Mathematics

Application of Random Effects in Dependent Compound Risk Model

Himchan Jeong (University of Connecticut)

Abstract: In ratemaking for general insurance, the calculation of a pure premium has traditionally been based on modeling both frequency and severity in an aggregated claims model. Additionally for simplicity, it has been a standard practice to assume the independence of loss frequency and loss severity. However, in recent years, there has been sporadic interest in the actuarial literature exploring models that departs from this independence. Besides, usual property and casualty insurance enables us to explore the benefits of using random effects for predicting insurance claims observed longitudinally, or over a period of time. Thus, in this article, a research work is introduced with utilizes random effects in dependent two-part model for insurance ratemaking, testing the presence of random effects via Bayesian sensitivity analysis with its own theoretical development as well as empirical results and performance measures using out-of-sample validation procedures.

Statistical inference for mortality models

Chen Ling (Georgia State University)

Abstract: Underwriters of annuity products and administrators of pension funds are under financial obligation to their policyholder until the death of counterparty. Hence, the underwriters are subject to longevity risk when the average lifespan of the entire population increases, and yet, such risk can be managed through hedging practices based on parametric mortality models. As a benchmark mortality model in insurance industry is Lee-Carter model, we first summarize some flaws regarding the model and inference methods derived from it. Based on these understandings we propose a modified Lee-Carter model, accompanied by a rigorous statistical inference with asymptotic results and satisfactory numerical and simulation results derived from a small sample. Then we propose bias corrected estimator which is consistent and asymptotically normally distributed regardless of the mortality index being a unit root or stationary AR(1) time series. We further extend the model to accommodate AR(2) process for mortality index, and, a bivariate dataset of U.S. mortality rates. Finally, we conclude by a detailed model validation and some discussions of potential hedging practices based on our parametric model.

Two-Part D-Vine Copula Models for Longitudinal Insurance Claim Data

Lu Yang (University of Amsterdam)

Abstract: Insurance companies keep track of each policyholder's claims over time, resulting in longitudinal data. Efficient modeling of time dependence in longitudinal claim data can improve the prediction of future claims needed, for example, for ratemaking. Insurance claim data have their special complexity. They usually follow a two-part mixed distribution: a probability mass at zero corresponding to no claim and an otherwise positive claim from a skewed and long-tailed distribution. We propose a two-part D-vine copula model to study longitudinal mixed claim data. We build two stationary D-vine copulas. One is used to model the time dependence in binary outcomes resulting from whether or not a claim has occurred, and the other studies the dependence in the claim size given occurrence over time. The proposed model can predict the probability of making claims and the quantiles of severity given occurrence straightforwardly. We use our approach to investigate a dataset from the Local Government Property Insurance Fund in the state of Wisconsin.

Insuring longevity risk and long-term care: Bequest, housing and liquidity

Mengyi Xu (University of New South Wales)

Abstract: The demand for life annuities and long-term care insurance (LTCI) varies among retirees with different preferences and financial profiles. This paper shows that bequest motives can enhance the demand for LTCI unless the opportunity cost of self-insurance through precautionary savings is low. This typically occurs when retirees have sufficient liquid wealth. If retirees tend to liquidate housing assets in the event of becoming disabled that requires sizeable costs, housing wealth is likely to enhance the demand for annuities and to crowd out the demand for LTCI. Cash-poor-asset-rich retirees show little interest in annuities, but they may purchase LTCI to preserve their bequests.

Local and global boundary rigidity

Plamen Stefanov   [email] (Purdue University)

Abstract: Abstract: The boundary rigidity problem consist of recovering a Riemannian metric in a domain, up to an isometry, from the distance between boundary points. We show that in dimensions three and higher, knowing the distance near a fixed strictly convex boundary point allows us to reconstruct the metric inside the domain near that point, and that this reconstruction is stable. We also prove semi-global and global results under certain an assumption of the existence of a strictly convex foliation. The problem can be reformulated as a recovery of the metric from the arrival times of waves between boundary points; which is known as travel-time tomography. The interest in this problem is motivated by imaging problems in seismology: to recover the sub-surface structure of the Earth given travel-times from the propagation of seismic waves. In oil exploration, the seismic signals are man-made and the problem is local in nature. In particular, we can recover locally the compressional and the shear wave speeds for the elastic Earth model, given local information. The talk is based on a joint work with G.Uhlmann (UW) and A.Vasy (Stanford). We will also present results for a recovery of a Lorentzian metric from red shifts motivated by the problem of observing cosmic strings. The methods are based on Melrose’s scattering calculus in particular but we will try to make the exposition accessible to a wider audience without going deep into the technicalities.

