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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Autumn Semester 2024

Date / Time Speaker Title Location
3 September 2024
13:30-14:20 		Prof. Dr. Sotirios Sabanis
University of Edinburgh
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Talks in Financial and Insurance Mathematics

Title Stochastic optimization and diffusion-based generative models
Speaker, Affiliation Prof. Dr. Sotirios Sabanis, University of Edinburgh
Date, Time 3 September 2024, 13:30-14:20
Location HG G 19.1
Abstract Diffusion processes have become one of the most significant mathematical frameworks in machine learning (ML), exerting considerable influence across various subfields, including generative modelling, sampling techniques, and nonconvex optimization. The exceptional empirical success of diffusion-based approaches has prompted a growing interest in exploring their mathematical foundations and has driven the development of novel diffusion-based algorithms. In the context of finance, one could argue that applications of diffusion-based generative models can benefit tasks such as scenario generation for risk management purposes and generation of approximate solutions to complex pricing models, such as those used for path-dependent options or exotic derivatives. We will review some recent progress in diffusion-based stochastic optimizers and diffusion-based/score-based generative models.
Stochastic optimization and diffusion-based generative modelsread_more
HG G 19.1
26 September 2024
17:15-18:15 		Prof. Dr. Rama Cont
University of Oxford
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Talks in Financial and Insurance Mathematics

Title Itô calculus without probability
Speaker, Affiliation Prof. Dr. Rama Cont, University of Oxford
Date, Time 26 September 2024, 17:15-18:15
Location HG G 19.1
Abstract It was shown by Hans Föllmer that the Ito formula may be derived as an analytical change of variable formula for smooth functions of irregular trajectories with finite quadratic variation, without making use of any probabilistic methods. We show how this idea may be pushed farther to construct an Ito calculus for causal functionals of irregular paths with finite (and non-zero) p-th order variation for any p>1. In particular we will discuss pathwise integration and change of variable formulae for functionals of irregular trajectories, as well as pathwise analogues of martingales and Doob-Meyer decompositions. These results have natural applications to pathwise optimal control and model-free formulations in mathematical finance. References: R Cont, R Jin (2024) Fractional Ito calculus, Transactions of the American Mathematical Society, Ser. B 11, 727-761. H Chiu, R Cont (2022) Causal Functional Calculus. Transactions of the London Mathematical Society, Volume 9, No. 1 December 2022, 237-269. R Cont, N Perkowski (2019) Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity, Transactions of the American Mathematical Society (Series B), Volume 6, 161-186.
Itô calculus without probabilityread_more
HG G 19.1
10 October 2024
17:15-18:15 		Dr. William Hammersley
Université Côte d'Azur
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Talks in Financial and Insurance Mathematics

Title Rearranged Stochastic Heat Equation: A Regularising Infinite Dimensional Common Noise for Mean Field Models
Speaker, Affiliation Dr. William Hammersley, Université Côte d'Azur
Date, Time 10 October 2024, 17:15-18:15
Location HG G 43
Abstract This talk will present a regularising diffusion in the space of square integrable probability measures over the reals. One begins by representing each probability measure via a uniquely chosen symmetric non-increasing random variable (over the circle) having that measure as its law under (unit-normalised) Lebesgue measure; this is akin to considering its quantile function representation. A diffusion on this space of functions is constructed via a discrete-in-time splitting scheme. Over one interval, one acts on the representative random variables (symmetrised quantiles) by an increment of stochastic heat driven by coloured noise, then at the end of the time interval, one transforms the output function to its symmetric non-increasing rearrangement. This operation ensures the Markovianity of the resulting evolution of the induced measures. The fine time-mesh limit of the schemes can be shown to admit a well-posed characterisation that we call the rearranged stochastic heat equation. This diffusion's regularisation effect is illustrated via its associated semigroup, which maps bounded functions to Lipschitz ones with an integrable small-time singularity for the Lipschitz constants. A simple-to-write minimisation problem of a non-convex functional of probability measure is used as a prototypical and motivational application for which the stochastic gradient descent driven by the rearranged stochastic heat is studied. Exponential convergence to a unique equilibrium is demonstrated under modest assumptions along with metastability properties exhibiting the same order as the finite dimensional setting. Time permitting, I will discuss open directions of inquiry.
Rearranged Stochastic Heat Equation: A Regularising Infinite Dimensional Common Noise for Mean Field Modelsread_more
HG G 43
17 October 2024
17:15-18:00
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Talks in Financial and Insurance Mathematics

