Insurance Mathematics and Stochastic Finance

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Talks in Financial and Insurance Mathematics

This is the regular weekly research seminar on Insurance Mathematics and Stochastic Finance.

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Spring Semester 2023

Date / Time Speaker Title Location
23 February 2023
17:15-18:15 		Jonas Blessing
Universität Konstanz
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Talks in Financial and Insurance Mathematics

Title Nonlinear semigroups and limit theorems for convex expectations
Speaker, Affiliation Jonas Blessing, Universität Konstanz
Date, Time 23 February 2023, 17:15-18:15
Location HG G 43
Abstract Based on the Chernoff approximation, we provide an approximation result for convex monotone semigroups and, in particular, for dynamic convex risk measures. Starting with a generating family $(I(t))_{t\geq 0}$ of operators on the space of bounded continuous functions, the semigroup is constructed as $S(t)f:=\lim_{n\to\infty}I(\frac{t}{n})^n f$ and is uniquely determined by the time derivative $I’(0)f$ for smooth functions $f$. Moreover, we identify explicit conditions for the generating family that are transferred to the semigroup and can easily be verified in applications. It turns out that there is a structural connection between Chernoff approximations for semigroups and LLN and CLT type results for convex risk measures and convex expectations. Other applications are scaling limits of discrete-time models to continuous models under model uncertainty.
Nonlinear semigroups and limit theorems for convex expectationsread_more
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2 March 2023
17:15-18:15 		Prof. Dr. Martin Larsson
Carnegie Mellon University, USA
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Talks in Financial and Insurance Mathematics

Title Propagation of chaos for maxima of particle systems with mean-field drift interaction
Speaker, Affiliation Prof. Dr. Martin Larsson, Carnegie Mellon University, USA
Date, Time 2 March 2023, 17:15-18:15
Location HG G 43
Abstract We study the asymptotic behavior of normalized maxima of real-valued particles with mean-field drift interaction. Our main result establishes propagation of chaos: in the large population limit, the normalized maxima behave as those arising in an i.i.d. system where each particle follows the associated McKean-Vlasov limiting dynamics. Because the maximum depends on all particles, our result does not follow from classical propagation of chaos, where convergence to an i.i.d. limit holds for any fixed number of particles but not all particles simultaneously. This is joint work with Nikos Kolliopoulos and Zeyu Zhang.
Propagation of chaos for maxima of particle systems with mean-field drift interactionread_more
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9 March 2023
17:15-18:15 		Prof. Dr. Lyudmila Grigoryeva
Universität St. Gallen
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Talks in Financial and Insurance Mathematics

Title Reservoir kernels and Volterra series
Speaker, Affiliation Prof. Dr. Lyudmila Grigoryeva, Universität St. Gallen
Date, Time 9 March 2023, 17:15-18:15
Location HG G 43
Abstract A universal kernel is constructed whose sections approximate any causal and time-invariant filter in the fading memory category with inputs and outputs in a finite-dimensional Euclidean space. This kernel is built using the reservoir functional associated with a state-space representation of the Volterra series expansion available for any analytic fading memory filter. It is hence called the Volterra reservoir kernel. Even though the state-space representation and the corresponding reservoir feature map are defined on an infinite-dimensional tensor algebra space, the kernel map is characterized by explicit recursions that are readily computable for specific data sets when employed in estimation problems using the representer theorem. We showcase the performance of the Volterra reservoir kernel in a popular data science application in relation to bitcoin price prediction.
Reservoir kernels and Volterra seriesread_more
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23 March 2023
17:15-18:15 		Prof. Dr. Stefano Marmi
Scuola Normale Superiore di Pisa
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Talks in Financial and Insurance Mathematics

