Stochastic Processes Syllabus

The syllabus closely follows the textbook Numerical Solution of Stochastic Differential Equations, by Peter E Kloeden and Eckhard Platen. We will take the chapters in the following order: 1, 2, 3, 4, and 6. I am only "promising" to get through these 5 chapters. But I also promise you there is a LOT of material in these 5 chapters. I will judge the pace of the course with student surveys and if we are able to go faster will will continue with 5.1, 8, 9, 10.1 through 10.5, 11.1 through 11.4, 12.1 through 12.4, 13, 14.1 through 14.4, 15.1 through 15.5, 16 and 17 for as far as we get (following Kloeden and Platen's suggested sequence for Engineers).

Chapter 1: Probability and Statistics

All prerequisite concepts from probability theory are presented and reviewed in Chapters 1 and 2. But not everything in these chapters will be a review --- I expect that a lot of material in the first two chapters will be new to students.

• Probabilities and Events. Sample space. Sure event. Frequency interpretation of probability. Facts about events and probabilities that will later make axioms. The challenges of uncountable sample spaces. Probability space. Conditional probability. Independent events.

• Random Variables and Distributions.

• Random Number Generators.

• Moments

• Convergence of Random Sequences

• Basic Ideas About Stochastic Processes. Discrete and continuous state and discrete and continuous time stochastic processes. Markov chains and accompanying theory.

• Diffusion Processes

• Wiener Process and White Noise

• Statistical Tests And Estimation.

Chapter 2: Probability and Stochastic Processes

• Aspects of Measure and Probability Theory

• Integration and Expectations

• Stochastic Processes

• Diffusion and Wiener Processes

Chapter 3: Ito Stochastic Calculus

• Introduction

• The Ito Stochastic Integral

• The Ito Formula

• Vector valued Ito Integrals

• Other Stochastic Integrals. The Stratonovich integral.

Chapter 4: Stochastic Differential Equations

• Introduction

• Linear Stochastic Differential Equations

• Reducible Stochastic Differential Equations

• Some Explicitly Solvable Equations

• The Existence and Uniqueness of Strong Solutions

• Strong Solutions as Diffusion Processes

• Diffusion Processes as Weak Solutions

• Vector Stochastic Differential Equations

• Stratonovich Stochastic Differential Equations

Chapter 6: Modeling with Stochastic Differential Equations

• Ito versus Stratonovich

• Diffusion Limits of Markov Chains

• Stochastic Stability

• Parametric Estimation

• Optimal Stochastic Control

• Filtering (e.g. Kalman filtering)