Stochastic Processes

Markov chains, Poisson processes, Brownian motion, and their actuarial applications.

Key Concepts

• •Markov chains: transition matrices, stationary distributions, and classification of states
• •Poisson process: homogeneous and non-homogeneous arrival processes
• •Compound Poisson process: modeling aggregate claims
• •Continuous-time Markov chains: transition rates and Kolmogorov equations
• •Brownian motion: properties and applications in financial modeling
• •Random walks: discrete-time models for surplus processes
• •Ruin theory: probability of ruin, adjustment coefficient, and Lundberg inequality
• •Renewal theory: renewal equations and their actuarial applications
• •Multi-state models: disability, death, and recovery transitions
• •Martingales: fair game interpretation and applications in pricing

Study Tips

• 1.Start with discrete-time Markov chains before moving to continuous-time.
• 2.Practice computing transition probabilities for small state spaces.
• 3.Understand the memoryless property and how it simplifies Markov and Poisson models.
• 4.Work through ruin theory problems with different claim distributions.
• 5.Connect multi-state Markov models to life contingency applications.

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