Title Turning lemmings into hogs: the Yoccoz-Birkeland (livestock) population model coupled with (random) price dynamics
Speaker, Affiliation Prof. Dr. Stefano Marmi, Scuola Normale Superiore di Pisa
Date, Time 23 March 2023, 17:15-18:15
Location HG G 43
Abstract Around 1996 Yoccoz and Birkeland introduced an endogenous lagged integral equation to explain the approximately periodic but chaotic lemming population fluctuations in the Arctic ecosystem. The evidence of cyclical but unpredictable behavior of fluctuations is also a well-known feature of livestock product prices (hog cycle) that has attracted the attention of economists for a long time. I will discuss deterministic and random versions of the population-market model proposed by Arlot, Marmi, and Papini in Arlot et al. (2019). The model is obtained by coupling the Yoccoz–Birkeland integral equation with a demand- and supply-dependent price dynamics as in Bélair and Mackey (1989). In the random model, we introduce a stochastic component into the price equation inspired by the Black-Scholes market model and prove the existence of a random attractor and a random invariant measure. We numerically compute the fractal dimension and the entropy of the random attractor and prove its convergence to the deterministic one when the volatility of the market equation tends to zero. We also numerically analyze in detail the dependence of the attractor on the choice of the temporal discretization parameter. We implement several statistical distances to quantify the similarity between the attractors of the discretized systems and the original ones. In particular, following a work by Cuturi (2013), we use the Sinkhorn distance. This distance is a discrete and penalized version of the optimal transport distance between two measures, given a transportation cost matrix. The work on the random dynamics and the investigation of the dependence on the discretization is joint with R. Ceccon and G. Livieri.
Turning lemmings into hogs: the Yoccoz-Birkeland (livestock) population model coupled with (random) price dynamicsread_more
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30 March 2023
17:15-18:15 		Prof. Dr. Nabil Kazi-Tani
Université de Lorraine
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Talks in Financial and Insurance Mathematics

Title The role of correlation in diffusion control ranking games
Speaker, Affiliation Prof. Dr. Nabil Kazi-Tani, Université de Lorraine
Date, Time 30 March 2023, 17:15-18:15
Location HG G 43
Abstract In this talk, we will study Nash equilibria in two player continuous time stochastic differential games with diffusion control, and where the Brownian motions driving the state processes are correlated. We consider zero-sum ranking games, in the sense that the criteria to optimize only depends on the difference of the two players' state processes. We explicitly compute the players' equilibrium strategies, depending on the correlation of the Brownian motions driving the two state equations: in particular, if the correlation coefficient is smaller than some explicit threshold, then the equilibrium strategies consist of strong controls, whereas if the correlation exceeds the threshold, then the equilibrium controls are mixed strategies. To characterize these equilibria, we rely on a relaxed formulation of the game based on solutions to martingale problems, allowing the players to randomize their actions. The talk is based on a joint work with Stefan Ankirchner (University of Jena) and Julian Wendt (University of Jena).
The role of correlation in diffusion control ranking gamesread_more
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6 April 2023
17:15-18:15 		Dr. Tobias Fissler
ETH Zurich, Switzerland
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Talks in Financial and Insurance Mathematics

Title Generalised Covariances and Correlations
Speaker, Affiliation Dr. Tobias Fissler, ETH Zurich, Switzerland
Date, Time 6 April 2023, 17:15-18:15
Location HG G 43
Abstract Pearson covariance of two random variables $X$ and $Y$ measures the average joint comovements around the respective means of $X$ and $Y$. We generalise this well known measure by replacing the means with other summary statistics of the marginal distributions of $X$ and $Y$ such as quantiles, expectiles, or absolute thresholds. Deviations from these quantities are defined via generalised errors, induced by identification or moment functions. As a normalised measure of dependence, a generalised correlation, is constructed. Replacing the common Cauchy--Schwartz normalisation by a novel Fréchet--Hoeffding normalisation, we obtain attainability of the entire interval $[-1,1]$ by the generalised correlation for any given marginals. After uncovering favourable properties of these new dependence measures and establishing consistent estimators, we construct function-valued distributional correlations, exhibiting the entire dependence structure. They give rise to tail correlations, which should arguably supersede the coefficients of tail dependence. The two quantities coincide for positive tail dependence, but the novel notion may distinguish between negative dependence and asymptotic independence. Finally, we construct summary covariances (correlations), which arise as (normalised) weighted averages of distributional covariances. We retrieve Pearson covariance and Spearman correlation as special cases. The talk is based on joint work with Marc-Oliver Pohle.
Generalised Covariances and Correlationsread_more
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13 April 2023
17:15-18:15
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Talks in Financial and Insurance Mathematics

Title No seminar (Easter break)
Speaker, Affiliation
Date, Time 13 April 2023, 17:15-18:15
Location HG G 43
No seminar (Easter break)
HG G 43
20 April 2023
17:15-18:15 		Prof. Dr. Julia Eisenberg
Vienna University of Technology
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Talks in Financial and Insurance Mathematics

