Table of Contents

Dedication
Preface to the Second Edition
Preface from the 1997 Edition

Acknowledgements


General Notation



1: Brownian Motions and Stochastic Integrals

1.1 INTRODUCTION
1.2 BASIC NOTATIONS OF PROBABILITY THEORY
1.3 STOCHASTIC PROCESSES
1.4 BROWNIAN MOTIONS
1.5 STOCHASTIC INTEGRALS
1.6 ITÔ’S FORMULA
1.7 MOMENT INEQUALITIES
1.8 GRONWALL-TYPE INEQUALITIES


2: Stochastic Differential Equations

2.1 INTRODUCTION
2.2 STOCHASTIC DIFFERENTIAL EQUATIONS
2.3 EXISTENCE AND UNIQUENESS OF SOLUTIONS
2.4 LP-ESTIMATES
2.5 ALMOST SURELY ASYMPTOTIC ESTIMATES
2.6 CARATHEODORY’S APPROXIMATE SOLUTIONS
2.7 EULER–MARUYAMA’S APPROXIMATE SOLUTIONS
2.8 SDE AND PDE: FEYNMAN–KAC’S FORMULA
2.9 THE SOLUTIONS AS MARKOV PROCESSES


3: Linear Stochastic Differential Equations

3.1 INTRODUCTION
3.2 STOCHASTIC LIOUVILLE’S FORMULA
3.3 THE VARIATION-OF-CONSTANTS FORMULA
3.4 CASE STUDIES
3.5 EXAMPLES


4: Stability of Stochastic Differential Equations

4.1 INTRODUCTION
4.2 STABILITY IN PROBABILITY
4.3 ALMOST SURE EXPONENTIAL STABILITY
4.4 MOMENT EXPONENTIAL STABILITY
4.5 STOCHASTIC STABILIZATION AND DESTABILIZATION
4.6 FURTHER TOPICS


5: Stochastic Functional Differential Equations

5.1 INTRODUCTION
5.2 EXISTENCE-AND-UNIQUENESS THEOREMS
5.3 STOCHASTIC DIFFERENTIAL DELAY EQUATIONS
5.4 EXPONENTIAL ESTIMATES
5.5 APPROXIMATE SOLUTIONS
5.6 STABILITY THEORY—RAZUMIKHIN THEOREMS
5.7 STOCHASTIC SELF-STABILIZATION


6: Stochastic Equations of Neutral Type

6.1 INTRODUCTION
6.2 NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS
6.3 NEUTRAL STOCHASTIC DIFFERENTIAL DELAY EQUATIONS
6.4 MOMENT AND PATHWISE ESTIMATES
6.5 Lp-CONTINUITY
6.6 EXPONENTIAL STABILITY


7: Backward Stochastic Differential Equations

7.1 INTRODUCTION
7.2 MARTINGALE REPRESENTATION THEOREM
7.3 EQUATIONS WITH LIPSCHITZ COEFFICIENTS
7.4 EQUATIONS WITH NON–LIPSCHITZ COEFFICIENTS
7.5 REGULARITIES
7.6 BSDE AND QUASILINEAR PDE


8: Stochastic Oscillators

8.1 INTRODUCTION
8.2 THE CAMERON–MARTIN-GIRSANOV THEOREM
8.3 NONLINEAR STOCHASTIC OSCILLATORS
8.4 LINEAR STOCHASTIC OSCILLATORS
8.5 ENERGY BOUNDS


9: Applications to Economics and Finance

9.1 INTRODUCTION
9.2 STOCHASTIC MODELLING IN ASSET PRICES
9.3 OPTIONS AND THEIR VALUES
9.4 OPTIMAL STOPPING PROBLEMS
9.5 STOCHASTIC GAMES


10: Stochastic Neural Networks

10.1 INTRODUCTION
10.2 STOCHASTIC NEURAL NETWORKS
10.3 STOCHASTIC NEURAL NETWORKS WITH DELAYS


11: Stochastic Delay Population Systems

11.1 INTRODUCTION
11.2 NOISE INDEPENDENT OP POPULATION SIZES
11.3 NOISE DEPENDENT ON POPULATION SIZES: PART I
11.4 NOISE DEPENDENT ON POPULATION SIZES: PART II
11.5 STOCHASTIC DELAY LOTKA–VOLTERRA FOOD CHAIN


Bibliographical Notes
References
Index

About the Author

Xuerong Mao, Strathclyde University, UK

Reviews

A helpful book for both experts and beginners in pure and applied mathematics, and in probability theory, systems dynamics, and control theory. An enjoyable read., (Review of the first edition) Professor Martynuk, Ukraine Academy of Sciences
…a welcome and important addition to stochastic differential equations. …giving a clear presentation of the fundamental underpinnings of stochastic differential equations [including the] known theory. …also the development of new results and methods. …both the depth and breadth of the coverage are remarkable., Professor G.G. Yin, Wayne State University, USA