Title Inaugural lecture of Prof. Dr. Johanna Ziegel: How good is their best guess? Constructing and evaluating predictions
Speaker, Affiliation
Date, Time 17 October 2024, 17:15-18:00
Location HG F 30
More information https://ethz.ch/en/news-and-events/events/inaugural-farewell-introductory-lectures.html
Inaugural lecture of Prof. Dr. Johanna Ziegel: How good is their best guess? Constructing and evaluating predictionsread_more
HG F 30
24 October 2024
17:15-18:15 		Dr. Lukas Gonon
Imperial College London
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Talks in Financial and Insurance Mathematics

Title Deep learning for pricing: from applications to theory and back
Speaker, Affiliation Dr. Lukas Gonon, Imperial College London
Date, Time 24 October 2024, 17:15-18:15
Location HG G 43
Abstract In the past years, deep learning algorithms have been applied to numerous classical problems from mathematical finance and financial services. For example, deep learning has been employed to numerically solve high-dimensional derivatives pricing and hedging tasks, to provide efficient volatility smoothing or to detect financial asset price bubbles. Theoretical foundations of deep learning for finance, however, are far less developed. In this talk, we start by revisiting some recently developed deep learning methods. We then present our recent results on theoretical foundations for approximating option prices, solutions to jump-diffusion PDEs and optimal stopping problems using (random) neural networks. We address neural network expressivity, highlight challenges in analysing optimization errors and show the potential of random neural networks for mitigating these difficulties. Our results allow to obtain more explicit convergence guarantees, thereby making employed neural network methods more trustable. Based on joint works with F. Biagini, A. Jacquier, A. Mazzon, T. Meyer-Brandis, C. Schwab, R. Wiedemann
Deep learning for pricing: from applications to theory and backread_more
HG G 43
31 October 2024
09:00-19:00
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Talks in Financial and Insurance Mathematics

Title One-Day Symposium: The Complexity of Social and Economic Systems. From Models to Measures
Speaker, Affiliation
Date, Time 31 October 2024, 09:00-19:00
Location HG F 30
More information https://www.sg.ethz.ch/events/sg-final-symposium-october-2024/
One-Day Symposium: The Complexity of Social and Economic Systems. From Models to Measuresread_more
HG F 30
7 November 2024
17:15-18:15 		Prof. Dr. Mitja Stadje
Ulm University
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Talks in Financial and Insurance Mathematics

Title Utility maximization under endogenous pricing
Speaker, Affiliation Prof. Dr. Mitja Stadje, Ulm University
Date, Time 7 November 2024, 17:15-18:15
Location HG G 43
Abstract We study the expected utility maximization problem of a large investor who is allowed to make transactions on tradable assets in an incomplete financial market with endogenous permanent market impacts. The asset prices are assumed to follow a nonlinear price curve quoted in the market as the utility indifference curve of a representative liquidity supplier. We show that optimality can be fully characterized via a system of coupled forward-backward stochastic differential equations (FBSDEs) which corresponds to a non-linear backward stochastic partial differential equation (BSPDE). We show existence of solutions to the optimal investment problem and the FBSDEs in the case where the driver function of the representative market maker grows at least quadratically or the utility function of the large investor falls faster than quadratically or is exponential. Furthermore, we derive smoothness results for the existence of solutions of BSPDEs. Examples are provided when the market is complete, the utility function is exponential or the driver is positively homogeneous.
Utility maximization under endogenous pricingread_more
HG G 43
14 November 2024
17:15-18:15 		Yifan Jiang
University of Oxford
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Talks in Financial and Insurance Mathematics

Title Sensitivity of causal distributionally robust optimization
Speaker, Affiliation Yifan Jiang, University of Oxford
Date, Time 14 November 2024, 17:15-18:15
Location HG G 43
Abstract In this talk, we study the causal distributionally robust optimization (DRO) in both discrete and continuous-time settings. The framework captures model uncertainty, with potential models penalized in function of their adapted Wasserstein distance to a given reference model. The strength of model uncertainty is parameterized via a penalization parameter, and we compute the first-order sensitivity of the value of causal DRO with respect to this parameter. Moreover, we investigate the case where a martingale constraint is imposed on the underlying model, as is the case for pricing measures in mathematical finance. We introduce different scaling regimes, which allow us to obtain the continuous-time sensitivities as nontrivial limits of their discrete-time counterparts. Our proofs rely on novel methods. In particular, we introduce a pathwise Malliavin derivative, which agrees with its classical counterpart under the Wiener measure, and we extend the adjoint operator, the Skorokhod integral, to regular martingale integrators and show it satisfies a stochastic Fubini theorem.
Sensitivity of causal distributionally robust optimizationread_more
HG G 43
21 November 2024
17:15-18:15 		Prof. Dr. Yucheng Yang
University of Zurich
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Talks in Financial and Insurance Mathematics

