STAT 230 · Year 2 · Semester 2 · 3 credits · Mathematics & Statistics

Stochastic Processes

The mathematical language of time, uncertainty, and ruin: from Markov jumps and Poisson arrivals to continuous diffusions and martingales.

Full practice exam · 30 Q · 120 min

Season 1 · 8 episodes

The Solvency Frontier: Stochastic Mechanics at Boreas Mutual

When an unseasonal storm cluster and a commercial fleet liability crisis threaten Boreas Mutual with regulatory receivership, actuary Maya Lin must deploy the full arsenal of modern stochastic processes—from Markovian bonus-malus systems and continuous jump models to Itô calculus, martingales, and Lundberg ruin bounds—to reconstruct the firm's capital adequacy before the state insurance commission intervenes.

Protagonist · Maya Lin, ASA, Senior Predictive Modeler and Capital Analyst at Boreas Mutual.

Setting · Boreas Mutual headquarters in Chicago, overlooking Lake Michigan during a high-volatility winter storm cycle, alongside regulatory audit chambers.

Stakes · A four-hundred-million-dollar capital deficit flagged by state regulators that threatens Boreas Mutual with insolvency, liquidation of commercial lines, and severe policyholder impairment.

Pick an episode

Episodes

Season finale: Maya solves the Lundberg defective renewal equation and deploys the exponential martingale adjustment coefficient to establish exact dynamic solvency capital requirements, proving the insurer's long-term survival probability to the regulatory board.

From the lecturer: Welcome to STAT 230. Stochastic processes form the engine room of modern actuarial science, financial mathematics, and machine learning. To master this course, you cannot be a passive spectator. When watching each film, keep a dedicated computation notebook open beside you. Pause at every prompt to draw the state-space transition diagram or write down the conditional expectation before I reveal the derivation. Stochastic models demand two parallel fluences: the geometric intuition of sample paths and the analytical rigor of generator matrices, martingales, and renewal equations. This course directly builds the foundations tested in CAS Exam MAS-I (Modern Actuarial Statistics I) and the multi-state contingency frameworks in SOA FAM and ASTAM. Keep a table of signature properties: memorylessness, filtration histories, and martingale stopping conditions. When we derive Lundberg's inequality or Itô's formula, verify every algebraic substitution line by line on your paper. If a path or transition makes conceptual sense but fails algebraically, re-examine your state conditioning immediately. Let us begin.

Cold open

Maya arrives at Boreas Mutual at six in the morning to find the Chief Actuary staring at a state audit notice: the regulator claims Boreas has severely under-reserved its five-level commercial auto bonus-malus book. If Maya cannot model policyholder transitions and long-term equilibrium premiums by midnight, the commission will freeze underwriting.

preparing lecture…

Transcript

Formulate discrete-time Markov chains and verify the Markov property in actuarial contexts.
Compute multi-step transition probabilities using Chapman-Kolmogorov equations and matrix powers.
Classify states into communicating classes, determine periodicity, and identify recurrence or transience.
Derive stationary distributions, mean first passage times, and absorption probabilities for reserving and bonus-malus systems.