existence Theorems 5 The Vanishing of Certain Integrals 6 Special Cases 7 Examples: Brownian Motions; Point Processes 8 Extension to the Itô-Belated IntegralIV. Continuity, Chain Rule, and Substitution 1 Continuity of Sample Functions 2 Differentiation of a Composite Function 3 Applications of Itô's Differentiation Formula 4 Substitution 5 Extension to Itô-Belated IntegralsV. Stochastic Differential Equations 1 Existence of Solutions of Stochastic Differential Equations 2 Linear Differential Equations and Their Adjoints 3 An Approximation Lemma 4 The Cauchy-Maruyama ApproximationVI. Equations in Canonical Form 1 Invariance Under Change of Coordinates 2 Runge-Kutta Approximations 3 Comparision of Ordinary and Stochastic Differential Equations 4 Rate of Convergence of Approximations to Solutions 5 Continuous Dependence of the Solution on the Disturbance 6 Justification of the Canonical Extension in Stochastic ModelingReferencesSubject Index