Carnegie Mellon University

The goal of sequential learning is to draw inference from data that is gathered gradually through time. This is a typical situation in many applications, including finance. A sequential inference procedure is `anytime-valid’ if the decision to stop or continue an experiment can depend on anything that has been observed so far, without compromising statistical error guarantees. A recent approach to anytime-valid inference views a test statistic as a bet against the null hypothesis. These bets are constrained to be supermartingales - hence unprofitable - under the null, but designed to be profitable under the relevant alternative hypotheses. This perspective opens the door to tools from financial mathematics. In this talk I will discuss how notions such as supermartingale measures, log-optimality, and the optional decomposition theorem shed new light on anytime-valid sequential learning. (This talk is based on joint work with Wouter Koolen (CWI), Aaditya Ramdas (CMU) and Johannes Ruf (LSE).)
 

In an incomplete continuous-time securities market with uncertainty generated by Brownian motions, we derive closed-form solutions for the equilibrium interest rate and market price of risk processes. The economy has a finite number of heterogeneous exponential utility investors, who receive partially unspanned income and can trade continuously on a finite time-interval in a money market account and a single risky security. Besides establishing the existence of an equilibrium, our main result shows that if the investors' unspanned income has stochastic countercyclical volatility, the resulting equilibrium can display both lower interest rates and higher risk premia compared to the Pareto efficient equilibrium in an otherwise identical complete market. This is joint work with Peter Ove Christensen.

Condition numbers are a central concept in numerical analysis.

They provide a natural parameter for studying the behavior of

algorithms, as well as sensitivity and geometric properties of a problem.

The condition number of a problem instance is usually a measure

of the distance to the set of ill-posed instances. For instance, the

classical Eckart and Young identity characterizes the condition

number of a square matrix as the reciprocal of its relative distance

to singularity.

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We present concepts of conditioning for optimization problems and

for more general variational problems. We show that the Eckart and

Young identity has natural extension to much wider contexts. We also

discuss conditioning under the presence of block-structure, such as

that determined by a sparsity pattern. The latter has interesting

connections with the mu-number in robust control and with the sign-real

spectral radius.

Modern mathematical approaches to plasticity result in non-convex variational problems for which the standard methods of the calculus of variations are not applicable. In this contribution we consider geometrically nonlinear crystal elasto-plasticity in two dimensions with one active slip system. In order to derive information about macroscopic material behavior the relaxation of the corresponding incremental problems is studied. We focus on the question if realistic systems with an elastic energy leading to large penalization of small elastic strains can be well-approximated by models based on the assumption of rigid elasticity. The interesting finding is that there are qualitatively different answers depending on whether hardening is included or not. In presence of hardening we obtain a positive result, which is mathematically backed up by Γ-convergence, while the material shows very soft macroscopic behavior in case of no hardening. The latter is due to the vanishing relaxation for a large class of applied loads.

This is joint work with Sergio Conti and Georg Dolzmann.

Clustering phenomena occur in numerous areas of science. This series of lectures will discuss:

(i) basic kinetic models for clustering- Smoluchowski's coagulation equation, random shock clustering, ballistic aggregation, domain-wall merging;

(ii) Criteria for approach to self-similarity- role of regular variation;

(iii) The scaling attractor and its measure representation. A particular theme is the use of methods and insights from probability in tandem with dynamical systems theory. In particular there is a

close analogy of scaling dynamics with the stable laws of probability and infinite divisibility.

Clustering phenomena occur in numerous areas of science.

This series of lectures will discuss:

(i) basic kinetic models for clustering- Smoluchowski's coagulation equation, random shock clustering, ballistic aggregation, domain-wall merging;

(ii) Criteria for approach to self-similarity- role of regular variation;

(iii) The scaling attractor and its measure representation.

A particular theme is the use of methods and insights from probability in tandem with dynamical systems theory. In particular there is a

close analogy of scaling dynamics with the stable laws of probability and infinite divisibility.

I will recount progress regarding the robustness of solitary waves in
nonintegrable Hamiltonian systems such as FPU lattices, and discuss 
a proof (with Shu-Ming Sun) of spectral stability of small
solitary waves for the 2D Euler equations for water of finite depth
without surface tension.

The question of local existence of a deformation of a simply connected body whose Left Cauchy Green Tensor matches a prescribed, symmetric, positive definite tensor field is considered. A sufficient condition is deduced after formulation as a problem in Riemannian Geometry. The compatibility condition ends up being surprisingly different from that of compatibility of a Right Cauchy Green Tensor field, a fact that becomes evident after the geometric formulation. The question involves determining conditions for the local existence of solutions to an overdetermined system of Pfaffian PDEs with algebraic constraints that is typically not completely integrable.

1. It can generate flexible credit spread curves.
2. It leads to flexible implied volatility curves, thus providing a link between credit spread and implied volatility.
3. It implies that high tech firms tend to have very little debts.
4. It yields analytical solutions for debt and equity values.