Title DeepHAM: A Global Solution Method for Heterogeneous Agent Models with Aggregate Shocks
Speaker, Affiliation Prof. Dr. Yucheng Yang, University of Zurich
Date, Time 21 November 2024, 17:15-18:15
Location HG G 43
Abstract We propose an efficient, reliable, and interpretable global solution method, the Deep learning-based algorithm for Heterogeneous Agent Models (DeepHAM), for solving high dimensional heterogeneous agent models with aggregate shocks. The state distribution is approximately represented by a set of optimal generalized moments. Deep neural networks are used to approximate the value and policy functions, and the objective is optimized over directly simulated paths. In addition to being an accurate global solver, this method has three additional features. First, it is computationally efficient in solving complex heterogeneous agent models, and it does not suffer from the curse of dimensionality. Second, it provides a general and interpretable representation of the distribution over individual states, which is crucial in addressing the classical question of whether and how heterogeneity matters in macroeconomics. Third, it solves the constrained efficiency problem as easily as it solves the competitive equilibrium, which opens up new possibilities for normative studies. As a new application, we study constrained efficiency in heterogeneous agent models with aggregate shocks. We find that in the presence of aggregate risk, a utilitarian planner would raise aggregate capital for redistribution less than in absence of it because poor households do more precautionary savings and thus rely less on labor income. Joint work with Jiequn Han and Weinan E
DeepHAM: A Global Solution Method for Heterogeneous Agent Models with Aggregate Shocksread_more
HG G 43
28 November 2024
17:15-18:15 		Dr. Purba Das
King's College London
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Talks in Financial and Insurance Mathematics

Title Understanding roughness – A Schauder expansion approach
Speaker, Affiliation Dr. Purba Das, King's College London
Date, Time 28 November 2024, 17:15-18:15
Location HG G 43
Abstract We study how to construct a stochastic process on a finite interval with given `roughness'. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of Hölder regularity of a function in terms of its Schauder coefficients. Using this characterization we provide a better (pathwise) estimator of Hölder exponent. Furthermore, We study the concept of (generalized) p-th variation of a real-valued continuous function along a sequence of partitions. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of ℓp-norm of the coefficients along a Schauder basis. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.
Understanding roughness – A Schauder expansion approachread_more
HG G 43
5 December 2024
17:15-18:15 		Prof. Dr. Walter Schachermayer
University of Vienna
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Talks in Financial and Insurance Mathematics

Title Optimal martingale transport on R^d
Speaker, Affiliation Prof. Dr. Walter Schachermayer, University of Vienna
Date, Time 5 December 2024, 17:15-18:15
Location HG G 43
Abstract In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. For mu,nu probability measures on R^d, increasing in convex order, stretched Brownian motion provides an analogue for the martingale version of this problem. We provide a characterization in terms of gradients of convex func- tions, similar to the characterization of optimizers in classical optimal transport. We use stretched Brownian motion to extend Kellerer’s theorem on the existence of Markov martingales for given marginal distributions to the d-dimensional case. While this celebrated theorem is known for 50 years in the one-dimensional case, this is the first version pertaining to the d-dimensional case. We also provide a gradient flow to find the optimizer of stretched Brownian motion in the irreducible case. Joint work with M. Beiglboeck, J. Backhoff, B. Robinson, and B. Tschiderer.
Optimal martingale transport on R^dread_more
HG G 43
19 December 2024
17:15-18:15 		Prof. Dr. Max Nendel
University of Waterloo
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Talks in Financial and Insurance Mathematics

Title Upper Comonotonicity and Risk Aggregation under Dependence Uncertainty
Speaker, Affiliation Prof. Dr. Max Nendel, University of Waterloo
Date, Time 19 December 2024, 17:15-18:15
Location HG G 43
Abstract In this talk, we study dependence uncertainty and the resulting effects on tail risk measures, which play a fundamental role in modern risk management. We introduce the notion of a regular dependence measure, defined on multi-marginal couplings, as a generalization of well-known correlation statistics such as the Pearson correlation. The first main result states that even an arbitrarily small positive dependence between losses can result in perfectly correlated tails beyond a certain threshold and seemingly complete independence before this threshold. In a second step, we focus on the aggregation of individual risks with known marginal distributions by means of arbitrary nondecreasing left-continuous aggregation functions. In this context, we show that under an arbitrarily small positive dependence, the tail risk of the aggregate loss might coincide with the one of perfectly correlated losses. A similar result is derived for expectiles under mild conditions. In a last step, we discuss our results in the context of credit risk, analyzing the potential effects on the value at risk for weighted sums of Bernoulli distributed losses. The talk is based on joint work with Corrado De Vecchi and Jan Streicher.
Upper Comonotonicity and Risk Aggregation under Dependence Uncertaintyread_more
HG G 43

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