Groupe de travail

Financial and Actuarial Mathematics, Numerical Probability


Day, hour and place

Thursday at 11:00, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016


Contact(s)

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Previous talks


Year 2025

Financial and Actuarial Mathematics, Numerical Probability
Thursday May 15, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Maxime Egéa (LPSM) Bayesian estimation with MCMC

My talk is motivated by the problem of sampling from a posterior distribution in Bayesian estimation, particularly when the posterior concentrates as the number of observations increases. In this context, I will present the Laplace approximation—a classical method for approximating concentrated posterior distributions. We will examine its application in the setting of log-concave measures, and then extend the discussion to more challenging cases involving multimodal distributions. The second part of the talk will be devoted to the Metropolis-Hastings algorithm, and how one can use the previously established results to improve its performance. This ongoing work is conducted in collaboration with Daniel Rudolf.

Financial and Actuarial Mathematics, Numerical Probability
Thursday April 3, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Jules Delemotte (Ecole Polytechnique) Smile dynamics and rough volatility

We investigate the dynamic properties of various stochastic, and notably rough, volatility models, with an emphasis on the dynamics of implied volatilities. While recent literature has extensively analyzed static properties, such as a model’s calibration power or the term structure of at-the-money skews, dynamic features have received less attention. We focus on the Skew-Stickiness Ratio (SSR), an industry-standard indicator of joint spot price and implied volatility dynamics, pursuing the analysis of [Bergomi, Smile Dynamics IV, Risk, 2009] and extending it to rough volatility models. Using different numerical estimators, we compare the behavior of the SSR generated by several models (not limited to the affine framework) with the empirical market SSR estimated for the SPX Index. Interestingly, we observe that different forward variance models—two-factor Bergomi, rough Bergomi, rough Heston, Heston—calibrated as best as possible to the same SPX smile generate SSR that (i) are close to one another, and (ii) display significant deviations from market data, failing to reproduce the term structure observed for the empirical SSR. These observations suggest a certain rigidity within the stochastic volatility family under consideration, and indicate that rough volatility alone does not significantly alter the joint spot-implied volatility dynamics.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 27, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Antonio Ocello (Ecole Polytechnique) Theoretical Advances in Score-Based Generative Models: Convergence Bounds and Noise Schedule Optimization

Score-based generative models (SGMs) have emerged as a state-of-the-art framework for sampling from complex data distributions, leveraging the estimation of score functions through noise-perturbed samples. A central challenge in understanding SGMs lies in quantifying their convergence to the target distribution. Various works have analyzed this convergence using Kullback-Leibler (KL) divergence and Wasserstein distances, often under restrictive assumptions.

In this talk, I will present an overview of existing convergence bounds for SGMs and introduce two recent contributions that provide a refined theoretical understanding of their performance. First, I will discuss our work on the impact of the noise schedule on generative performance, where we establish explicit upper bounds for the KL divergence and Wasserstein-2 distance under mild assumptions. This result not only improves upon prior state-of-the-art bounds but also provides practical insights into hyperparameter selection. Building on this framework, I will then present recent advances in Wasserstein-2 convergence analysis. By leveraging the regularization properties of the Ornstein–Uhlenbeck (OU) process, we relax the traditional assumptions of log-concavity and score regularity. Our approach reveals that weakly log-concave distributions evolve towards log-concavity, and we establish a novel characterization of the dynamics of score function contraction and non-contraction. This enables more general and widely applicable convergence results, particularly for complex distributions such as Gaussian mixtures.

This talk is based on joint work with Stanislas Strasman, Claire Boyer, Sylvain Le Corff, and Vincent Lemaire, recently accepted to Transactions on Machine Learning Research (https://openreview.net/forum?id=BlYIPa0Fx1, as well as a recent preprint in collaboration with Marta Gentiloni-Silveri (https://arxiv.org/abs/2501.02298)

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 20, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Maxime Guellil (LPSM & CACIB) Volterra Stein-Stein Fourier-Laplace transform and Fredholm determinant

We study complex discontinuities arising from the miscomputation of the Fourier-Laplace transform in the Volterra Stein-Stein model, which involves the complex square root of a Fredholm determinant. Discontinuities occur when the determinant crosses the negative real axis. We characterize these crossings for the joint Fourier-Laplace transform of the integrated variance and log-price. Additionally, we derive a corrected formula for the Fourier-Laplace transform and develop efficient numerical techniques to detect and compute these crossings. Applying our algorithms to Fourier-based pricing in the fractional Stein-Stein model, we achieve a remarkable increase in accuracy while drastically reducing computational cost compared to existing methods.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 20, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Othman Mazhar (LPSM) An optimal interpolation diffusion model for time series: bridging Schrödinger and Bass

This talk presents the Schrödinger Bass Bridge (SBB), a novel framework for generating a continuous semi-martingale with a prescribed joint distribution at successive times 0 = t0 < . . . < tN = T. This is formulated as an optimal interpolation problem that unifies the Schrödinger bridge and Bass frameworks, allowing for the construction of a diffusion process that simultaneously calibrates both drift and volatility to times series data. By decomposing the problem into a sequence of optimal transport problems between successive time steps, where mass is transported from a Dirac measure to a given conditional density, we derive a simple quasi-analytic formula to compute the drift and volatility of the optimal diffusion sequentially over each time interval. Our approach relies on proving the attainment of the dual problem for which we provide an explicit resolution. Much like for the Martingale optimal transport problem, we also show that the solution to the SBB optimal transport is obtained via a simple process that we call a Bass semi-Martingale, and we provide an efficient fix point scheme for computing its drift and volatility.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 13, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
François-Xavier Vialard (Université Gustave Eiffel) Entropic semi-martingale optimal transport.

After a brief introduction to standard tools in optimal transport and to entropic regularization, we present how to use the tool of entropic regularization to design numerical schemes to learn the diffusion and drift coefficients. Our proposal relies on renormalizing the entropy of a discretized process. We formulate a time-discrete minimization problem for which we show a sort of gamma-convergence result to a continuous functional that makes appear a Bregman divergence associated to the entropy on gaussian processes.

We finish with some numerical simulations illustrating our main result.

Financial and Actuarial Mathematics, Numerical Probability
Thursday February 13, 2025, 11:15AM, Sophie Germain salle 1016
Sergio Pulido (ENSIIE/LaMME) Stochastic Volterra Equations in Finance

In this presentation, I will provide an overview of several works related to the theory of Stochastic Volterra Equations (SVEs) and their applications in finance. I will discuss the well-posedness and invariance properties of these equations, examine the relevance of incorporating jumps from a modeling perspective, and explore factor approximation methods for control problems such as optimal stopping. Additionally, I will present potential future research directions that can be further developed in these areas.

Financial and Actuarial Mathematics, Numerical Probability
Thursday February 6, 2025, 11:15AM, Sophie Germain salle 1016
Ahmed Kebaier (Université d'Evry / LaMME) A pure dual approach for hedging Bermudan options.

