Stochastic Differential Equations

*Module1 (6.5 hp): Theory.* The module starts with a review of the necessary prerequisites in probability theory, including an introduction to measure theory and stochastic processes . Thereafter (local) martingales and the quadratic variation are introduced with its most famous example being the Brownian motion.  The Ito integral and the Ito calculus are introduced.,  This is applied to solving certain stochastic differential equations (SDE) analytically . Furthermore, the existence- and uniqueness theory for SDE is treated in the Lipschitz case, which naturally leads to numerical methods for simulating solutions to SDEs. The connection between SDE and partial differential equations (PDE) is investigated (e.g.,  the Feynman-Kac equation), which gives the possibility to simulate solutions of PDEs in separate points by using simulations of SDEs.  Additionally, Girsanov's theorem and the martingale representation theorem are discussed, as well as a quick introduction to optimal stopping problems. *Module 2 (1 hp) Computer labs*. The module covers implementation of some numerical method for simulating solutions, fitting model parameters to given data, and the Least-Square-Monte-Carlo (LSMC) method for solving optimal stopping problems.