First Variation: computing the first variation, Euler-Lagrange equation, extensions; Applications: brachistochrone, Lagrangian and Hamiltonian dynamics; Second Variation: computing the second variation, Ricatti equation, convexity and minimisers; Multivariable Variational Problems: eigenpairs, minimal surfaces, gradient flows; Optimal Control Theory: time-optimal linear control, Pontryagin Maximum Principle; Applications: linear-quadratic regulator, production-consumption, optimal harvesting; Dynamic Programming: Hamilton-Jacobi-Bellmann equation, general linear-quadratic regulator; Further Topics on Differential Games, Stochastic Control Theory
For further information see the academic catalog: IAM773
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
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
Probability, Random Processes, and Statistics; Markov Chains; Sampling and Monte Carlo Methods; Parameter Estimation; Uncertainty Propagation in Models; Stochastic Spectral Methods; Surrogate Models and Advanced Topics.
For further information see the academic catalog: IAM768
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
\( \LaTeX \) and Matlab; Basic Commands and Syntax of \( \LaTeX \) and Matlab; Working within a Research Group via Subversion; Arrays and Matrices; Scripts and Function in Matlab; Commands and Environments in \( \LaTeX \); More on Matlab Functions; Toolboxes of Matlab; Packages in \( \LaTeX \); Graphics in Matlab; Handling Graphics and Plotting in \( \LaTeX \); Advanced Techniques in Matlab: memory allocation, vectoristaion, object orientation, scoping, structures, strings, file streams.
For further information see the academic catalog: IAM591
Review of linear algebra and multivariate calculus, sequences of series and functions, Reimann-Stieltjes and Lebesgue integration.
For further information see the academic catalog: IAM527
Review of Programming and Toolboxes, Packages, Modules; Iterative Linear Algebra Problems; Root Finding Programs; Recursive Functions and Algorithms; Optimisation Algorithms; Data Fitting and Interpolation; Extrapolation; Numerical Integration; Numerical Solutions of Differential Equations: IVPs and BVPs; Selected Topics (algorithms and coding in different fields).
For further information see the academic catalog: IAM592
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 Hamiltion-Jacobi-Bellman equations, computational methods.
For further information see the academic catalog: IAM614
This course introduces time series methodology emphasizing the data analytic aspects related to financial applications. Topics that will be discussed are as follows: Univariate linear stochastic models: ARMA and ARIMA models building and forecasting using these models. Univariate non-linear stochastic models: Stochastic variance models, ARCH processes and other non-linear univariate models. Topics in the multivariate modeling of financial time series. Applications of these techniques to finance such as time series modeling of equity returns, trading day effects and volatility estimations will be discussed.
For further information see the academic catalog: IAM526
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