In this work we develop a new dual approach to compute the hedging portfolio of a Bermudan option and its initial value. It gives a “purely dual” algorithm following the spirit of Rogers (2010) in the sense that it only relies on the dual pricing formula. The key is to rewrite the dual formula as an excess reward representation and to combine it with a strict convexification technique. The hedging strategy is then obtained by using a Monte Carlo method, solving backward a sequence of least square problems. We show convergence results for our algorithm and test it on many different Bermudan options. Beyond giving directly the hedging portfolio, the strength of the algorithm is to assess both the relevance of including financial instruments in the hedging portfolio and the effect of the rebalancing frequency.

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 23, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Ludovic Goudenège (CNRS LaMME / Université Paris Saclay) Numerical scheme for SDE with singular drifts : Application to rough Heston model.

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 9, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Peter Bank (Technische Universität Berlin) How much should we care what others know? Jump signals in optimal investment under relative performance concerns

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 9, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Olivier Guéant (Université Paris 1 Panthéon-Sorbonne) Market-Making Models: Overview and Applications to Precious Metals Markets

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 9, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Julien Guyon (Cermics / ENPC) Fast Exact Joint S&P 500/VIX Smile Calibration in Discrete and Continuous Time

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 9, 2025, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Mathieu Laurière (NYU Shanghai) Deep Learning for Stackelberg Mean Field Games via Single-Level Reformulation


Year 2024

Financial and Actuarial Mathematics, Numerical Probability
Thursday December 19, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Charles Bertucci (CNRS Ecole Polytechnique) Stochastic optimal transport and HJB equations on the set of probability measures

Financial and Actuarial Mathematics, Numerical Probability
Thursday December 12, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Paul Gassiat (Université Paris Dauphine et IUF) Martingale property and moments in signature volatility models

Financial and Actuarial Mathematics, Numerical Probability
Thursday December 5, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Zhenjie Ren (Univ Paris Saclay) Self-interacting approximation to McKean-Vlasov long time limit

Financial and Actuarial Mathematics, Numerical Probability
Thursday November 28, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Nabil Kazi-Tani (Université de Lorraine) Stochastic differential equations with normal reflection on time dependent domains

Financial and Actuarial Mathematics, Numerical Probability
Thursday November 21, 2024, 10AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Christian Yeo (LPSM) Convex ordering for stochastic control: the swing contracts case.

We investigate propagation of convexity and convex ordering on a typical stochastic optimal control problem, namely the pricing of “Take-or-Pay” swing option, a financial derivative product commonly traded on energy markets. The dynamics of the underlying asset is modelled by an ARCH model with convex coefficients. We prove that the value function associated to the stochastic optimal control problem is a convex function of the underlying asset price. We also introduce a domination criterion offering insights into the monotonicity of the value function with respect to parameters of the underlying ARCH coefficients. We particularly focus on the one-dimensional setting where, by means of Stein’s formula and regularization techniques, we show that the convexity assumption for the ARCH coefficients can be relaxed with a semi-convexity assumption. To validate the results presented in this paper, we also conduct numerical illustrations. An ArXiv version of the paper is available here: http://www.arxiv.org/abs/2406.07464.

Financial and Actuarial Mathematics, Numerical Probability
Thursday November 21, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Noufel Frikha (Université Paris 1) To be announced.

Financial and Actuarial Mathematics, Numerical Probability
Thursday October 3, 2024, 11AM, Sophie Germain salle 1013
Charles Albert Lehalle (Ecole Polytechnique) Machine Learning and Data Sciences for Financial Markets

The last ten years have seen increased use of machine learning and new kinds of data in financial markets. In this talk I will explain how these techniques and new sources of information can improve the traditional “quantitative finance” approaches. After a general introduction, the talk will be structured in 3 parts: - customisation of services to users of the financial ecosystem; this will cover Robo-advisors and recommendation systems - improvement of risk management; including off-line reinforcement learning to solve HJBs - better connection with the real economy, that will introduce the use of alternative data (texts, satellite images, geolocation, etc).

Financial and Actuarial Mathematics, Numerical Probability
Thursday June 20, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Susana Gomes (University of Warwick) Mean field limits for interacting particle systems

In this talk, I will present a class of interacting particle systems which are systems of Stochastic Differential Equations (SDEs) evolving in a confining potential, that can have multiple wells, interacting via their mean and in the presence of white or coloured noise. I will discuss how to obtain the mean-field limit of this system, as well as how to use this to analyse its longtime behaviour. If time permits, I will then move on to discuss inference for this type of SDEs, in particular parameter estimation for coefficients of interest, and will explain this framework with an example of parameter estimation for a pedestrian dynamics problem using data from individual trajectories (rather than aggregate data such as density).

Financial and Actuarial Mathematics, Numerical Probability
Thursday June 13, 2024, 11AM, Sophie Germain salle 1016
René Aid (Paris dauphine PSL) Dynamic regulation of carbon emissions market

We deal with optimal dynamic carbon emission regulation of a set of firms. The regulator dynamically allocates emission allowances to each firm. Firms face idiosyncratic, as well as common, economic shocks on emissions, and they have linear quadratic abatement costs. Firms can trade allowances so as to minimize total expected costs, which arise from abatement, trading, and terminal penalty. We find a closed-form expression of the non-unique optimal dynamic allocation policies that allow a desired expected emission reduction. The optimal policies are fully responsive, and therefore induce a constant abatement effort and a constant price of allowances. We apply these results to an extension where the regulator wishes also to limit carbon inflation. Joint work with Maria Arduca, Sara Biagini and Luca Taschini.

Financial and Actuarial Mathematics, Numerical Probability
Thursday June 6, 2024, 11:30AM, Sophie Germain salle 1016
Filippo De Feo (LUISS) Optimal control of stochastic delay differential equations and applications to portfolio optimization and optimal advertising

In this talk, we consider an optimal control problem of stochastic delay differential equations. To regain Markovianity and use the dynamic programming approach, we lift the problem on a suitable Hilbert space. Then we characterize the value function of the problem as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, which is a fully non-linear second-order partial differential equation on a Hilbert space with an unbounded operator. Since no regularity results are available for viscosity solutions of these kinds of HJB equations, via a new finite-dimensional reduction procedure and using the regularity theory for finite-dimensional PDEs, we prove a partial $C^{1,\alpha}$-regularity result of the value function. When the diffusion is independent of the control, this regularity result allows us to define a candidate optimal feedback control. However, due to the lack of $C^2$-regularity of the value function, we cannot prove a verification theorem using standard techniques based on Ito’s formula. Then, using a technical double approximation procedure, we construct regular approximants of the value function, which are supersolutions of perturbed HJB equations and are regular enough to satisfy a non-smooth Ito’s formula. This allows us to prove a verification theorem and construct optimal feedback controls. We discuss applications to optimal advertising and portfolio optimization.

