Preface | p. vii |
Notation | p. xv |
Brownian Motions and Stochastic Integrals | p. 1 |
Introduction | p. 1 |
Probability Theory | p. 4 |
Stochastic Processes | p. 12 |
Brownian Motions | p. 18 |
Stochastic Integrals | p. 23 |
Ito's Formula | p. 38 |
Markov Processes | p. 43 |
Generalised Ito's Formula | p. 47 |
Exercises | p. 49 |
Inequalities | p. 51 |
Introduction | p. 51 |
Frequently Used Inequalities | p. 51 |
Gronwall-Type Inequalities | p. 54 |
Matrices and Inequalities | p. 58 |
Linear Matrix Inequalities | p. 62 |
M-Matrix Inequalities | p. 67 |
Stochastic Inequalities | p. 69 |
Exercises | p. 75 |
Stochastic Differential Equations with Markovian Switching | p. 77 |
Introduction | p. 77 |
Stochastic Differential Equations | p. 77 |
Existence and Uniqueness of Solutions | p. 81 |
SDEs with Markovian Switching | p. 88 |
L[superscript p]Estimates | p. 96 |
Almost Surely Asymptotic Estimates | p. 101 |
Solutions as Markov Processes | p. 104 |
Exercises | p. 110 |
Approximate Solutions | p. 111 |
Introduction | p. 111 |
Euler-Maruyama's Approximations | p. 111 |
Global Lipschitz Case | p. 114 |
Local Lipschitz Case | p. 118 |
More on Local Lipschitz Case | p. 121 |
Caratheodory's Approximations | p. 126 |
Split-Step Backward Euler Scheme | p. 134 |
Backward Euler Scheme | p. 146 |
Stochastic Theta Method | p. 149 |
Exercises | p. 154 |
Boundedness and Stability | p. 155 |
Introduction | p. 155 |
Asymptotic Boundedness | p. 157 |
Exponential Stability | p. 164 |
Nonlinear Jump Systems | p. 178 |
Multi-Dimensional Linear Equations | p. 180 |
Scalar Linear Equations | p. 182 |
Examples | p. 187 |
Moment and Almost Sure Asymptotic Stability | p. 191 |
Stability in Probability | p. 204 |
Asymptotic Stability in Distribution | p. 210 |
Exercises | p. 226 |
Numerical Methods for Asymptotic Properties | p. 229 |
Introduction | p. 229 |
Euler-Maruyama's Method and Exponential Stability | p. 230 |
Euler-Maruyama's Method and Lyapunov Exponents | p. 239 |
Generalised Results and Stochastic Theta Method | p. 241 |
Asymptotic Stability in Distribution of the EM Method: Constant Step Size | p. 249 |
Stability in Distribution of the EM Method | p. 249 |
Sufficient Criteria for Assumptions 6.16-6.18 | p. 256 |
Convergence of Stationary Distributions | p. 265 |
Asymptotic Stability in Distribution of the EM Method: Variable Step Sizes | p. 267 |
Exercises | p. 270 |
Stochastic Differential Delay Equations with Markovian Switching | p. 271 |
Introduction | p. 271 |
Stochastic Differential Delay Equations | p. 273 |
SDDEs with Markovian Switching | p. 277 |
Moment Properties | p. 282 |
Asymptotic Boundedness | p. 285 |
Exponential Stability | p. 289 |
Approximate Solutions | p. 294 |
Exercises | p. 300 |
Stochastic Functional Differential Equations with Markovian Switching | p. 301 |
Introduction | p. 301 |
Stochastic Functional Differential Equations | p. 301 |
SFDEs with Markovian Switching | p. 303 |
Boundedness | p. 305 |
Asymptotic Stability | p. 308 |
Razumikhin-Type Theorems on Stability | p. 311 |
Examples | p. 314 |
Exercises | p. 317 |
Stochastic Interval Systems with Markovian Switching | p. 319 |
Introduction | p. 319 |
Interval Matrices | p. 320 |
SDISs with Markovian Switching | p. 322 |
Razumikhin Technology on SDISs | p. 328 |
Delay Independent Criteria | p. 328 |
Delay Dependent Criteria | p. 334 |
Examples | p. 341 |
SISs with Markovian Switching | p. 346 |
Exercises | p. 349 |
Applications | p. 351 |
Introduction | p. 351 |
Stochastic Population Dynamics | p. 351 |
Global Positive Solutions | p. 353 |
Ultimate Boundedness | p. 356 |
Moment Average in Time | p. 358 |
Stochastic Financial Modelling | p. 360 |
Non-Negative Solutions | p. 361 |
The EM Approximations | p. 363 |
Stochastic Volatility Model | p. 370 |
Options Under Stochastic Volatility | p. 375 |
Stochastic Stabilisation and Destabilisation | p. 379 |
Stochastic Neural Networks | p. 387 |
Exercises | p. 394 |
Bibliographical Notes | p. 395 |
Bibliography | p. 397 |
Index | p. 407 |
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