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Title: Noncausal Affine Processes with Applications to Derivative Pricing
Speaker: Dr. Yang Lu (Université Paris 13, France)
Date: Tuesday, January 21, 2020
Time: 11:30 am
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
Abstract: Linear factor models, where the factors are affine processes, play a key role in Finance, since they allow for quasi-closed form expressions of the term structure of risks. We introduce the class of noncausal affine linear factor models by considering factors that are affine in reverse time. These models are especially relevant for pricing sequences of speculative bubbles. We show that they feature much more complicated non affine dynamics in calendar time, while still providing (quasi) closed form term structures and derivative pricing formulas. The framework is illustrated with zero-coupon bond and European call option pricing examples.
Title: Optimal Reinsurance-Investment Strategy for a Dynamic Contagion Claim Model
Speaker: Ms. Jingyi Cao (University of Waterloo, ON)
Date: Friday, January 17, 2020
Time: 10:00 am
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
Abstract: We study the optimal reinsurance-investment problem for the compound dynamic contagion process introduced by Dassios and Zhao (2011). This model allows for self-exciting and externally-exciting clustering effect for the claim arrivals, and includes the well-known Cox process with shot noise intensity and the Hawkes process as special cases. For tractability, we assume that the insurer’s risk preference is the time-consistent mean-variance criterion. By utilizing the dynamic programming and extended HJB equation approach, a closed-form expression is obtained for the equilibrium reinsurance-investment strategy. An excess-of-loss reinsurance type is shown to be optimal even in the presence of self-exciting and externally-exciting contagion claims, and the strategy depends on both the claim size and claim arrivals assumptions. Further, we show that the self-exciting effect is of a more dangerous nature than the externally-exciting effect as the former requires more risk management controls than the latter. In addition, we find that the reinsurance strategy does not always become more conservative (i.e., transferring more risk to the reinsurer) when the claim arrivals are contagious. Indeed, the insurer can be better off retaining more risk if the claim severity is relatively light-tailed. This is a joint work with David Landriault and Bin Li (both from University of Waterloo).
Title: Application of Random Effects in Dependent Compound Risk Model
Speaker: Mr. Himchan Jeong (University of Connecticut, CT)
Date: Friday, January 10, 2020
Time: 10:00 am
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
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.
Title: Mixture of Experts Regression Models for Insurance Ratemaking and Reserving
Speaker: Mr. Tsz Chai Fung (University of Toronto, ON)
Date: Thursday, January 9, 2020
Time: 11:30 am
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)

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.

Title: Compressive Sensing and its Applications in Data Science and in Computational Mathematics
Speaker: Dr. Simone Brugiapaglia (Simon Fraser University, BC)
Date: Monday, January 21, 2019
Time: 11:45 am
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
Abstract: Compressive sensing (CS) is a general paradigm that enables us to measure objects (such as images, signals, or functions) by using a number of linear measurements proportional to their sparsity, i.e. to the minimal amount of information needed to represent them with respect to a suitable system. The vast popularity of CS is due to its impact in many practical applications of data science and signal processing, such as magnetic resonance imaging, X-ray computed tomography, or seismic imaging.

In this talk, after presenting the main theoretical ingredients that made the success of CS possible and discussing recovery guarantees in the noise-blind scenario, we will show the impact of CS in computational mathematics. In particular, we will consider the problem of computing sparse polynomial approximations of functions defined over high-dimensional domains from pointwise samples, highly relevant for the uncertainty quantification of PDEs with random inputs. In this context, CS-based approaches are able to substantially lessen the curse of dimensionality, thus enabling the effective approximation of high-dimensional functions from small datasets. We will illustrate a rigorous noise-blind recovery error analysis for these methods and show their effectiveness through numerical experiments. Finally, we will present some challenging open problems for CS-based techniques in computational mathematics.  
Title: The Algorithmic Hardness Threshold for Continuous Random Energy Models
Speaker: Dr. Pascal Maillard (Institut de Mathématique d’Orsay, France)
Date: Friday, January 18, 2019
Time: 10:00 am
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
Abstract: He will report on recent work with Louigi Addario-Berry on algorithmic hardness for finding low-energy states in the continuous random energy model of Bovier and Kurkova. This model can be regarded as a toy model for strongly correlated random energy landscapes such as the Sherrington--Kirkpatrick model. We exhibit a precise and explicit hardness threshold: finding states of energy above the threshold can be done in linear time, while below the threshold this takes exponential time for any algorithm with high probability. If time permits, I further discuss what insights this yields for understanding algorithmic hardness thresholds for random instances of combinatorial optimization problems.
Title: Rapid Mixing Bounds for Hamiltonian Monte Carlo under Strong Log-Concavity
Speaker: Dr. Oren Manbougi (Ecole polytechnique fédérale de Lausanne [EPFL], Switzerland)
Date: Thursday, January 10, 2019
Time: 9:30 am
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)

Sampling from a probability distribution is a fundamental algorithmic problem.  We discuss applications of sampling to several areas including machine learning, Bayesian statistics and optimization. In many situations, for instance when the dimension is large, such sampling problems become computationally difficult.  

Markov chain Monte Carlo (MCMC) algorithms are among the most effective methods used to solve difficult sampling problems.  However, most of the existing guarantees for MCMC algorithms only handle Markov chains that take very small steps and hence can oftentimes be very slow.  Hamiltonian Monte Carlo (HMC) algorithms – which are inspired from Hamiltonian dynamics in physics – are capable of taking longer steps. Unfortunately, these long steps make HMC difficult to analyze.  As a result, non-asymptotic bounds on the convergence rate of HMC have remained elusive.