Analytic Grothendieck Riemann Roch Theorem

Xiang Tang   [email] (Washington University St. Louis)

Abstract: Abstract: In this talk, we will introduce an interesting index problem naturally associated to the Arveson-Douglas conjecture in functional analysis. This index problem is a generalization of the classical Toeplitz index theorem, and connects to many different branches of Mathematics. In particular, it can be viewed as an analytic version of the Grothendieck Riemann Roch theorem. This is joint work with R. Douglas，M. Jabbari, and G. Yu.

From Polaris Variable Annuities to Regression-based Monte Carlo

Zhiyi (Joey) Shen (University of Waterloo)

Abstract: In this talk, I will first discuss the no-arbitrage pricing of Polaris variable annuities (VAs), which were issued by the American International Group in recent years. Variable annuities are prevailing equity-linked insurance products that provide the policyholder with the flexibility of dynamic withdrawals, mortality protection, and guaranteed income payments against a market decline. The Polaris allows a shadow account to lock in the high watermark of the investment account over a monitoring period that depends on the policyholder’s choice of his/her first withdrawal time. This feature makes the insurer’s payouts depend on policyholder’s withdrawal behaviours and significantly complicates the pricing problem. By prudently introducing certain auxiliary state variables, we manage to formulate the pricing problem into solving a convoluted stochastic optimal control framework and developing a computationally efficient algorithm to approach the solution. Driven by the challenges from the pricing Polaris VAs, in the second part of the talk, I will introduce a regression-based Monte Carlo algorithm, which we propose to solve a class of general stochastic optimal control problems numerically. The algorithm has three pillars: a construction of auxiliary stochastic control model, an artificial simulation of the post-action value of state process, and a shape-preserving sieve estimation method. The algorithm enjoys many merits, including obviating forward simulation and control randomization, eliminating in-sample bias, evading extrapolating the value function, and alleviating the computational burden of the tuning parameter selection. This talk is based on two joint works with Chengguo Weng from the University of Waterloo.

Mixture of Experts Regression Models for Insurance Ratemaking and Reserving

Tsz Chai "Samson" Fung (University of Toronto)

Abstract: Understanding the effect of policyholders' risk profile on the number and the amount of claims, as well as the dependence among different types of claims, are critical to insurance ratemaking and IBNR-type reserving. To accurately quantify such features, it is essential to develop a regression model which is flexible, interpretable and statistically tractable. In this presentation, I will discuss a highly flexible nonlinear regression model we have recently developed, namely the logit-weighted reduced mixture of experts (LRMoE) models, for multivariate claim frequencies or severities distributions. The LRMoE model is interpretable as it has two components: Gating functions to classify policyholders into various latent sub-classes and Expert functions to govern the distributional properties of the claims. The model is also flexible to fit any types of claim data accurately and hence minimize the issue of model selection. Model implementation is illustrated in two ways using a real automobile insurance dataset from a major European insurance company. We first fit the multivariate claim frequencies using an Erlang count expert function. Apart from showing excellent fitting results, we can interpret the fitted model in an insurance perspective and visualize the relationship between policyholders' information and their risk level. We further demonstrate how the fitted model may be useful for insurance ratemaking. The second illustration deals with insurance loss severity data that often exhibits heavy-tail behavior. Using a Transformed Gamma expert function, our model is applicable to fit the severity and reporting delay components of the dataset, which is ultimately shown to be useful and crucial for an adequate prediction of IBNR reserve. This project is joint work with Andrei Badescu and Sheldon Lin.

