Calculus I Ordinary Differential Equations Probability

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: diffusive processes and Fokker-Planck equation 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 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: basic theory 1. Stochastic diffusive processes 1.1 Brownian motion, models of Einstein and Langevin 1.2 White noise and Wiener process 1.3 Fokker-Planck equation 2. Itô's calculus 2.1 Stochastic integral 2.2 Stochastic differential equation and Itô calculus 2.3 Properties of stochastic differential equations 2.4 Relation to partial differential equations: Feynman-Kac formula 3. Numerical methods for stochastic equations. 3.1 Euler-Maruyama method 3.2 Higher-order methods 3.3 Weak and strong convergence of numerical algorithms 3.4 Extension to bounded diffusions 3.5 Langevin simulations Part Two: applications 4. Biochemical kinetics 5. Black-Scholes model; option pricing 6. Stochastic optimal control; Merton's optimal portfolio 7. Biological evolution

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.