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Description: 
The course covers the theoretical
foundations that are necessary for modeling stochastic processes in
the financial world. 


Topics: 



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 RadonNikodym
derivative;

Laplace and Fourier transforms;

independence and uncorrelatedness;

conditional expectation, Bayes
formula;

stopping times.




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, FeynmanKac theorem;

stochastic control, HamiltonJacobiBellman
equation.




contingent claims, the payoff function, derivatives products;

selffinancing 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.




the existence and uniqueness of the
risk neutral measure;

contingent claims risk neutral
valuation, options pricing, BlackScholesMerton equation ,
BlackScholesMerton formula;

optimal portfolio and consumption
choice, the Merton problem;

extensions of the model using
stochastic volatility, Heston model.




Teaching: 
42 hours during second semester 


