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Stochastic Differential Equations and Applications

2nd Edition - December 30, 2007

Author: X Mao

eBook ISBN:

9 7 8 - 0 - 8 5 7 0 9 - 9 4 0 - 2

This advanced undergraduate and graduate text has now been revised and updated to cover the basic principles and applications of various types of stochastic systems, with much on… Read more

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This advanced undergraduate and graduate text has now been revised and updated to cover the basic principles and applications of various types of stochastic systems, with much on theory and applications not previously available in book form. The text is also useful as a reference source for pure and applied mathematicians, statisticians and probabilists, engineers in control and communications, and information scientists, physicists and economists.

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 L^P-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 L^p-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