Data-driven methods for model identification and parameter estimation of dynamical systems

Niall Mangan   [email] (Northwestern University)

Abstract: Inferring the structure and dynamical interactions of complex systems is critical to understanding and controlling their behavior. I am interested in discovering models from the time-series in order to understand biological systems, material behavior, and other dynamical systems. One can frame the problem as selecting which interactions, or model terms, are most likely responsible for the observed dynamics from a library of possible terms. Several challenges make model selection and parameter estimation difficult including nonlinearities, varying parameters or equations, and unmeasured state variables. I will discuss methods for reframing these problems so that sparse model selection is possible including implicit formulation and data clustering. I will also discuss preliminary results for parameter estimation and model selection for deterministic and chaotic systems with hidden or unmeasured variables. We use a variational annealing strategy that allows us to estimate both the unknown parameters and the unmeasured state variables.

Digits

Frank Calegari (University of Chicago)

Abstract: We discuss some results concerning the decimal expansion of 1/p for primes p, some due to Gauss, and some from the present day. This talk will be accessible to undergraduates.

Counting

Frank Calegari (University of Chicago)

Abstract: What can one say about a system of polynomial equations with integer coefficients simply by counting the number of solutions to these equations modulo primes? We begin with the case of polynomials in one variable and relate this to how the polynomial factors and to Galois theory. We then discuss what happens in higher dimensions, and are led to a conjectural notion of the "Galois group" of an algebraic variety. This will be a colloquium style talk and will be independent of the first talk.

Coble

Frank Calegari (University of Chicago)

Abstract: Coble is known (in part) for his work on invariant theory and the geometry of certain of exceptional moduli spaces in low dimension. We discuss the quest to find explicit equations for one particular family of moduli spaces. An important role is played by a number of exceptional geometrical coincidences and also the theory of complex reflection groups. This will be a colloquium style talk and will be independent of the first two talks.

Plane Trees and Algebraic Numbers

George Shabat (Russian State University for the Humanities and Independent University of Moscow)

Abstract: The main part of the talk will be devoted to an elementary version of the deep relations between the combinatorial topology and the arithmetic geometry. Namely, an object defined over the field of algebraic numbers, a polynomial with algebraic coefficients and only two finite critical values, will be associated to an arbitrary plane tree. Some applications of this construction will be presented, including polynomial Pell equations and quasi-elliptic integrals (going back to N.-H. Abel). The relations with finite groups and Galois theory will be outlined. At the end of the talk the possible generalizations will be discussed, including the dessins d'enfants theory initiated by Grothendieck.

Stability of roll wave solutions in inclined shallow-water flow

Kevin Zumbrun   [email] (Indiana University Bloomington)

Abstract: We review recent developments in stability of periodic roll-wave solutions of the Saint Venant equations for inclined shallow-water flow. Such waves are well-known instances of hydrodynamic instability, playing an important role in hydraulic engineering, for example, flow in a channel or dam spillway. Until recently, the analysis of their stability has been mainly by formal analysis in the weakly unstable or ``near-onset'' regime. However, hydraulic engineering applications are mainly in the strongly unstable regime far from onset. We discuss here a unified framework developed together with Blake Barker, Mat Johnson, Pascal Noble, Miguel Rodrigues, and Zhao Yang for the study of roll wave stability across all parameter regimes, by a combination of rigorous analysis and numerical computation. The culmination of our analysis is a complete stability diagram, of which the low-frequency stability boundary is, remarkably, given explicitly as the solution of a a cubic equation in the parameters of the solution space.

Trapping, resonances, and the decay of waves [to be rescheduled Fall 2020]

Jared Wunsch   [email] (Northwestern University)

Abstract: I will discuss some results, new and old, involving the influence of the geometry on the decay of waves. The quantum correspondence principle dictates that at high frequency, the dynamics of particle trajectories should be related to the rate at which the energy of a solution to the wave or Schrödinger equation decays. This relationship is mediated by the existence of resonances, which correspond to states with a finite (but possibly long) lifetime that ultimately decay owing to tunneling effects. I will discuss what we know about the existence and nonexistence of resonances, and focus on some recent results about resonances associated to the subtle effects of diffraction in classical and quantum problems that have singular structures in a metric or potential.

To Be Rescheduled Fall 2020

Kevin Purbhoo   [email] (University of Waterloo)

Abstract: To come.

To Be Rescheduled Fall 2020

Sami Assaf   [email] (University of Southern California)

Abstract: To come.

To Be Announced

Marius Junge   [email] (University of Illinois at Urbana-Champaign)

Abstract: To come.

