# Syllabus ASRM 499

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

Text

1. Sheldon M. Ross (2010), Introduction to Probability Models. 10th Edition. Elsevier.
2. Struppeck, T. (2015), Life Contingencies, CAS Study Note.

Prerequisite: Stat 408 or Math 461

Learning Objectives

1. Students will be able to demonstrate their understanding of discrete-time and continuous-time Markov chains and apply this knowledge to simple multiple decrement life insurance models and binomial tree option pricing.

2. Students will be able to demonstrate their understanding of Brownian motion and Ito’s formula and apply this knowledge to simple European call options and variable annuity guaranteed minimum maturity products.

Course topics: 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.

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

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