## ASRM 595. Adv Topics in Act Sci & Risk

**Fall 2018: Stochastic Processes for Finance and Insurance**

**Text**

- Sheldon M. Ross (2010),
*Introduction to Probability Models*. 10^{th}Edition. Elsevier. - Struppeck, T. (2015),
*Life Contingencies*, CAS Study Note.

**Course topics:** This course emphasizes techniques of stochastic processes and introductory applications to actuarial science, finance and economics. Topics include conditional probability and expectation, Markov Chains, reliability theory, Brownian motion and simulations. We will quickly review the materials from the first 3 chapters. We will then cover most of materials in chapters 4, 5, 6, 9 and 10 of the book.

**Learning Outcomes**

1. Students will be able to demonstrate understanding of stochastic processes including Markov Chains, reliability theory and Brownian motion.

2. Students will be able to demonstrate understanding of stochastic processes by solving problems from the fields of actuarial science, finance and economics.

**Chapter 1 Introduction to Probability Theory (1.5 hours)**

- Sample Space and Events
- Probabilities Defined on Events
- Conditional Probabilities
- Independent Events
- Bayes’ Formula

**Chapter 2 Random Variables (3 hours)**

- Discrete Random Variables
- Continuous Random Variables
- Expectation of a Random Variable
- Jointly Distributed Random Variables
- Moment Generating Functions
- Limit Theorems
- Stochastic Processes

**Chapter 3 Conditional Probability and Conditional Expectation (3 hours)**

- The Discrete Case
- The Continuous Case
- Computing Expectations by Conditioning
- Computing Probabilities by Conditioning
- An Identity for Compound Random Variables

**Chapter 4 – Markov Chain (12 hours)**

- Chapman–Kolmogorov Equations
- Classification of States
- Limiting Probabilities
- Some Applications
- Branching Processes
- Time Reversible Markov Chains
- Markov Chain Monte Carlo Methods

**Chapter 5 – The Exponential Distribution and the Poisson Process (6 hours)**

- The Exponential Distribution
- The Poisson Process
- Generalizations of the Poisson Process

**Chapter 6 – Continuous-Time Markov Chains (6 hours)**

- Continuous-Time Markov Chains
- Birth and Death Processes
- The Transition Probability Function
- Limiting Probabilities

**Chapter 9 – Reliability Theory (4.5 hours)**

- Structure Functions
- Reliability of Systems of Independent Components
- System Life as a Function of Component Lives
- Expected System Lifetime

**Chapter 10 Brownian Motion and Stationary Processes (4.5 hours)**

- Brownian Motion
- Variations on Brownian Motion
- Pricing Stock Options

**Struppeck, T., Life Contingencies, CAS Study Note, September 2015 (1.5 hours)**

Briefly covers applications of Markov Chains in life insurance

For graduate credits, the following topics will be covered through group projects.

**Chapter 11 Simulations **

- General Techniques for Simulating Continuous Random Variables
- Simulating from Discrete Distributions
- Stochastic Processes
- Variance Reduction Techniques

**Midterm Exams (1 hour)**

**Total: 43 hours**