## Probability Seminar, Spring 2021

Time: Tuesdays, 3:00pm - 4:00pm

### Title: Linear Regression and Its Application in Finance. Part 2

As one of the most widely used statistical methods, linear regression is a basic statistical tool in empirical financial data analysis. In this talk, we briefly review standard linear regression theory from the statistical point of view, including statistical inference (confidence intervals and ANOVA), model selection, and regression diagnostics. By extending the theory to stochastic regressors, one can bridge the gap between standard linear regression theory and its application in financial data analysis. We will use R for computation, data analysis, and data visualization.

Slides: 2021-03-30_Wang.pdf

### Title: A crash course on Dirichlet processes. Part 3

In the third talk in this series, we consider the problem of the "broken pressed penny machine" from the previous talk. Imagine a machine at a museum that produces a "pressed penny", by mangling and distorting a penny that you provide. The resulting pressed penny will have a certain (unknown) probability of heads. Now imagine the machine is broken so that every penny comes out different. This adds a new layer of unknown: the unknown distribution of the unknown probabilities of heads for the pennies it produces.

We treat this toy problem with the Dirichlet process in order to introduce the important conceptual components of importance sampling and sequential imputation. Ultimately, we want to treat the more general problem, in which the "machine" produces objects whose possible behavior is more complicated than simply heads or tails.

Slides: 2021-04-06_Swanson.pdf

### Title: Linear Regression and Its Application in Finance

As one of the most widely used statistical methods, linear regression is a basic statistical tool in empirical financial data analysis. In this talk, we briefly review standard linear regression theory from the statistical point of view, including statistical inference (confidence intervals and ANOVA), model selection, and regression diagnostics. By extending the theory to stochastic regressors, one can bridge the gap between standard linear regression theory and its application in financial data analysis. We will use R for computation, data analysis, and data visualization.

Slides: 2021-03-30_Wang.pdf

### Title: Large deviations and a non-Markovian vanishing viscosity result. Part 2

See Mar 2 abstract below.

Slides: 2021-03-09_Keller.pdf

### Title: Large deviations and a non-Markovian vanishing viscosity result. Part 1

Recently, Backhoff-Veraguas, Lacker, and Tangpi [Annals of Applied Probabability, 30 (2020), pages 1321-1367] established a non-exponential generalization of the classical Schilder theorem of the theory of large deviations. As applications they obtained new limit theorems for backward stochastic differential equations (BSDEs) and partial differential equations (PDEs). The results for PDEs correspond to a Markovian setting. The results for BSDEs are more general and are valid in a non-Markovian setting. The possibility of a corresponding non-Markovian PDE result, i.e., a vanishing viscosity limit theorem for path-dependent PDEs, has been addressed but the problem has been left open.

In this series of talks, I present a possible solution to this problem. I start with an overview of BSDEs and path-dependent PDEs. Then I discuss the mentioned open problem. Finally, I introduce appropriate notions of generalized solutions for two classes of path-dependent PDEs, under which we have existence and uniqueness, as well as a non-Markovian vanishing viscosity result.

Slides: 2021-03-02_Keller.pdf

### Title: A crash course on Dirichlet processes. Part 2

A stochastic process is simply an indexed collection of random variables. Typically, the index represents time, but this is not at all necessary. For instance, we could have a collection of random variables indexed by the Borel measurable subsets of the real line. As an example, let $\mu$ be a random Borel probability measure on the real line. Then, for each Borel set, $A$, the quantity $\mu(A)$ is a random variable. In this sense, $\mu$ is a stochastic process.

The Dirichlet process is a particular kind of random probability measure. It was first defined in the literature in a 1973 paper by Thomas Ferguson. It is often used in the context of Bayesian inference and learning to model the distribution of a random variable whose distribution is unknown. When used in this way, it serves as a prior that is then updated through data and observations.

This talk will be the second in a series of talks in which I will introduce the Dirichlet process and its properties, and show how it is used to facilitate Bayesian learning. The talks will be delivered at a level understandable to students who have completed Measure and Probability I.

Slides: 2021-02-09_Swanson.pdf

### Title: A crash course on Dirichlet processes. Part 1

A stochastic process is simply an indexed collection of random variables. Typically, the index represents time, but this is not at all necessary. For instance, we could have a collection of random variables indexed by the Borel measurable subsets of the real line. As an example, let $\mu$ be a random Borel probability measure on the real line. Then, for each Borel set, $A$, the quantity $\mu(A)$ is a random variable. In this sense, $\mu$ is a stochastic process.

The Dirichlet process is a particular kind of random probability measure. It was first defined in the literature in a 1973 paper by Thomas Ferguson. It is often used in the context of Bayesian inference and learning to model the distribution of a random variable whose distribution is unknown. When used in this way, it serves as a prior that is then updated through data and observations.

This talk will be the first in a series of talks in which I will introduce the Dirichlet process and its properties, and show how it is used to facilitate Bayesian learning. The talks will be delivered at a level understandable to students who have completed Measure and Probability I.

Slides: 2021-01-26_Swanson.pdf