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Description:	The course covers the theoretical foundations that are necessary for modeling stochastic processes in the financial world.
Topics:	Elements of measure and probability theory measure, probability, measure and probability spaces, measurable functions; discrete, continuous and absolutely continuous random variables: the distribution, the distribution function, the density function; the Lebesgue integral, the expectation of a random variable, Fubini theorem, the Radon-Nikodym derivative; Laplace and Fourier transforms; independence and uncorrelatedness; conditional expectation, Bayes formula; stopping times. Elements of stochastic calculus   stochastic processes, Markov processes, martingales, Brownian motion; the stochastic integral, the change of variables formula (Ito formula); Girsanov theorem, the martingale representation theorem, Levy theorem; stochastic differential equations, diffusions, Feynman-Kac theorem; stochastic control, Hamilton-Jacobi-Bellman equation. Continuous time financial market models contingent claims, the payoff function, derivatives products; self-financing portfolios, arbitrage portfolio; efficient (free of arbitrage opportunities) markets, complete markets; the risk neutral measure, the fundamental theorem of asset pricing; martingale measures, the change of numeraire. Black-Scholes-Merton financial market model the existence and uniqueness  of the risk neutral measure; contingent claims risk neutral valuation, options pricing, Black-Scholes-Merton equation , Black-Scholes-Merton formula; optimal portfolio and consumption choice, the Merton problem; extensions of the model using stochastic volatility, Heston model.
Teaching:	42 hours during second semester