Financial and Actuarial Mathematics, Numerical Probability
Thursday May 30, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Sergey Nadtochiy (Illinois Institute of Technology) Cascade equation for Stefan problem as a mean field game

The solutions to Stefan problem with Gibbs-Thomson law (i.e., with surface tension effect) are well known to exhibit singularities which, in particular, lead to jumps of the associated free boundary along the time variable. The correct times, directions and sizes of such jumps are well understood under the assumption of radial symmetry, under which the free boundary is a sphere with varying radius. The characterization of such jumps in a general multidimensional setting has remained an open question until recently. In our ongoing work with M. Shkolnikov and Y. Guo, we have derived a separate (hyperbolic) partial differential equation — referred to as the cascade equation — whose solutions describe the jumps of the solutions to the Stefan problem without any symmetry assumptions. It turns out that a solution of the cascade equation corresponds to a maximal element of the set of all equilibria in a family of (first-order local) mean field games. In this talk, I will present and justify the cascade equation, will show its connection to the mean field games, and will prove the existence of a solution to the cascade equation. If time permits, I will also show how these results can be used to construct a solution to the Stefan problem itself.

Financial and Actuarial Mathematics, Numerical Probability
Thursday May 23, 2024, 10AM, CACIB Montrouge
Julien Guyon & Bouazza Saadeddine (CERMICS & CACIB) Séance GT chez CACIB Montrouge

Julien Guyon : term-structure du ATM skew

Bouazza Saadeddine : Calibration through regression.

Financial and Actuarial Mathematics, Numerical Probability
Thursday May 2, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Fausto Gozzi (LUISS Roma) Regular solutions for HJB in infinite dimension through partial smoothing.

In this talk we first provide some examples of applied control problems which motivates the study of regular solutions for HJB equations in infinite dimensional spaces or in spaces of probability measures. We then discuss the methods to get such results focusing on the “partial smoothing” technique. In particular we present, depending also on timing: 1) recent results with F. Masiero on the control of delay equations and on boundary control; 2) recent results with F. Masiero and M. Rosestolato, on mean-field control problems where the HJB equation can be simplified and solved explicitly; 3) going on results (only partial results and some ideas) with A.Calvia, F. Masiero and G. Tessitore to HJB with measures coming from mean-field control.

Financial and Actuarial Mathematics, Numerical Probability
Thursday April 25, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Thierry Roncalli (Amundi) Modeling Climate Risk with Input-output Models

Input-output models are a type of economic analysis that quantify the interdependencies between different sectors of an economy. An economy is then summarized by a matrix of technical coefficients, which has the mathematical property of being a doubly substochastic nonnegative matrix. Input-output analysis can be extended to include environmental externalities such as pollution and greenhouse gas emissions. In this presentation, we show how this framework can be used to estimate upstream Scope 3 emissions at the country, sector and issuer levels. By calculating imported and exported carbon emissions, we will then illustrate how a consumption-based inventory provides a different carbon footprint picture than a production-based inventory. We will also show how to extend the traditional cost-push price model to account for pass-through mechanisms and apply this framework to calculate the economic cost of a carbon tax. Finally, we will illustrate how to define a climate stress test of an investment portfolio within input-output models. The analysis of the different results will be linked to the mathematics of input-output models, in particular to the density/sparsity of the supply chain matrix and the Markov properties of directed graphs.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 28, 2024, 9AM, Jussieu, Salle Paul Lévy, 16-26 209
Matinée Anr Reliscope To be announced.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 21, 2024, 11AM, Sophie Germain salle 1016
Athena Picarelli (Université de Verone) A deep solver for BSDEs with jumps

The aim of this work is to propose an extension of the deep solver by Han, Jentzen, E (2018) to the case of forward backward stochastic differential equations (FBSDEs) with jumps. As in the aforementioned  solver, starting from a discretized version of the FBSDE and parametrizing the (high dimensional) control processes by means of a family of artificial neural networks (ANNs), the FBSDE is viewed as model-based reinforcement learning problem and the ANN parameters are fitted so as to minimize a prescribed loss function. We take into account both finite and infinite jump activity by introducing, in the latter case, an approximation  with finitely many  jumps of the forward process. We successfully apply our algorithm to option pricing problems in low and high dimension and discuss the applicability in the context of counterparty credit risk.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 14, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Sergio Pulido (ENSIIE, Univ. Paris Saclay) Polynomial Volterra processes

Recent studies have extended the theory of affine processes to the stochastic Volterra equations framework. In this talk, I will describe how the theory of polynomial processes extends to the Volterra setting. In particular, I will explain the moment formula and an interesting stochastic invariance result in this context. This is joint work with Eduardo Abi Jaber, Christa Cuchiero, Luca Pelizzari and Sara Svaluto-Ferro.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 7, 2024, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Cyril Benezet (ENSIIE, Univ. Paris Saclay) Hedging Valuation Adjustment et Risque de Modèle

En 2021, Burnett introduit la notion de “Hedging Valuation Adjustment”, dans le but de prendre en compte les frictions sur le hedge dynamique telles que les coûts de transaction. En plus de ces frictions, nous incorporons la notion de risque de modèle à cette réserve, via la formalisation du risque de modèle Darwinien introduit par Albanese, Crépey et Labichino en 2021. La HVA résultante permet de quantifier ce risque, et établit un pont entre un modèle global “fair valuation” (supposé existant) et les modèles locaux utilisés par les différents desks de la banque. Ces modèles locaux sont quotidiennement recalibrés, ce qui induit un risque sur la couverture dynamique, risque également capturé par notre HVA. La réserve est enfin ajustée au risque au travers d'une contribution à la KVA (Capital Valuation Adjustment) de la banque. Nous calculons explicitement ou numériquement ces réserves sur des modèles jouets. Les ordres de grandeur des réserves suggèrent que les traders devraient utiliser de bons modèles, plutôt que de mauvais compensés par des réserves. Les résultats sont issus de deux travaux, en collaboration d'une part avec C. Albanese et S. Crépey, et d'autre part avec S. Crépey et D. Essaket.

Financial and Actuarial Mathematics, Numerical Probability
Thursday February 29, 2024, 10AM, CREDIT AGRICOLE, Pl. des États Unis, Montrouge
Vincent Lemaire Et Huyên Pham (LPSM) Séance Chaire CACIB

Programme:

10h accueil 

10h30-11h15 Vincent Lemaire (Sorbonne Université / LPSM), Denoising Diffusion Probabilistic Models, introduction et quelques résultats théoriques.

Résumé : Introduits récemment, les modèles génératifs basés sur une dynamique de bruitage/débruitage des données se révèlent très performants. On exposera le cadre mathématique à temps continu qui se base sur les équations différentielles stochastiques et le score matching. On donnera quelques résultats théoriques de convergence et on s'intéressera au comportement de la borne de l'erreur en fonction de la façon dont on bruite (noise schedule). Ce dernier point est un travail en commun avec Claire Boyer, Sylvain Le Corff, Antonio Ocello et Stanislas Strasman.