Title On Some Multi-Regime (and even beyond) Optimisation Problems in Non-Life Insurance
Speaker, Affiliation Prof. Dr. Julia Eisenberg, Vienna University of Technology
Date, Time 20 April 2023, 17:15-18:15
Location HG G 43
Abstract We consider a company who models the dependence of its surplus - a Brownian motion with drift - on business cycles by a Markov chain. Depending on the chosen target functional and the impact of the control on the surplus, the value function and the optimal strategy can be found explicitly, recursively or not at all. The constant "once and forever" strategies, optimal for the case of only one state, turn out to be suboptimal in most multi-state problems. In this talk, we consider some "explicit"- and "recursive"-solution cases. As a pudding, we look at the "beyond" case, where the model features a stochastic continuous discounting rate - an infinite-regime setting.
On Some Multi-Regime (and even beyond) Optimisation Problems in Non-Life Insuranceread_more
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27 April 2023
17:15-18:15 		Prof. Dr. Daniela Tonon
Università degli Studi di Padova
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Talks in Financial and Insurance Mathematics

Title Mean Field Games planning problems with general initial and final measures
Speaker, Affiliation Prof. Dr. Daniela Tonon, Università degli Studi di Padova
Date, Time 27 April 2023, 17:15-18:15
Location HG G 43
Abstract The planning problem in Mean Field Games (MFG) was introduced by P.-L. Lions in his lessons, to describe models in which a central planner would like to steer a population to a predetermined final configuration while still allowing individuals to choose their own strategies. In a recent variational approach, see Graber, Mészáros, Silva and Tonon (2019) and Orrieri, Porretta and Savaré (2019) the authors studied the well-posedness of this problem in case of merely summable initial and final measures, using techniques, coming from optimal transport, introduced by Benamou and Brenier in 2000, extended to the congestion case in Carlier, Cardaliaguet and Nazaret (2013), and already used to show the existence and uniqueness of weak solutions for classical MFGs by Cardaliaguet and collaborators. The case of less regular initial and final measures is now studied via techniques introduced by Jimenez in 2008, for the analogous problem in optimal transport.
Mean Field Games planning problems with general initial and final measuresread_more
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4 May 2023
17:15-18:15 		Prof. Dr. Vincent Tassion
ETH Zurich, Switzerland
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Talks in Financial and Insurance Mathematics

Title Noise sensitivity of percolation
Speaker, Affiliation Prof. Dr. Vincent Tassion, ETH Zurich, Switzerland
Date, Time 4 May 2023, 17:15-18:15
Location HG G 43
Abstract Consider critical site Bernoulli percolation on the triangular lattice, where each vertex is colored black or white with probability 1/2, independently of the other vertices. In 1999, Benjamini, Kalai and Schramm proved that crossing probabilities are noise sensitive: resampling a small proportion of the vertices lead to an independent percolation picture. Ten years later, Garban, Pete and Schramm obtained a sharp quantitative version of this result. These works rely on Fourier analysis, and are restricted to Bernoulli percolation (i.e. product measure) and the independent resampling dynamics. In this talk, we will first introduce and discuss the general question of noise sensitivity, focusing mostly on percolation applications. Then we will present a recent and robust approach that relies on geometrical arguments and not on spectral methods. Based on joint works with Hugo Vanneuville.
Noise sensitivity of percolationread_more
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11 May 2023
17:15-18:15 		Prof. Dr. Ariel Neufeld
Nanyang Technological University, Singapore
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Talks in Financial and Insurance Mathematics

Title Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its proof of overcoming the curse of dimensionality
Speaker, Affiliation Prof. Dr. Ariel Neufeld, Nanyang Technological University, Singapore
Date, Time 11 May 2023, 17:15-18:15
Location HG G 43
Abstract In this talk, we first provide a brief introduction to quantum computing from a mathematical perspective. No prior knowledge of quantum computing is necessary. We then introduce a quantum Monte Carlo algorithm to solve high-dimensional Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension d of the PDE and the reciprocal of the prescribed accuracy ε and so demonstrate that our quantum Monte Carlo algorithm does not suffer from the curse of dimensionality. This talk is based on a joint work with Yongming Li.
Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its proof of overcoming the curse of dimensionalityread_more
HG G 43
18 May 2023
17:15-18:15
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Talks in Financial and Insurance Mathematics

Title No seminar (Ascension Day)
Speaker, Affiliation
Date, Time 18 May 2023, 17:15-18:15
Location HG G 43
No seminar (Ascension Day)
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25 May 2023
17:15-18:15 		Dr. Daniel Bartl
Universität Wien
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Talks in Financial and Insurance Mathematics

