Calculus I Ordinary Differential Equations Partial Differential Equations Probability Basic programming

CB6, CB7, CB9, CB10 CG1, CG2, CG3, CG5, CG6 CE1, CE3, CE5, CE6, CE7, CE8, CE9, CE11 Understand the basic aspects of stochastic modelling: discrete time models; descriptions of random motion; Brownian motion, models of Einstein and Langevin Get acquainted with stochastic processes in continuous time, in particular the Wiener process. Grasp the motivation and subtleties behind the definitions of stochastic integrals, as well as the definition and properties of stochastic differential equations. Get acquainted with Itô's calculus and its relation with partial differential equations via the Feynman-Kac formula Understand and know how to program the basic numerical methods for stochastic differential equations and Langevin simulations, as well as the arising numerical errors Know the most paradigmatic applications of stochastic differential equations in finance and biology

Part One: introduction to stochastic calculus 1.1 Recap of probability; characteristic functions 1.2 The Law of Large Numbers and the Central Limit Theorem 1.3 Brownian motion; models of Einstein and Langevin 1.4 Wiener process and stochastic integral 1.5 Stochastic differential equations and Itô calculus; paradigmatic SDEs 1.6 Euler-Maruyama method; weak and strong convergence 1.7 Feynman-Kac formula Part Two: applications in biology 2.1 Stochastic models of population 2.2 Stochastic models of epidemics Part Three: financial applications 3.1 Financial options; arbitrage; hedging 3.2 Black-Scholes equation; martingales; risk-neutral probability 3.3 Analytical solution; by finite differences; by stochastic simulation 3.4 Implied volatility; Heston's and other advanced models

Class hours will be devoted to the following supervised learning activities: * Master classes / teacher presentations, in which the main concepts of the course are developed, that students are expected to learn. In order to facilitate this, students will be provided with class notes. Bibliography is also provided to complement the students' learning and enable them to dive further in those topics more interesting to them. * Practical classes, in which problems are didactically solved, supervised computer practice is carried out in the computer room, or students publicly present their work. These classes help develop specific skills. Additionally, there will be 2 office hours devoted to tutoring students, consisting in individualised teaching activities of theoretical and practical type, such that they call for closer supervision of a teacher even though they might be carried out autonomously by the student. Such activities may be, among other: scheduled tutorials, correction of student's work, and student mentoring. The remaining credits are earmarked for student's self-study or group study without teacher supervision. During this time, the student solves proposed exercises and reads supplementary texts suggested by the teacher, as well as other texts from the subject's syllabus. During the time, the student may use the computer room.