11h15-11h45 pause café

11h45-12h30 Huyen Pham (Université Paris Cité / LPSM), Nonparametric generative modeling for time series via Schrödinger bridge.

Résumé: We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence respecting  the temporal dynamics of the time series distribution. We  estimate the drift function from data samples by nonparametric, e.g. kernel regression methods,  and the simulation of the SB diffusion  yields new synthetic data samples of the time series.The performance of our generative model is evaluated through a series of numerical experiments.  First, we test with autoregressive models, a GARCH Model, and the example of fractional Brownian motion,  and measure the accuracy of our algorithm with marginal, temporal dependencies metrics, and predictive scores. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets. 

Financial and Actuarial Mathematics, Numerical Probability
Thursday February 8, 2024, 11AM, Sophie Germain salle 1013
Yadh Hafsi (Univ. Paris Saclay) Uncovering Market Disorder and Liquidity Trends Detection

We propose a new methodology to detect notable changes in liquidity within an order-driven market. As part of our approach, we employ Marked Hawkes processes to model trades-through which constitute our liquidity proxy. Subsequently, our focus lies in accurately identifying the moment when a significant increase or decrease in its intensity takes place. We consider the minimax quickest detection problem of unobservable changes in the intensity of a doubly-stochastic Poisson process. The goal is to develop a stopping rule that minimizes the robust Lorden criterion, measured in terms of the number of events until detection. We prove our procedure's optimality in the case of a Cox process with simultaneous jumps while considering a finite time horizon. Finally, this novel approach is empirically validated using real market data analyses.

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 25, 2024, 11AM, Sophie Germain salle 1013
Fanny Cartelier (ENSAE) Can investors curb greenwashing

We show how investors with pro-environmental preferences and who penalize revelations of past environmental controversies impact corporate greenwashing practices. Through a dynamic equilibrium model with information asymmetry, we characterize firms' optimal environmental communication, emissions reduction, and greenwashing policies, and we explain the forces driving them. Notably, under a condition that we explicitly characterize, companies greenwash to inflate their environmental score above their fundamental environmental value, with an effort and impact increasing in investors’ pro-environmental preferences. However, investment decisions that penalize greenwashing, policies increasing transparency, and environment-related technological innovation contribute to mitigating corporate greenwashing. We provide empirical support for our results.

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 18, 2024, 11AM, Sophie Germain salle 1016
Ioannis Gasteratos (Imperial College London) Transportation-cost inequalities for nonlinear Gaussian functionals

In this talk, we study concentration properties for laws of non-linear Gaussian functionals on metric spaces. Our focus lies on measures with non-Gaussian tail behaviour which are beyond the reach of Talagrand’s classical Transportation- Cost Inequalities (TCIs). Motivated by solutions of Rough Differential Equations and relying on a suitable contraction principle, we prove generalised TCIs for functionals that arise in the theory of regularity structures and, in particular, in the cases of rough volatility and the two-dimensional Parabolic Anderson Model. Our work also extends existing results on TCIs for diffusions driven by Gaussian processes. This is joint work with Antoine Jacquier.

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 11, 2024, 11AM, Sophie Germain salle 1016
Olivier Guéant (Paris 1) Incorporating Variable Liquidity in Optimal Market Making and Inventory Management Models: A Comparison of Hawkes Processes and Markov-Modulated Poisson Processes

Since Avellaneda and Stoikov's seminal work, market making models have evolved to incorporate increasingly realistic aspects, such as complex price dynamics, price differentiation, adverse selection, or even parameter ambiguity. However, the dynamics of liquidity have been less frequently addressed. While Hawkes processes are a natural choice for modelling beyond constant request and trade intensities, this talk introduces an alternative approach using Markov-modulated Poisson processes (MMPPs). We will explore the benefits and limitations of MMPPs and Hawkes processes in the context of algorithmic market making and inventory management model development.


Year 2023

Financial and Actuarial Mathematics, Numerical Probability
Thursday December 14, 2023, 11AM, Sophie Germain salle 1016
Camilo Garcia Trillos (University College London) Adversarial Distributional Robustness from Wasserstein Ascent-Descent Particle Dynamics

In the context of machine learning, adversarial distributional attacks aim to deceive machine learning models by making imperceptible changes to the distribution of input data. These subtle modifications can cause the model to make incorrect predictions, even though the input data appears normal to a human observer. One technique to enhance the robustness of a given learner is through adversarial training, where the model is exposed to adversarial attacks during the training process. Mathematically, this problem can be formulated as a minimax problem. In this talk, we present a particle-based algorithm to solve adversarial training problems in various supervised learning scenarios. Our algorithm draws inspiration from ascent-descent dynamics in a projected Wasserstein space. We show that the particle dynamics converge towards mean-field limit equations as the number of particles increases. Furthermore, the mean-field dynamics tend to epsilon-Nash equilibria of the original adversarial learning problem as time approaches infinity. We provide results for both nonconvex-nonconcave (in the natural geometry of the dynamics) and nonconvex-concave settings. We will also highlight some other areas of application of the algorithm, for example in the context of mathematical finance. Based on joint work with N. García Trillos.

Financial and Actuarial Mathematics, Numerical Probability
Thursday December 7, 2023, 11AM, Sophie Germain salle 1016
Caroline Hillairet (ENSAE) Bi-Revealed Utilities in a defaultable universe

This talk investigates the inverse problem of bi-revealed utilities in a defaultable universe, defined as a standard universe (represented by a filtration F) perturbed by an exogenous defaultable time τ. We assume that the standard universe does not take into account

the possibility of the default, thus τ adds an additional source of risk. The defaultable universe is represented by the filtration G up to time τ (τ included), where G stands for the progressive enlargement of F by τ . The basic assumption in force is that τ avoids F-stopping times. The bi-revealed problem consists in recovering a consistent dynamic utility from the observable characteristic of an agent. The general results on bi-revealed utilities, first given in a general and abstract framework, are translated in the defaultable G-universe and then are interpreted in the F-universe. The decomposition of G-adapted processes XG provides an interpretation of a G-characteristic XτG stopped at τ as a reserve process. Thanks to the characterization of G-martingales stopped at τ in terms of F-martingales, we establish a correspondence between G-bi-revealed utilities from characteristic and F-bi-revealed pair of utilities from characteristic and reserves. In a financial framework, characteristic can be interpreted as wealth and reserves as consumption. This result sheds a new light on the consumption in utility criterion: the consumption process can be interpreted as a certain quantity of wealth, or reserves, that are accumulated for the financing of losses at the default time.

This is a joint work with N. El Karoui and M. Mrad.

