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Brownian motion and stochastic differential equations

The Brownian motion is a canonical example of a stochastic process which is both a martingale and a Markov process. Using the Brownian motion we develop stochastic integrals as well as stochastic differential equations, which are then used to formulate and study problems of stochastic control and stopping.

The course treats Brownian motion (the Wiener process), Itô integrals, Itô's formula and stochastic differential equations as well as their properties and relations to partial differential equations. The course also covers optimal stopping theory and the theory of optimal stochastic control, as well as applications. In addition, some basic concepts of measure theoretic probability theory is treated.