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Description of Basic Stochastic Processes |
This is a final year undergraduate text on stochastic processes, a tool used widely by statisticians and researchers working in the mathematics of finance. It provides a detailed treatment of conditional expectation and probability, a topic which in principle belongs to probability theory, but is essential as a tool for stochastic processes.
Although the book is a final year text, the author has chosen to use exercises as the main means of explanation for the various topics, and the book will have a strong self-study element.
The author has concentrated on the major topics within stochastic analysis:
- stochastic processes
- Markov chains
- spectral theory
- renewal theory
- Martingales and its stochastic processes
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Contents of Basic Stochastic Processes |
1. Review of Probability
1.1 Events and Probability
1.2 Random Variables
1.3 Conditional Probability and
Independence
1.4 Solutions
2. Conditional Expectation
2.1 Conditioning on an Event
2.2 Conditioning on a Discrete Random
Variable
2.3 Conditioning on an Arbitrary Random
Variable
2.4 Conditioning on a Alpha-Field
2.5 General Properties
2.6 Various Exercises on Conditional Expectation
2.7 Solutions
3. Martingales in Discrete Time
3.1 Sequences of Random Variables
3.2 Filtrations
3.3 Martingales
3.4 Games of Chance
3.5 Stopping Times
3.6 Optional Stopping Theorem
3.7 Solutions
4. Martingale Inequalities and Convergence
4.1 Doob's Martingale Inequalities
4.2 Doob's Martingale Convergence Theorem
4.3 Uniform Integrability and L(1)
Convergence of Martingales
4.4 Solutions
5. Markov Chains
5.1 First Examples and Definitions
5.2 Classification of States
5.3 Long-Time Behaviour of Markov Chains:
General Case
5.4 Long-Time Behaviour of Markov Chains with Finite State Space
5.5 Solutions
6. Stochastic Processes in Continuous Time
6.1 General Notions
6.2 Poisson Process
6.2.1 Exponential Distribution and Lack
of Memory
6.2.2 Construction of the Poisson Process
6.2.3 Poisson Process Starts from Scratch at Time t
6.2.4 Various Exercises on the Poisson Process
6.3 Brownian Motion
6.3.1 Definition and Basic Properties
6.3.2 Increments of Brownian Motion
6.3.3 Sample Paths
6.3.4 Doob's Maximal L(2) Inequality for
Brownian Motion
6.3.5 Various Exercises on Brownian
Motion
6.4 Solutions
7. Ito Stochastic Calculus
7.1 Ito Stochastic Integral: Definition
7.2 Examples
7.3 Properties of the Stochastic Intergral
7.4 Stochastic Differential and Ito Formula
7.5 Stochastic Differential Equations
7.6 Solutions
Index
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