Financial and Actuarial Mathematics, Numerical Probability
Thursday November 30, 2023, 11AM, Sophie Germain salle 1016
Grégoire Loeper (BNP Parisba) Black and Scholes, Legendre and Sinkhorn

This talk will be a unified overview of some recent contributions in financial mathematics. The financial topics are option pricing with market impact and model calibration. The mathematical tools are fully non-linear partial differential equations and semi-martingale optimal transport. Some new and fun results will be a Black-Scholes-Legendre formula for option pricing with market impact, a Measure Preserving Martingale Sinkhorn algorithm for martingale optimal transport, and a lognormal version of the Bass Martingale.

Financial and Actuarial Mathematics, Numerical Probability
Thursday November 23, 2023, 11AM, Sophie Germain salle 1016
Eduardo Abi Jaber (CMAP) From the Quintic model that jointly calibrates SPX/VIX to Signature Volatility models

The quintic Ornstein-Uhlenbeck volatility model is a stochastic volatility model where the volatility process is a polynomial function of degree five of a single Ornstein-Uhlenbeck process with fast mean reversion and large vol-of-vol. The model is able to achieve remarkable joint fits of the SPX-VIX smiles with only 6 effective parameters and an input curve that allows to match certain term structures. Even better, the model remains very simple and tractable for pricing and calibration: the VIX squared is again polynomial in the Ornstein-Uhlenbeck process, leading to efficient VIX derivative pricing by a simple integration against a Gaussian density and simulation of the volatility process is exact.

For pricing SPX products, we show that the Quintic model is part of a larger class of stochastic volatility model where the volatility is driven by a linear function of the path signature of a Brownian motion enhanced with the running time. For this larger class of models, we develop pricing and hedging methodologies using Fourier inversion techniques on the characteristic function which is known up to an infinite-dimensional Riccati equation. We illustrate our method on numerical examples for pricing, hedging and calibration of vanilla and path-dependent options in several classes of Markovian and Non-Markovian models.

Financial and Actuarial Mathematics, Numerical Probability
Thursday November 16, 2023, 11AM, Sophie Germain salle 1013
Samuel Daudin (Univ. Nice) On the optimal rate for the convergence problem in mean-field control

The goal of this work is to obtain optimal rates for the convergence problem in mean field control. Our analysis covers cases where the solutions to the limiting problem may not be unique nor stable. Equivalently the value function of the limiting problem might not be differentiable on the entire space. The key argument in our approach consists in mollifying the value function of the limiting problem in order to produce functions that are almost classical sub-solutions to the limiting Hamilton-Jacobi equation (which is a PDE set on the space of probability measures). These sub-solutions can be projected onto finite dimensional spaces and then compared with the value functions associated with the particle systems. In the end, this comparison is used to prove the most demanding bound in the estimates. The key challenge therein is thus to exhibit an appropriate form of mollification. We do so by employing sup-convolution within a convenient functional Hilbert space. This is a joint work with François Delarue and Joe Jackson.

Financial and Actuarial Mathematics, Numerical Probability
Thursday October 19, 2023, 11AM, INRIA 2 Rue Simone Iff, 75012 Paris, France
Robert Denkert (HU Berlin) xtended Mean Field Control Problems with Singular Controls

We consider a novel class of extended mean field control (MFC) problems with singular controls, where the costs depend on the current state, control, and the joint law of the state-control-process. We prove an approximation of general singular controls with purely regular controls. Subsequently, we derive a dynamic programming principle and use it to establish a quasi-variational inequality (QVI) for the value function in the Wasserstein space. Finally, using the master equations of the approximating regular MFC problems, we establish a uniqueness result for our QVI characterisation of the value function of the MFC problem with singular controls. This presentation is based on joint work with Ulrich Horst.

Financial and Actuarial Mathematics, Numerical Probability
Thursday October 19, 2023, 9AM, Inria 2 Rue Simone Iff, 75012 Paris, France
Gudmund Pammer (ETH, Zurich) Stretched Brownian Motion: Analysis of a Fixed-Point Scheme

The fitting problem is a classical challenge in mathematical finance about finding martingales that satisfy specific marginal constraints. Building on the Bass solution to the Skorokhod embedding problem and optimal transport, Backhoff, Beiglböck, Huesmann, and Källblad propose a solution for the two-marginal problem: the stretched Brownian motion. Notably rich in structure, this process is an Ito diffusion and a continuous, strong Markov martingale. Following a similar approach, Conze and Henry-Larbordère recently introduced a novel local volatility model. This model, rooted in an extension of the Bass construction, is efficiently computable through a fixed-point scheme. In our presentation, we reveal the fixed-point scheme's intricate connection to the stretched Brownian motion and analyse its convergence. This presentation is based on joint work with Beatrice Acciaio and Antonio Marini.

Financial and Actuarial Mathematics, Numerical Probability
Thursday October 19, 2023, 11:45AM, Inria 2 Rue Simone Iff, 75012 Paris, France
Aurélien Alfonsi (ENPC) Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation.

We define and study convolution kernels that preserve nonnegativity. When the past dynamics of a process is integrated with a convolution kernel like in Stochastic Volterra Equations or in the jump intensity of Hawkes processes, this property allows to get the nonnegativity of the integral. We give characterizations of these kernels and show in particular that completely monotone kernels preserve nonnegativity. We then apply these results to analyze the stochastic invariance of a closed convex set by Stochastic Volterra Equations. We also get a comparison result in dimension one. Last, when the kernel is a positive linear combination of decaying exponential functions, we present a second order approximation scheme for the weak error that stays in the closed convex domain under suitable assumptions. We apply these results to the rough Heston model and give numerical illustrations.

Financial and Actuarial Mathematics, Numerical Probability
Thursday October 19, 2023, 9:45AM, Inria 2 Rue Simone Iff, 75012 Paris, France
Mehdi Talbi (LPSM) Sannikov’s contracting problem with many Agents

This work aims to study an extension of the celebrated Sannikov’s Principal-Agent problem to the multi-Agents case. In this framework, the contracts proposed by the Principal consist in a running payment, a retirement time and a final payment at retirement. After discussing how the Principal may derive optimal contracts in the N-Agents case, we explore the corresponding mean field model, with a continuous infinity of Agents. We then prove that the Principal’s problem can be reduced to a mixed control-and-stopping mean field problem, and we derive a semi-explicit solution of the first best contracting problem. This is a joint work with Thibaut Mastrolia and Nizar Touzi.

Financial and Actuarial Mathematics, Numerical Probability
Thursday October 12, 2023, 11AM, Sophie Germain salle 1016
Céline Labart (Université de Savoie) To be announced.

Financial and Actuarial Mathematics, Numerical Probability
Wednesday July 12, 2023, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Mortiz Voss (UCLA) On Adaptive Robust Optimal Execution and Machine Learning Surrogates

We study a stochastic optimal order execution problem in discrete time in the presence of price impact. First, we propose a unified price impact model which nests various existing models in the literature. Second, we propose a numerical algorithm based on dynamic programming and neural network surrogates to solve the resulting optimal stochastic control problem. Third, we extend our modeling framework to account for model uncertainty in the sense of unknown model parameters and propose an adaptive robust optimization approach that combines dynamic learning with a robust (worst-case) optimization.

