Stochastic Calculus and Stochastic Models

Probability and Mathematical Statistics: A Series of Monographs and Textbooks: Stochastic Calculus and Stochastic Models focuses on the properties, functions, and applications of stochastic integrals. The publication first ponders on stochastic integrals, existence of stochastic integrals, and continuity, chain rule, and substitution. Discussions focus on differentiation of a composite function, continuity of sample functions, existence and vanishing of stochastic integrals, canonical form, elementary properties of integrals, and the Itô-belated integral. The book then examines stochastic differential equations, including existence of solutions of stochastic differential equations, linear differential equations and their adjoints, approximation lemma, and the Cauchy-Maruyama approximation. The manuscript takes a look at equations in canonical form, as well as justification of the canonical extension in stochastic modeling; rate of convergence of approximations to solutions; comparison of ordinary and stochastic differential equations; and invariance under change of coordinates. The publication is a dependable reference for mathematicians and researchers interested in stochastic integrals.

Preface

Acknowledgments

I. Introduction

0 Motivation, and a Forward Look

1 Random Variables

2 Conditional Expectations

3 Stochastic Processes

II. 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 Integral

III. 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 Integral

IV. 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 Integrals

V. 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 Approximation

VI. 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 Modeling

References

Subject Index