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Herbstsemester 2024

Datum / Zeit Referent:in Titel Ort
3. September 2024
13:30-14:20 		Prof. Dr. Sotirios Sabanis
University of Edinburgh
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Talks in Financial and Insurance Mathematics

Titel Stochastic optimization and diffusion-based generative models
Referent:in, Affiliation Prof. Dr. Sotirios Sabanis, University of Edinburgh
Datum, Zeit 3. September 2024, 13:30-14:20
Ort 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

Titel Itô calculus without probability
Referent:in, Affiliation Prof. Dr. Rama Cont, University of Oxford
Datum, Zeit 26. September 2024, 17:15-18:15
Ort 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. Oktober 2024
17:15-18:15 		Dr. William Hammersley
Université Côte d'Azur
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Talks in Financial and Insurance Mathematics

Titel Rearranged Stochastic Heat Equation: A Regularising Infinite Dimensional Common Noise for Mean Field Models
Referent:in, Affiliation Dr. William Hammersley, Université Côte d'Azur
Datum, Zeit 10. Oktober 2024, 17:15-18:15
Ort 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. Oktober 2024
17:15-18:00
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Talks in Financial and Insurance Mathematics

Titel Inaugural lecture of Prof. Dr. Johanna Ziegel: How good is their best guess? Constructing and evaluating predictions
Referent:in, Affiliation
Datum, Zeit 17. Oktober 2024, 17:15-18:00
Ort HG F 30
Mehr Informationen 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. Oktober 2024
17:15-18:15 		Dr. Lukas Gonon
Imperial College London
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Talks in Financial and Insurance Mathematics

Titel Deep learning for pricing: from applications to theory and back
Referent:in, Affiliation Dr. Lukas Gonon, Imperial College London
Datum, Zeit 24. Oktober 2024, 17:15-18:15
Ort 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. Oktober 2024
09:00-19:00
Details

Talks in Financial and Insurance Mathematics

Titel One-Day Symposium: The Complexity of Social and Economic Systems. From Models to Measures
Referent:in, Affiliation
Datum, Zeit 31. Oktober 2024, 09:00-19:00
Ort HG F 30
Mehr Informationen 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

Titel Utility maximization under endogenous pricing
Referent:in, Affiliation Prof. Dr. Mitja Stadje, Ulm University
Datum, Zeit 7. November 2024, 17:15-18:15
Ort 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

Titel Sensitivity of causal distributionally robust optimization
Referent:in, Affiliation Yifan Jiang, University of Oxford
Datum, Zeit 14. November 2024, 17:15-18:15
Ort 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

Titel DeepHAM: A Global Solution Method for Heterogeneous Agent Models with Aggregate Shocks
Referent:in, Affiliation Prof. Dr. Yucheng Yang, University of Zurich
Datum, Zeit 21. November 2024, 17:15-18:15
Ort 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

Titel Understanding roughness – A Schauder expansion approach
Referent:in, Affiliation Dr. Purba Das, King's College London
Datum, Zeit 28. November 2024, 17:15-18:15
Ort 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. Dezember 2024
17:15-18:15 		Prof. Dr. Walter Schachermayer
University of Vienna
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Talks in Financial and Insurance Mathematics

Titel Optimal martingale transport on R^d
Referent:in, Affiliation Prof. Dr. Walter Schachermayer, University of Vienna
Datum, Zeit 5. Dezember 2024, 17:15-18:15
Ort 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
12. Dezember 2024
17:15-18:15
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Talks in Financial and Insurance Mathematics

Titel Title T.B.A.
Referent:in, Affiliation
Datum, Zeit 12. Dezember 2024, 17:15-18:15
Ort HG G 43
Title T.B.A.
HG G 43
19. Dezember 2024
17:15-18:15 		Prof. Dr. Max Nendel
Bielefeld University
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Talks in Financial and Insurance Mathematics

Titel Title T.B.A.
Referent:in, Affiliation Prof. Dr. Max Nendel, Bielefeld University
Datum, Zeit 19. Dezember 2024, 17:15-18:15
Ort HG G 43
Title T.B.A.
HG G 43

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