Financial and Actuarial Mathematics, Numerical Probability
Thursday June 15, 2023, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Thomas Kruse (Bergischen Universität Wuppertal) Multilevel Picard approximations for high-dimensional semilinear parabolic PDEs and further applications

We present the multilevel Picard approximation method for high-dimensional semilinear parabolic PDEs. A key idea of our method is to combine multilevel approximations with Picard fixed-point approximations. We prove in the case of semilinear heat equations with Lipschitz continuous nonlinearities that the computational effort of the proposed method grows polynomially both in the dimension and in the reciprocal of the required accuracy. Moreover, we present further applications of the multilevel Picard approximation method and illustrate its efficiency by means of numerical simulations. The talk is based on joint works with Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Tuan Nguyen and Philippe von Wurstemberger.

Financial and Actuarial Mathematics, Numerical Probability
Thursday May 11, 2023, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Botao Li (LPSM) Simplified Models and Optimization Algorithms in Deep Learning

In general, deep learning models are considered black boxes and are not able to be solved analytically. However, some simplified models are easy enough to analyze while sharing characteristics with realistic deep-learning models. In this talk, we present results obtained in the simplified models that aim at providing insight into deep learning in general. We demonstrate that, in some toy models, the stochastic gradient descent algorithm (SGD) with a constant learning rate can converge to local maxima; adaptive methods can have the same problem; SGD can escape a saddle point arbitrarily slow; it may prefer a sharp minimum rather than a flat one. We also present the expression for the global minima of a deep linear network with weight decay and dropout.

Financial and Actuarial Mathematics, Numerical Probability
Thursday April 13, 2023, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Xavier Erny (CMAP) Propagation du chaos conditionnelle de modèles en champ moyen

Nous étudions un système de particules en interaction (de type champ moyen) dans une normalisation diffusive. Chaque particule interagit avec les autres de manière ponctuelle à un taux aléatoire. A chaque interaction, la dynamique des particules est perturbée par une quantité aléatoire centrée de l'ordre de N^{−1/2}. Nous prouvons que ce système converge, quand N tend vers l'infini, vers une équation différentielle stochastique dirigée par un mouvement brownien W, qui est créé par le théorème central limite. Ce mouvement brownien régit les mouvements de toutes les particules, et créé un bruit commun dans le système limite. Conditionnellement à W, les particules sont indépendantes dans le système limite. C'est la propriété de propagation du chaos conditionnelle. Pour prouver la convergence en loi du système fini vers le système limite, nous introduisons un nouveau type de problème de martingale adapté à notre cadre de travail. Les techniques utilisées dans les preuves reposent sur le fait que l'on étudie des systèmes échangeables.

Financial and Actuarial Mathematics, Numerical Probability
Thursday April 13, 2023, 10AM, Jussieu, Salle Paul Lévy, 16-26 209
Pierre Lavigne (Institut Louis Bachelier) Decarbonization of financial markets: a mean field game approach

We present a model of a financial market where a large number of firms determine their dynamic emission strategies under climate transition risk in the presence of both green-minded and neutral investors. The firms aim to achieve a trade-off between financial and environmental performance, while interacting through the stochastic discount factor, determined in equilibrium by the investors’ allocations.

We formalize the problem in the setting of mean-field games and prove the existence and uniqueness of a Nash equilibrium for firms. We then present a convergent numerical algorithm for computing this equilibrium and illustrate the impact of climate transition risk and the presence of green-minded investors on the market decarbonization dynamics and share prices.

We show that uncertainty about future climate risks and policies leads to higher overall emissions and higher spreads between share prices of green and brown companies. This effect is partially reversed in the presence of environmentally concerned investors, whose impact on the cost of capital spurs companies to reduce emissions.

Joint work with Peter Tankov.

Financial and Actuarial Mathematics, Numerical Probability
Thursday April 6, 2023, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Mahmoud Khabou (INSA Toulouse) The normal approximation of compound Hawkes functionals

We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals of deterministic and non-negative integrands with respect to Hawkes processes by a normally distributed random variable. Our results are specifically applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time, or have been obtained only for specific kernels such as the exponential or Erlang functions.

Financial and Actuarial Mathematics, Numerical Probability
Thursday April 6, 2023, 10AM, Jussieu, Salle Paul Lévy, 16-26 209
Ziad Kobeissi (INRIA ILB) Temporal Difference Learning with Continuous Time and State in the Stochastic Setting

Classical methods in reinforcement learning allow to efficiently approximate discrete-time value functions using observations. In this talk, we investigate the extensions of such methods to the continuous-time case in a stochastic regime. Mathematically, this boils down to numerically solving BSDEs through learning from observations. Here, we focus our attention on the problem of policy evaluation and more precisely on the basic method TD(0) (for temporal difference). This corresponds to learning the value function of an uncontrolled SDE associated with a reward function. Using vanishing time steps, we propose two adaptations of the TD(0) algorithm: the first is similar to the one in discrete time and is model-free; the second is model-based and is obtained by adding a zero-expectation term, resulting in a reduction of its variance. In the linear setting, we prove multiple convergence results for the two algorithms, the model-based one is more flexible and enjoys better convergence rates. In particular, using the Polyak-Juditsky averaging method and a constant learning step, we obtain a convergence rate similar to the state of the art on the simpler problem of linear regression using SGD. Finally, we present simulations showing numerical evidence of our theoretical analysis.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 30, 2023, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Mehdi Talbi (ETH Zurich) Mean field games of stopping times

We are interested in the study of stochastic games for which each player faces an optimal stopping problem. In our setting, the players may interact through the criterion to optimize as well as through their dynamics. After briefly discussing the N-players game, we formulate the corresponding mean field problem. In particular, we introduce a weak formulation of the game for which we are able to prove existence of Nash equilibria for a large class of criteria. We also prove that equilibria for the mean field problem provide approximated Nash equilibria for the N-players game, and we formally derive the master equation associated with our mean field game. Joint work with Dylan Possamaï.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 16, 2023, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Rudy Morel (Ecole Normale Supérieure) A statistical model of financial time-series through Scattering Spectra

We introduce the wavelet scattering spectra which provide non-Gaussian models of time-series having stationary increments.

These spectra are an extension of the standard wavelet spectrum and are defined as the diagonal of a certain non-linear correlation matrix on wavelet coefficients.

They characterize a wide range of non-Gaussian properties of multi-scale processes. This is analyzed for a variety of processes in the Finance literature.

We prove that self-similar processes have scattering spectra which are scale invariant. This property can be tested statistically on a single realization and defines a class of wide-sense self-similar processes.

We build maximum entropy models conditioned by scattering spectra coefficients, and generate new time-series with a microcanonical sampling algorithm.

