Brief introduction to Statistical Learning: Regression versus Classification; Linear Regression: simple and multiple Linear Regression; Classification: Logistic Regression, Discriminant Analysis; Resampling Methods: Cross-Validation, the Bootstrap; Regularization: Subset Selection, Ridge Regression, the Lasso, Principle Components and Partial Least Squares Regression; Nonlinear Models: Polynomial; Splines; Generalized Additive Models; Tree-Based Models: Decision Trees, Random Forest, Boosting; Support Vector Machines; Unsupervised Learning: Principle Component Analysis, Clustering Methods.
For further information see the academic catalog: IAM557 - Statistical Learning and Simulation
Part I: Probability spaces, random variables, probability distributions and probability densities, conditional probability, Bayes' formula, mathematical expectation, moments. Part II: Sampling distributions, decision theory, estimation (theory and applications), hypothesis testing (theory and applications), regression and correlation, analysis of variance, non-parametric tests.
For further information see the academic catalog: IAM530 - Elements of Probability and Statistics
FEM for one dimensional problems. Variational formulation and weak solutions. FEM for elliptic equations. FEM spaces. Error analysis and adaptivity. Diffusion-convection equations. Time dependent problems. Iterative solution techniques and preconditioning.
For further information see the academic catalog: IAM572 - Finite Element Methods for Partial Differential Equations: Theory and Applications
Numerical Methods for Discrete Time Models: binomial method for options; discrete time optimal control problems. Reminders on Continuous Models: Ito process and its applications in stock market, Black-Scholes equation and its solution; Hedging, Volatility smile. Monte Carlo Method for Options: generating random numbers, transformation of random variables and generating normal variates; Monte Carlo integration; pricing by Monte Carlo integration; variance reduction techniques, quasi-random numbers and quasi-Monte Carlo method. Finite Difference Methods for Options: explicit and implicit finite difference schemes, Crank-Nicolson method; Free-Boundary Problems for American options. Finite Difference Methods for Control Problems: Markov Chain approximation method, elliptic Hamilton-Jacobi-Bellman equations, computational methods.
For further information see the academic catalog: IAM614 - Methods of Computational Finance
Generating Random Numbers; Basic Principles of Monte Carlo; Numerical Schemes for Stochastic Differential Equations; Simulating Financial Models; Jump-Diffusion and Levy Type Models; Simulating Actuarial Models; Markov Chain Monte Carlo Methods.
For further information see the academic catalog: IAM757 - Monte Carlo Methods in Finance and Insurance