The lecture covers the following topics:

Semimartingales and stochastic integration
Finite variation processes, Lebesgue-Stieltjes integration, square-integrable martingales, local martingales, quadratic variation, semimartingales, stochastic integration, Ito’s formula.

Option pricing in semimartingale markets
Financial markets, trading strategies, arbitrage, equivalent martingale measures, risk neutral pricing, complete markets.

Standard Brownian market models
Fundamental theorem of standard markets, Levy’s characterization, Girsanov’s theorem, arbitrage, martingale representation, completeness, PDE pricing, diffusion markets, stochastic differential equations.

Stochastic volatility and interest rate theory
Implied volatility, volatility surfaces, Dupire’s local volatility model, overview of stochastic volatility models, fixed income markets, types of interest rates, short rate models, instantaneous forward rates, HJM framework.

Optimal investment and stochastic control
Investor preferences, utility functions, Merton’s optimal investment problem, stochastic optimal control, dynamic programming, Merton problem with power utility.