Besides capturing statistical properties of observed time-series, these models can be used to predict future volatility and are shown to capture non-trivial statistical properties of the option smile.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 9, 2023, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
Ofelia Bonesini (Imperial College) Correlated equilibria for mean field games with progressive strategies

In a discrete space and time framework, we study the mean field game limit for a class of symmetric N-player games based on the notion of correlated equilibrium. We give a definition of correlated solution that allows to construct approximate N-player correlated equilibria that are robust with respect to progressive deviations. We illustrate our definition by way of an example with explicit solutions.

Financial and Actuarial Mathematics, Numerical Probability
Thursday February 16, 2023, 11AM, Jussieu, Salle Paul Lévy, 16-26 209
William Hammersley (Univ. Nice) A prospective regularising common noise for mean field systems

Financial and Actuarial Mathematics, Numerical Probability
Thursday February 2, 2023, 4PM, Jussieu, Salle Paul Lévy, 16-26 209
Nabil Kazi-Tani (Université de Lorraine) The role of correlation in diffusion control ranking games

We study Nash equilibriums in 2-players continuous time stochastic differential games, where the players are allowed to control the diffusion coefficient of their state process. 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 optimal 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 optimal strategies consist of strong controls, whereas if the correlation exceeds the threshold, then the optimal controls are mixed strategies. To characterize these equilibria, we rely on a relaxed formulation of the game, allowing the players to randomize their actions. This is a joint work with Stefan Ankirchner and Julian Wendt (University of Jena).

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 26, 2023, 4:30PM, Jussieu, Salle Paul Lévy, 16-26 209
Manal Jakani (Le Mans Université) Approximation of reflected SDEs in time-dependent domains and applications to Generalized BSDEs and PDE in time-dependent domain

We consider a class of reflected SDEs in non-smooth convex time-dependent domains. We provide a strong approximation for this type of equations using a sequence of standard SDEs. As a consequence, we obtain an approximation scheme for generalized BSDEs using standard BSDEs. As a by-product, we get an approximation for the solution of a system of PDEs with nonlinear boundary conditions in time-dependent domains.

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 19, 2023, 4:30PM, Jussieu, Salle Paul Lévy, 16-26 209
Andrea Mazzon (LMU München) Detecting asset price bubbles using deep learning

In this paper we employ deep learning techniques to detect financial asset bubbles by using observed call option prices. The proposed algorithm is widely applicable and model-independent. We test the accuracy of our methodology in numerical experiments within a wide range of models and apply it to market data of tech stocks in order to assess if asset price bubbles are present. Under a given condition on the pricing of call options under asset price bubbles, we provide a theoretical foundation of our approach for positive and continuous stochastic processes. When such a condition is not satisfied, we focus on local volatility models. To this purpose, we give a new necessary and sufficient condition for a process with time-dependent local volatility function to be a strict local martingale. This is a joint work with Francesca Biagini, Lukas Gonon and Thilo Meyer-Brandis.

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 5, 2023, 4PM, Jussieu, Salle Paul Lévy, 16-26 209
Ahmed Kebaier (Université d'Evry) The interpolated drift implicit Euler scheme Multilevel Monte Carlo method for pricing Barrier options and applications to the CIR and CEV models

Recently, Giles et al. [2019] proved that the efficiency of the Multilevel Monte Carlo (MLMC) method for evaluating Down-and-Out barrier options for a diffusion process (Xt)t∈[0,T] with globally Lipschitz coefficients, can be improved by combining a Brownian bridge technique and a conditional Monte Carlo method provided that the running minimum inft∈[0,T]Xt has a bounded density in the vicinity of the barrier. In the present work, thanks to the Lamperti transformation technique and using a Brownian interpolation of the drift implicit Euler scheme of Alfonsi [2013], we show that the efficiency of the MLMC can be also improved for the evaluation of barrier options for models with non-Lipschitz diffusion coefficients under certain moment constraints. We study two example models: the Cox-Ingersoll-Ross (CIR) and the Constant of Elasticity of Variance (CEV) processes for which we show that the conditions of our theoretical framework are satisfied under certain restrictions on the models parameters. In particular, we develop semi-explicit formulas for the densities of the running minimum and running maximum of both CIR and CEV processes which are of independent interest. Finally, numerical tests are processed to illustrate our results.

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 5, 2023, 5PM, Jussieu, Salle Paul Lévy, 16-26 209
Thomas Wagenhofer (TU Berlin) Weak error estimates for rough volatility models

We consider a rough volatility model where the volatility is a (smooth) function of a Riemann-Liouville Brownian motion with Hurst parameter H in (0,1/2). When simulating these models, one often uses a discretization of stochastic integrals as an approximation. These integrals can be interpreted as log-stock-prices and in financial applications such as in pricing, the most relevent quantities are expectations of (payoff) functions.

Our main result is that moments of these integrals have a weak error rate of order 3H+1/2 if H<1/6 and order 1 otherwise. For this we first derive a moment formula for both the discretization and the true stochastic integral. We then use this formula and properties of Gaussian random variables to prove our main theorems. Furthermore, we show that this convergence rate also holds for slightly more general payoffs and also provide a lower bound. Note that our rate of 3H+1/2 is in stark contrast to the strong error rate which is of order H.


Year 2022

Financial and Actuarial Mathematics, Numerical Probability
Thursday December 15, 2022, 4PM, Jussieu, Salle Paul Lévy, 16-26 209
Nizar Touzi (CMAP, Ecole Polytechnique) Arrêt optimal en champ moyen

Financial and Actuarial Mathematics, Numerical Probability
Thursday December 15, 2022, 5PM, Jussieu, Salle Paul Lévy, 16-26 209
Olivier Bokanowski (Univ. Paris Cité, LJLL) Neural Networks for First Order HJB Equations

We consider a deterministic optimal control problem, in a finite horizon context, and propose deep neural network approximations for Bellman's dynamic programming principle, corresponding also to some first-order Hamilton-Jacobi-Bellman (HJB) equations. This work follows the work of Huré, Pham, Bachouch, and Langrené (SIAM J. Numer. Anal., vol. 59 (1), 2021, pp. 525-557) where algorithms are proposed in a stochastic context. However, we need to develop an entirely new approach in order to deal with the propagation of errors in the deterministic setting, where no diffusion is present in the dynamics. Our analysis gives precise error estimates in an average norm. The algorithms are then illustrated on several academic numerical examples related to front propagations models in the presence of obstacle constraints (modelized by an optimal maximum running cost problem) showing the relevance of the approach for average dimensions (e.g. from $2$ to $8$), in particular in situations where the value functions is non-smooth. This is a joint work with Xavier Warin and Averil Prost.

