Seminars

Nuttanan Wichitaksorn

On the Matrix Copula with Application to Portfolio Analysis

1.30pm, Friday, 2 December 2016
Room 446, Erskine Building

Abstract. In this talk, I will present a new matrix-copula construction, and show an application in finance.

The matrix copula is an implicit copula such that its construction is based on the associated matrix-variate distribution. The matrix copula based on the Gaussian distribution is presented to illustrate the formulation and estimation. The major and important feature of this matrix copula construction is that it embodies two dependence structures, among-row and among-column, that can be conveniently estimated via Markov chain Monte Carlo methods.

As an application, I propose a new approach to analyze monthly stock loss data, whereby the model construction is based primarily on the so-called insurance collective risk model. The two dependence structures, for loss sizes and numbers of days in loss for stocks in a portfolio, are then modeled using the matrix Gaussian copula. Simulation and empirical studies are implemented to illustrate the performance and applicability of the proposed model and method.

Fabian Dunker

Nonparametric estimation in stochastic differential equations with low-frequency observations

2.00pm, Thursday, 1 December 2016
Room 446, Erskine Building

Abstract. In this talk we present nonparametric estimators for coefficients in time homogeneous stochastic differential equations with multivariate \(X_t\)

\(dX_t = \mu(X_t)dt + \sigma(X_t)dW_t\).

We assume i.i.d. observations. The estimator does not relay on the common assumption that the distance between observations decreases when the sample size increases.

The problem is formulated as a nonlinear ill-posed operator equation with a deterministic forward operator described by the Fokker-Planck equation.

We derive convergence rates of the risk for a Newton-type method based on penalized maximum likelihood with convex penalty terms. The assumptions of our general convergence results are verified for estimation of the drift coefficient \(\mu\).

The advantages of log-likelihood compared to a quadratic loss function are demonstrated in Monte-Carlo simulations. The method is applied to exchange rate data for evaluation of parametric specifications in exchange rate models.

Eugenia Saorín Gómez (University of Magdeburg)

Inequalities which characterize operators in Convex Geometry: an approach

2.00pm, Tuesday, 30 August 2016
Room 446, Erskine Building

Abstract. We will begin the talk introducing the Brunn-Minkowski inequality. Our ambient space is the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\). For a measurable set \(M \subset \mathbb{R}^n\) we denote by \(V_n(M)\) the volume, i.e., the Lebesgue measure of the \(M\). Let \(A,B \subset \mathbb{R}^n\) be compact sets and \(\lambda \in [0, 1]\), the classical Brunn-Minkowski inequality establishes that

\(V_n(\lambda A + (1 - \lambda)B)^{1/n} \geq \lambda V_n(A)^{1/n} + (1 - \lambda)V_n(B)\),

where for \(A,B \subset \mathbb{R}^n\), \(A + B = \{a + b : a \in A; \in B\}\), i.e., \(+\) is the vectorial addition of sets in \(\mathbb{R}^n\) and \(\mu A = \{\mua : a \in A\}\) for \(\mu\geq0\). We will briefly discuss some equivalent formulations of the inequality, as well as a classical consequence of it.

Next we will deal with the Rogers-Shephard inequality for the difference body of a convex body. Let \(K^n\) denote the class of convex bodies in \(\mathbb{R}^n\), that is, compact and convex sets and for \(K \in K^n\), let \(-K := \{k : -k \in K\}\) be the reflection of \(K\) in the origin. The difference body of \(K\) is the convex body \(K - K := K + (-K)\). Its fundamental affine inequality, the Rogers-Shephard inequality states the upper bound in the following inequality; the lower bound is a consequence of the Brunn-Minkowski inequality:

\(2^n V_n(K) \leq V_n(K - K) \leq \begin{pmatrix} 2n \\ n \end{pmatrix} V_n (K)\).

Motivated by both inequalities we say that a functional \(F : K^n \longrightarrow \mathbb{R}\), satisfies a Brunn-Minkowski type inequality if there exists \(\alpha > 0\) such that for all \(K,L \in K^n\) and \(\lambda \in [0, 1]\)

\(F(\lambdaA + (1 - \lambda) B )^{1/\alpha} \geq \lambda F(A)^{1/\alpha}+ (1 - \lambda) F (B)^{1/\alpha}\).

Analogously, we say that an operator \(\Phi : K^n \longrightarrow K^n\) satisfies a Brunn-Minkowski type inequality if there exists \(c > 0\) such that

\(c V_n(K) \leq V_n(\PhiK)\)   for all \(K \in K^n\).

The operator \(\Phi : K^n \longrightarrow K^n\) satisfies a Rogers-Shephard type inequality if there exists \(C > 0\) such that

\(Vn(\PhiK) \leq C V_n(K)\)   for all \(K \in K^n\).

We will give some examples of functionals and operators satisfying these inequalities.

Then we will focus on the interplay of satisfying a Rogers-Shephard type and/or a Brunn-Minkowski type inequality with other properties, such as continuity, monotonicity or covariance under the action of a group, and present some results which characterize special functionals and operators within Convex Geometry under the assumption of (some of) these inequalities.

Brendan Creutz (University of Canterbury)

Local-global principles for divisibility

2.00pm, Monday, 29 August 2016
Room 446, Erskine Building

Abstract. An integer \(x\) is said to be divisible by \(N\) if there is some other integer \(y\) such that \(Ny = x\). This notion makes sense in any group, but it can behave in rather unexpected ways when one takes topology into account. I will discuss a classical example of this, the Grunwald-Wang theorem, and analogous phenomena for elliptic curves.

Daniel Turetsky (Victoria University of Wellington)

Algorithmic randomness and Turing reducibility

2:00pm, Friday 26 August 2016
Room 446, Erskine Building

Abstract. Algorithmic randomness uses the tools of computability theory to study the question, "When is a sequence of numbers random?". Several different definitions of a random sequence are studied, with the most common being Martin-Löf randomness.

Randomness has a rich interaction with the notion of Turing reducibility which is studied in more classical computability theory. One longstanding question in this area was the covering problem, which concerns the relationship between random sequences, Turing reducibility, and very non-random sequences. Somewhat surprisingly, the solution to the covering problem involves effective analysis, which is a different area with interesting interactions with randomness.

I will introduce the appropriate notions and discuss the covering problem and similar questions. I will also discuss the various ingredients necessary for the solution to the covering problem.

Geertrui Van de Voorde (Ghent University)

Finite Projective Geometry

2.00pm, Tuesday, 23 August 2016
Room 446, Erskine Building

Abstract. It takes only three very natural axioms to define an axiomatic projective plane, but from these three axioms follow an interesting theory with some famous open problems. In the first part of this talk I will provide a basic introduction to the theory of finite projective geometry and mention some of these open problems.

I will then show how André, Bruck and Bose used a certain geometric structure, later called a linear representation of a spread, to construct a broad class of projective planes which possess a lot of symmetry. In order to find spreads, the technique of field reduction will be introduced.

Finally, I will explore the links between linear representations and other research areas such as graph theory and coding theory.