In this talk, we obtain rapid mixing bounds for HMC in an important class of strongly log-concave target distributions encountered in statistical and Machine learning applications.  Our bounds show that HMC is faster than its main competitor algorithms, including the Langevin and random walk Metropolis algorithms, for this class of distributions.

Finally, we consider future directions in sampling and optimization.  Specifically, we discuss how one might design adaptive online sampling algorithms for applications to reinforcement learning.  We also discuss how Markov chain algorithms can be used to solve difficult non-convex sampling and optimization problems, and how one might be able to obtain theoretical guarantees for the MCMC algorithms that can solve these problems.

Title: Suboptimality of Local Algorithms for Optimization on Sparse Graphs  
Speaker: Dr. Mustazee Rahman (KTH Royal Institute of Technology, Sweden)
Date: Tuesday, January 8, 2019
Time: 11:15 am
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
Abstract: Suppose we want to find the largest independent set or maximal cut in a large yet sparse graph, where the average vertex degree is asymptotically constant.  These are two basic optimization problems relevant to both theory and practice.  For typical, or randomized, sparse graphs, many natural algorithms proceed by way of local decision rules. Examples include Glauber dynamics, Belief propagation, etc.  I will explain a form of local algorithms that captures many of these. I will then explain how they provably fail to find optimal independent sets or cuts once the average degree of the graph becomes large.  This answers a question that traces back to computer science, probability and statistical physics.  Along the way, we will find connections to entropy and spin glasses.
Title: Ranks of Elliptic Curves, Limiting Distributions and Moments of Arithmetical Sequences
Speaker: Dr. Daniel Fiorilli (University of Ottawa, ON)
Date: Friday, February 3, 2017
Time: 10:30 a.m. - 12:00 p.m.
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
Abstract: Dr. Fiorilli will begin with an introduction to elliptic curves, with some historical comments. Concepts such as the rank of an elliptic curve will be discussed, as well as famous open problems such as extreme ranks and the Birch and Swinnerton-Dyer conjecture. My work links such problems to statements about the involved limiting distributions, which allow to apply tools such as the central limit theorem and the theory of large deviations. In particular, I will describe how one can combine probability, analytic number theory, Galois theory and algebraic geometry to understand elliptic curves over function fields. Finally, I will describe my work on moments of arithmetical sequences in progressions, in particular bounds and asymptotics for the first and second moments (both in the classical case and with the circle method), as well as a probabilistic study of the second moment.
Title: Quantum Chaos and Arithmetic
Speaker: Dr. Stephen Lester (Concordia University & CRM-ISM, QC)
Date: Tuesday, January 24, 2017
Time: 1:30 - 3:00 p.m.
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
Abstract: In this talk I will discuss some problems in Quantum Chaos and describe results in arithmetic settings where more can be proved. Given a compact, smooth Riemannian manifold (M,g) a central problem in Quantum Chaos is to understand the behavior of eigenfunctions of the Laplace-Beltrami operator in the limit as the eigenvalue tends to infinity.  The Quantum Ergodicity Theorem of Shnirelman, Colin de Vediere, and Zelditch asserts that if the geodesic flow on M is ergodic then the  mass of almost all of the eigenfunctions equidistributes.  I will discuss problems which go beyond the Quantum Ergodicity Theorem such as quantum unique ergodicity and small scale quantum ergodicity in the setting of arithmetic surfaces such as the torus and modular surface.  Limitations on equidistribution will also be discussed.  I will also indicate how these problems are related to arithmetic objects such as L-functions, modular forms, and multiplicative functions.
Title: The Arithmetic of L-functions and their P-Adic Properties
Speaker: Dr. Giovanni Rosso (University of Cambridge, UK)
Date: Monday, January 23, 2017
Time: 11:00 a.m. - 12:30 p.m.
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
Abstract: L-functions are complex functions which are defined from interesting arithmetic objects, such as number fields and elliptic curves. These functions contain a lot of interesting information about the objects that interest us. During the talk, I shall give some examples of how this information can be recovered, stressing the importance of the p-adic and mod p behaviour of these functions. I shall conclude with an overview of related conjectures (Iwasawa Main Conjecture, Greenberg's conjecture on trivial zeroes, existence of eigenvarieties,...) and my results on these topics.
Title: Families of Modular Forms
Speaker: Dr. John Bergdall (Boston University, MA)
Date: Thursday, January 19, 2017
Time: 2:00 - 3:30 p.m.
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
Abstract: Modular forms are central objects in number theory. They famously appear in the proof of Fermat's Last Theorem and more generally they play a role in the so-called Langlands program. This talk is concerned with the eigenforms, modular forms that are more algebraic in nature, and our ability to deform eigenforms into geometric families. My goal is to explain the history as well as the motivation for modular forms and families thereof. In addition, I will discuss recent results on the geometric properties of these families. This talk is intended for a general audience. A portion of this work is joint with David Hansen.
Title: The Geometry of Modular Forms, from Ramanujan to Moonshine
Speaker: Dr. Luca Candelori (Louisianna State University, LA)
Date: Tuesday, January 10, 2017
Time: 2:00 - 3:30 p.m.
Location: LB 921-4 (Concordia University, Library Building, 1400 de Maisonneuve West)
Abstract: Modular forms often appear as generating series of interesting mathematical data, from Ramanujan's work on integer partitions to John McKay's computations on the representation theory of the Monster group. In this talk we introduce new geometric perspectives on these classical topics, or rather on some of the mathematics they have directly inspired: the theory of mock modular forms and that of vector-valued modular forms associated to vertex operator algebras. Specifically, we employ the geometry of modular curves to study the fields of definition of coefficients of mock modular forms and to classify the structure of graded modules of vector-valued modular forms.
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