This graduate course is changes from term-to-term depending on the topic and will be updated to reflect the special topics in terms when this course is offered.

This course focuses on advanced theory and modeling of financial derivatives. The topics include, but are not limited to: HJM interest rate models, LFM and LSM market models; foreign exchange options; defaultable bonds; credit default swaps, equity default swaps and collateralized debt obligations; intensity and structural based models; jump processes and stochastic volatility; commodity models. As well, students are required to complete a project, write a report, and present a topic of current research interest.

The problem of minimizing an expected value is ubiquitous in machine learning, from approximate Bayesian inference to acting optimally in a Markov decision process. Progress on this problem may drive advances in methods for generating novel images, unsupervised discovery of object relations, or continuous control. This course will introduce students to various methodological issues at stake in this problem and lead them in a discussion of its modern developments. Introductory topics may include stochastic gradient descent, gradient estimation, policy and value iteration, and variational inference. The class will have a major project component.

The course discusses modern developments in modeling statistical dependence. Emphasis will be placed on copula models, particularly on conditional copula models that can be used in regression settings.

Tentative topics include: Random Effects; Copula Models for Continuous Data; Dependence measures; Types of Dependence; Conditional Copulas for Continuous Data; Copula Models for Discrete/Mixed Data; Conditional Copula Models for Discrete/Mixed Data; Vines.

Functional data analysis (FDA) has received substantial attention in recent years, with applications arising from various disciplines, such as engineering, public health, finance, etc. In general, the FDA approaches focus on nonparametric underlying models that often assume the data are observed from realizations of stochastic processes with smooth trajectories. This course will cover general issues in functional data analysis, such as functional principal component analysis, functional regression models, curve clustering, and classification. An introduction to smoothing methods will also be included at the beginning of class to provide a basic view of nonparametric regression (kernel and spline types) and serve as the basis of FDA approaches. The course will involve some computing and data analysis using R or matlab.

This course will focus on convergence rates and other mathematical properties of Markov chains on both discrete and general state spaces. Specific methods to be covered will include coupling, minorization conditions, spectral analysis, and more. Applications will be made to card shuffling and to MCMC algorithms.

With the availability of high frequency financial data, new areas of research in stochastic modeling and stochastic control have opened up. This six-week course will introduce students to the basic concepts, questions, and methods that arise in this domain. We will begin with the classical market microstructure models, understand different theories of price formation and price discovery, identify different types of market participants, and then move on to reduced form models. Next, we will investigate some of the typical algorithmic trading strategies employed in industry for different asset classes. Finally, we will develop stochastic optimal control problems for solving optimal liquidation and high frequency market making problems and demonstrate how to solve those problems using the principles of dynamic programming leading to Hamilton-Jacobi-Bellman equations. Students will also have a chance to work with historical limit order book data, develop Monte Carlo simulations and gain a working knowledge of the models and methods. and methods.

Tentative topics include: Market Microstructure; Overview of Stochastic Calculus; Dynamic Programming & HJB -Dynamics of LOB-Optimal Liquidation; Market Making; Risk Measures.

The course will cover modeling, estimation and inference of non-stationary time series. In particular, we will deal with statistical inference of trends, quantile curves, time-varying spectra and functional linear models related to non-stationary time series. With the recent advances in various fields, a systematic account of non-stationary time series analysis is needed.

Modeling the behaviour of extreme values is important in a variety of disciplines, from finance to environmental science, since catastrophes almost inevitably arise from extreme conditions. This course will cover both theoretical and applied aspects of extreme value modeling. Some of the topics to be covered are: extreme value types, point process methodology, the Hill and other estimators of the tail index, estimating extreme quantiles, multivariate extremes, estimators of tail dependence.

Inference based on the likelihood function has a prominent role in both theoretical and applied statistics. This course will introduce some of the more recent developments in likelihood-based inference, with an emphasis on adaptations developed for models with complex structure or large numbers of nuisance parameters. Special emphasis will be given to applications in biology and medicine throughout the course. Tentative topics to be covered include: review of likelihood inference and asymptotic results; adjustments to profile likelihood; misspecified models — composite likelihood; partially specified models — quasi-likelihood; properties and limitations of penalized likelihood.

The aim of this course is to provide an introduction to advanced insurance risk theory. This course covers frequent and severity models, aggregate losses, and compound distributions, EM algorithm, model selection, and estimation.

This course aims to discuss the latest research in insurance risk modelling in general insurance. It covers insurance data analysis, probababilty, and statistical models for insurance ratemaking and reserving, and their estimation procedures.

The general mathematics and logical foundations for statistical inference: geometric, algebraic, and topological symmetries that arise naturally in the solution to the inference problem, including rigorous comparison of the bayesian and frequentist approaches, and the group theoretic considerations of invariance (algebraic and logical symmetry), both on the sample space as well as on the parameter space (and both either implicit or manifest) that must be taken into account in the analysis. Unusual for the development, but fundamental to the inherent logic of such considerations, the finite-finite case is given special attention in respect of both sample space and parameter space.

This is an advanced course in models and methods for spatial data, with an emphasis on data which are not normally distributed. The course will cover different types of random spatial processes and how to incorporate them into mixed effects models for normal and non-Normal data, with maximum likelihood and Bayesian inference used for the two types of data respectively. Spatial point processes, where dare are random locations rather than measurements at fixed locations, will be dealt with extensively. Following the course, students will be able to undertake a variety of analyses on spatially dependent data, understand which methods are appropriate for various research questions, and interpret and convey results in the light of the original questions posed.

