Explain key stochastic processes such as Poisson processes, Markov chains, Brownian motion, and Lévy processes, and describe their properties and applications in insurance and finance.

Model insurance claim processes using compound Poisson processes and related risk models, including the classical Cramér–Lundberg model.

Apply Markov processes (discrete and continuous time) to model transitions between financial or insurance-related states (e.g., credit ratings, health states, policyholder statuses).

Derive and analyze ruin probabilities and survival functions in the context of risk theory.

Use Brownian motion and stochastic calculus, including Itô’s lemma, to model and analyze financial markets, including asset pricing and interest rate modeling.

Construct and analyze stochastic differential equations (SDEs) for dynamic modeling of financial and insurance phenomena.

Apply martingale techniques to problems in financial mathematics, such as arbitrage-free pricing and hedging of contingent claims.

Implement numerical methods and simulations (e.g., Monte Carlo methods) to solve problems involving stochastic models.

Evaluate and interpret real-world financial and insurance data using stochastic process models.

Communicate mathematical and statistical findings effectively, both in written reports and oral presentations, with relevance to actuarial and financial contexts.