Fair representation

Noga Alon (Princeton University and Tel Aviv)

Abstract: A substantial number of results and conjectures deal with the existence of a set of prescribed type which contains a fair share from each member of a finite collection of objects in a space. Examples include the Ham-Sandwich Theorem in Measure Theory, the Hobby-Rice Theorem in Approximation Theory, the Necklace Theorem and the Ryser Conjecture in Discrete Mathematics, and more. The techniques in the study of these results combine combinatorial, topological, geometric and algebraic tools. I will describe the topic, focusing on several recent results.

The Topology of Circuit-Field Coupling

Ralf Hiptmair   [email] (ETH Zurich)

Abstract: Imagine an electromagnetic wave impinging on a small electric device. The wave is governed by Maxwell's equations and propagates in a "field domain", whereas the device occupies a "circuit domain" and is described by a circuit model with well-defined ports located on the common interface. I am going to discuss how to incorporate ports into (finite-element discretized) variational formulations of Maxwell's equations. It turns out that two types of ports have to be distinguished, electric and magnetic. Each of them is linked to two port quantities, one "potential" and one "current". A key insight concerns the close relationship of the port quantities with generating fundamental cycles of the relative cohomology of the common interface minus the port areas. These cycles become instrumental for linking the circuit to Maxwell field models also in the context of finite-element discretization. Therefore I am going to elaborate the construction of fundamental cycles on triangulated surfaces by means of spanning-tree techniques. Another key insight is that in case of non-trivial topology of circuit domain ports alone are not sufficient to connect fields and circuits. "Linked fluxes" associated with handles of circuit domain also have to be taken into account and this entails knowledge about the topological properties of the circuit beyond the scope of customary descriptions. For demonstration I am going to give a striking numerical example. (Joint work with J. Ostrowski). > NOTE: This Colloquium talk is at 12:00 PM Central time, NOT the usual Colloquium time due to the time difference with speaker's time zone. The talk will be on Zoom for which the link will be sent out to the department and other mailing lists. If you don't receive such an email, please email hirani@illinois.edu from your Illinois email address to receive the Zoom link.

Computational complexity meets algebraic combinatorics

Greta Panova (University of Southern California)

Abstract: How hard is a problem? How nice is a solution? Such questions can actually be formalized using the theory of Computational Complexity. Yet, distinguishing the different computational complexity classes, like P vs NP, are major problems. Algebraic Combinatorics studies discrete structures originating in Algebra/Representation Theory via combinatorial methods and vice versa. Some of the main longstanding open problems concern the “combinatorial interpretation” of structure constants and multiplicities originally defined via representation theory like Kronecker and plethysm coefficients. In this talk we will discuss the two-way interaction between the fields via such structure constants. First, how Kronecker coefficients appear in the distinction of algebraic complexity classes via the Geometric Complexity Theory. Second, how computational complexity explains why the problem of finding a combinatorial interpretation is hard.

Meeting Info: please email ayong@illinois.edu

Geometry of transportation cost (also known as Earth Mover or Wasserstein distance)

Mikhail Ostrovskii (St. John's University)

Abstract: (Note: 3:30-4 p.m. will be an open/informal talk with the speaker) We consider (finitely supported) transportation problems on a metric space M. They form a vector space TP(M). The optimal transportation cost for such transportation problems is a norm on this space. Such normed spaces were introduced and studied by Kantorovich and his students in 1940-1950s years. This development lead to terms: Kantorovich distance, Kantorovich-Rubinstein distance, and transportation cost. Simultaneously, another group of researchers, starting with Markov (1941) started the study of algebraically “free” topological (or metric) structures which contain a given topological (or metric) structure as a substructure. On these lines Arens and Eells (1956) constructed the space which coincides with the space of transportation problems with the norm equal to the optimal transportation cost. In this connection you can meet such terms as Arens-Eells space and Lipschitz-free space. I am going to talk about geometry of such normed spaces. My results presented in this talk, mentioned in it, or related to it, can be found in joint papers with Stephen Dilworth, Seychelle Khan, Denka Kutzarova, Mutasim Mim, and Sofiya Ostrovska (available on arXiv).

For zoom info, please email currid@illinois.edu