A central issue in many current large-scale scientific studies is how to assess statistical significance while taking into account the inherent multiple hypothesis testing issue. This graduate course will provide an in-depth understanding of the topic in the context of data science with a focus on statistical 'omics.' We start with an insightful revisit of single hypothesis testing, the building block of multiple hypothesis testing. We then study the fundamental elements of multiple hypothesis testing, including the control of family-wise error rate and false discovery rate. We will also touch upon various more advanced topics such as data integration, selective inference and fallacy of p-values. The course will provide both analytical arguments and empirical evidence.

Students are evaluated based on class participation and one final research report on a suggested or self-selected project related to multiple hypothesis testing.

Basic concepts in nonstandard analysis, including infinitesimal and infinite numbers, and descriptions of basic concepts like continuity and integration in terms of these notions. Advanced topics, including Loeb measure theory. Applications to stochastic processes and statistics.

This course introduces the research area of causal inference in the intersection of statistics, social science, and artificial intelligence. A central theme of this course will be that without a formal theory of causation, intuition alone can be misleading for drawing causal conclusions. Topics include: potential outcomes and counterfactuals, measures of treatment effects, causal graphical models, confounding adjustment, instrumental variables, principal stratification, mediation, and interference. Concepts will be illustrated with applications in a wide range of subjects, such as computer science, social science, and biomedical data science.

This course will give an overview of robust statistical methods, that is, methods that are insensitive to outliers or other data contamination. Topics will include theoretical notions such as qualitative robustness and breakdown point, robust estimation of location (minimax variance and bias) and scale parameters, robust estimation in regression and multivariate analysis, and applications (including in computer vision).

Optimal transport is a vast subject and has deep connections with analysis, probability, and geometry. In recent years optimal transport has found widespread applications in data science (a notable example is the Wasserstein GAN). In this course we offer a balanced treatment featuring both the theory and applications of the subject. After laying down the theoretical foundation including the Kantorovich duality, we turn to numerical methods and their applications to data science. Possible topics include entropic regularization, dynamic formulations, gradient flows, statistical divergences and the W-GAN. Our main reference is the recent book Computational Optimal Transport by Gabriel Peyré and Marco Cuturi.

The concept of statistical evidence is central to the field of statistics. In spite of many references to "the evidence" in statistical applications, it is fair to say that there is no definition of this that achieves broad support in the sense of serving as the core of a theory of statistics. The course will examine the various attempts made to measure evidence in the statistical literature and why these are not entirely satisfactory. A proposal to base the theory of statistical inference on a particular measure, the relative belief ratio, is discussed and how this fits into a general theory of statistical reasoning.

This course provides an overview of the core areas of demography (fertility, mortality and migration) and the techniques to model such processes. The course will cover life table analysis, measures of fertility and nuptiality, mortality and migration models, and statistical methods commonly used in demography, such as Poisson regression, survival analysis, and Bayesian hierarchical models. The goal of the course is to equip students with a range of demographic techniques to use in their own research.

The course will introduce students to the basic theory of stochastic optimal control. We will cover both the analytic approach, including an introduction to viscosity solution theory, and the probabilistic approach which is based on BSDE and the stochastic maximum principle. Applications to portfolio optimization and contract theory will be discussed.

Random matrix theory is now a big subject with applications in many disciplines of science, engineering, and statistics. This course will cover fundamental concepts, principal and theory in random matrix theory, orienting towards the needs and interests in statistics. Applications to big data analytics and geometric data analysis are provided.

This course introduces the theory of modelling dependence in statistical/stochastic models, including copulas and factor models. Typically, data of joint (rare) events are scarce making dependence modelling highly challenging. In financial and insurance risk management, however rare events are prevalent, and misspecification in the dependence structure may greatly impact risk management decisions.

This course provides, additional to copulas and factor models, an overview of financial risk management including risk measures and regulation, such as the Basel accords, with a focus on dependence modelling. It further covers risk assessment and risk management under dependence uncertainty.

Nonstandard analysis provides a rigorous foundation for carrying out mathematical analysis with the aid of infinitesimal numbers and other structures that appear in so-called saturated models of the real numbers. This course introduces nonstandard analysis using concepts and examples from statistics and probability. Topics include: extension, transfer, and saturation; infinitesimal and infinite numbers; hyperfinite sets and measures; hyperfinite models of stochastic processes; nonstandard Bayesian decision theory and connections to frequentism. Background in real analysis, probability theory, and statistics recommended. No background will be assumed in mathematical logic.

This course explores the replication of financial derivatives from the standpoint of investment banks ("sell-side") and the application of derivatives from the standpoint of pension funds, insurers, hedge funds, mutual funds and private equity funds ("buy-side").

The course is structured into three components: 1. The first module analyses how trading and structuring desks at investment banks use vanilla options to create bespoke payouts for institutional investors, corporates, and retail investors. 2. The second module examines how the buy-side uses derivatives for: Hedging: e.g., protecting traditional balanced portfolios, managing currency risk; Outperforming benchmarks: currency and equity overlay; Expressing "macro" views on equity indices, rates, currencies, and commodities; Expressing "micro" views on sectors and single stocks; And addresses why investor preferences give rise to risk premia, and how derivatives can be structured to take advantage of persistent behavioural biases in the market. 3. The third module synthesizes the key learnings from 1. and 2. into case studies.

Information geometry is the geometric study of statistical manifolds which are spaces of probability or nonnegative measures. This course is an introduction to information geometry and some of its recent developments. Mathematical prerequisites such as di erential geometry and convex analysis will be introduced as needed.