Title Statistical estimation of stochastic optimization problems
Speaker, Affiliation Dr. Daniel Bartl, Universität Wien
Date, Time 25 May 2023, 17:15-18:15
Location HG G 43
Abstract We develop a novel procedure for estimating the optimizer of general convex stochastic optimization problems from an iid sample. This procedure is the first one that exhibits the optimal statistical performance in heavy tailed situations and also applies in highdimensional settings. We discuss the portfolio optimization problem as a special instance. Joint work with Shahar Mendelson.
Statistical estimation of stochastic optimization problemsread_more
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1 June 2023
17:15-18:15 		Prof. Dr. Guillaume Carlier
Université Paris Dauphine
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Talks in Financial and Insurance Mathematics

Title A variant of Hewitt and Savage theorem for finitely exchangeable laws and applications to optimal transport
Speaker, Affiliation Prof. Dr. Guillaume Carlier, Université Paris Dauphine
Date, Time 1 June 2023, 17:15-18:15
Location HG G 43
Abstract In this talk, I will describe a variant of the Hewitt and Savage theorem for laws of exchangeable finite families of random variables as well as their marginals. This representation reveals the role of some universal polynomials of measures which contain correlated correction terms and capture the geometry of extreme points of symmetric laws and their marginals. I will also describe possible applications to some multi-marginal optimal problems with symmetries. This is based on a joint work with Gero Friesecke and Daniela Vögler (TU Munich).
A variant of Hewitt and Savage theorem for finitely exchangeable laws and applications to optimal transportread_more
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21 June 2023
17:15-18:15 		Prof. Dr. Arnulf Jentzen
The Chinese University of Hong Kong, Shenzhen and University of Münster
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Talks in Financial and Insurance Mathematics

Title Overcoming the curse of dimensionality: from nonlinear Monte Carlo to the training of neural networks
Speaker, Affiliation Prof. Dr. Arnulf Jentzen, The Chinese University of Hong Kong, Shenzhen and University of Münster
Date, Time 21 June 2023, 17:15-18:15
Location HG G 43
Abstract Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. Nearly all traditional approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula. In this talk we present an efficient machine learning algorithm to approximate solutions of high-dimensional PDE and we also prove that deep artificial neural network (ANNs) do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs. Moreover, we specify concrete examples of smooth functions which can not be approximated by shallow ANNs without the curse of dimensionality but which can be approximated by deep ANNs without the curse of dimensionality. In the final part of the talk we present some recent mathematical results on the training of neural networks.
Overcoming the curse of dimensionality: from nonlinear Monte Carlo to the training of neural networksread_more
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3 July 2023
15:15-16:15 		Prof. Dr. Jim Dai
Cornell University and The Chinese University of Hong Kong, Shenzhen
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Talks in Financial and Insurance Mathematics

Title Queueing Network Controls via Deep Reinforcement Learning
Speaker, Affiliation Prof. Dr. Jim Dai, Cornell University and The Chinese University of Hong Kong, Shenzhen
Date, Time 3 July 2023, 15:15-16:15
Location HG G 43
Abstract Conservative update methods such as Trust Region policy optimization and Proximal policy optimization (PPO) have become the dominant reinforcement learning algorithms because of their ease of implementation and good practical performance. A conventional setup for notoriously difficult queueing network control problems is a Markov decision problem (MDP) that has three features: infinite state space, unbounded costs, and long-run average cost objective. We extend the theoretical framework of these conservative update methods for such MDP problems. The resulting PPO algorithm is tested on a parallel-server system and large-size multiclass queueing networks. The algorithm consistently generates control policies that outperform state-of-art heuristics in literature in a variety of load conditions from light to heavy traffic. These policies are demonstrated to be near-optimal when the optimal policy can be computed. A key to the successes of our PPO algorithm is the use of three variance reduction techniques in estimating the relative value function via sampling. First, we use a discounted relative value function as an approximation of the relative value function. Second, we propose regenerative simulation to estimate the discounted relative value function. Finally, we incorporate the approximating martingale-process method into the regenerative estimator. This is joint work with Mark Gluzman at Meta.
More information https://math.ethz.ch/imsf/courses/talks-in-imsf.html
Queueing Network Controls via Deep Reinforcement Learningread_more
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