Financial and Actuarial Mathematics, Numerical Probability
Thursday November 24, 2022, 4:30PM, Jussieu, Salle Paul Lévy, 16-26 209
Pierre Bras (LPSM, Sorbonne Université) Total variation convergence of the Euler-Maruyama scheme in small time with unbounded drift

We give bounds for the total variation distance between the solution of a stochastic differential equation in $R^d$ and its one step Euler-Maruyama scheme in small time. We show that for small t, the total variation is of order $t^{1/3}$ , and more generally of order $t^{r/(2r+1)}$ if the noise coefficient $\sigma$ of the SDE is elliptic and $C^2r_b$, $r \in N$, using multi-step Richardson-Romberg extrapolation. The Richardson-Romberg extrapolation is a method used in numerical analysis to improve the convergence rates of numerical schemes and relies on a linear combination of Taylor expansions with null coefficients up to some order; we adapt this method to our case for theoretical purposes.

Financial and Actuarial Mathematics, Numerical Probability
Thursday October 20, 2022, 4PM, Jussieu, Salle Paul Lévy, 16-26 209
Damien Lamberton (Université Gustave Eiffel) Régularité de la frontière libre d'un problème d'arrêt optimal : une approche probabiliste

Dans ce travail en collaboration avec Tiziano De Angelis (de l'Université de Turin), nous proposons une approche probabiliste de la dérivabilité de la frontière libre d'un problème d'arrêt optimal d'une diffusion uni-dimensionnelle.

Financial and Actuarial Mathematics, Numerical Probability
Thursday October 20, 2022, 5PM, Jussieu, Salle Paul Lévy, 16-26 209
Aurélien Alfonsi (Ecole des Ponts) Approximation of Stochastic Volterra Equations with kernels of completely monotone type (Joint work with Ahmed Kebaier)

In this work, we develop a multifactor approximation for d-dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an L2-estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in (0, 1/2), we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.

Financial and Actuarial Mathematics, Numerical Probability
Thursday October 6, 2022, 4:30PM, Jussieu, Salle Paul Lévy, 16-26 209
Michaël Allouche (Ecole Polytechnique) Estimation of extreme quantiles from heavy-tailed distributions with neural networks

We propose new parametrizations for neural networks in order to estimate extreme quantiles in both non-conditional and conditional heavy-tailed settings. All proposed neural network estimators feature a bias correction based on an extension of the usual second-order condition to an arbitrary order. The convergence rate of the uniform error between extreme log-quantiles and their neural network approximation is established. The finite sample performances of the non-conditional neural network estimator are compared to other bias-reduced extreme-value competitors on simulated data. The source code is available at https://github.com/michael-allouche/nn-quantile-extrapolation.git. Finally, the conditional neural network estimators are implemented to investigate the behavior of extreme rainfalls as functions of their geographical location in the southern part of France.

Financial and Actuarial Mathematics, Numerical Probability
Thursday July 7, 2022, 4PM, Sophie Germain salle 1016
Anthony Reveillac (INSA Toulouse) Malliavin calculus for Hawkes functionals and application to Insurance

In this talk we will present the basics of the Malliavin calculus we developed for the Hawkes process. We will recall this mathematical object together with one key representation which is at the core of our analysis. As an application of our theoretical findings, we will present some quantifications of the risk for (re-)Insurance contracts. This talk is based on several joint works with Caroline Hillairet; Lorick Huang; Mahmoud Khabou and Mathieu Rosenbaum.

Financial and Actuarial Mathematics, Numerical Probability
Thursday June 23, 2022, 5PM, Sophie Germain salle 1016
Boualem Djehiche (KTH Stockholm) On zero-sum Dynkin games of mean field type.

I will review recent results on a class of zero-sum game problems of mean-field type which extend the classical zero-sum Dynkin game problems. The results I will highlight include sufficient conditions under which such a game admits a value and a saddle point and a characterization of the value of the game in terms of a specific class of doubly reflected backward stochastic differential equations (BSDEs) of mean-field type. Moreover, a corresponding system of weakly interacting values of zero-sum Dynkin games is introduced and is shown to converge to the value of the zero-sum mean-field Dynkin game. In particular, a propagation of chaos result is derived.

This is a joint work with Roxana Dumistrescu.

Financial and Actuarial Mathematics, Numerical Probability
Thursday June 2, 2022, 5PM, Jussieu, Salle Paul Lévy, 16-26 209 / Sophie Germain salle 1016
Marcos Lopes De Prado (ADIA) Open problems in Finance

Financial and Actuarial Mathematics, Numerical Probability
Thursday May 12, 2022, 5PM, Sophie Germain salle 1016
Maximilien Germain (Université Paris Cité, LPSM) A level-set approach to the control of state-constrained McKean-Vlasov equations: application to portfolio selection

We consider the control of McKean-Vlasov dynamics (or mean-field control) with probabilistic state constraints. We rely on a level-set approach which provides a representation of the constrained problem in terms of an unconstrained one with exact penalization and running maximum or integral cost. Our work extends (Bokanowski, Picarelli, and Zidani, SIAM J. Control Optim. 54.5 (2016), pp. 2568–2593) and (Bokanowski, Picarelli, and Zidani, Appl. Math. Optim. 71 (2015), pp. 125–163) to a mean-field setting. The reformulation as an unconstrained problem is particularly suitable for the numerical resolution of the problem, that is achieved from an extension of a machine learning algorithm from (Carmona, Laurière, arXiv:1908.01613 to appear in Ann. Appl. Prob., 2022). An application focuses on a mean-variance portfolio selection problem with probabilistic constraints on the wealth. We also illustrate our approach for a direct numerical resolution of the primal Markowitz continuous-time problem without relying on duality.

Financial and Actuarial Mathematics, Numerical Probability
Thursday April 28, 2022, 4PM, Sophie Germain salle 1016
Nabil Khazi-Tani (IECL, université de Lorraine) To be announced.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 17, 2022, 4PM, Sophie Germain salle 1016
Pierre Cardaliaguet (Ceremade, Université Paris-Dauphine) On the convergence rate for the optimal control of McKean-Vlasov dynamics

In this talk I will report on a joint work with S. Daudin (Paris Dauphine), J. Jackson (U. Texas) and P. Souganidis (U. Chicago). We are interested in the convergence problem for the optimal control of McKean-Vlasov dynamics, also known as mean field control. We establish an algebraic rate of convergence of the value functions of N-particle stochastic control problems towards the value function of the corresponding McKean-Vlasov problem. This convergence rate is established in the presence of both idiosyncratic and common noise, and in a setting where the value function for the McKean-Vlasov problem need not be smooth. Then I will discuss a quantitative propagation of chaos property for the optimal trajectories of the optimal control of the particle system. It relies on the regularity of the value function of the limit problem on an open and dense subset of the state space.

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 17, 2022, 5PM, Sophie Germain salle 1016
Haoyang Cao (École Polytechnique) Identifiability in Inverse Reinforcement Learning

Inverse reinforcement learning attempts to reconstruct the reward function in a Markov decision problem, using observations of agent actions. As already observed in Russell [1998] the problem is ill-posed, and the reward function is not identifiable, even under the presence of perfect information about optimal behavior. We provide a resolution to this non-identifiability for problems with entropy regularization. For a given environment, we fully characterize the reward functions leading to a given policy and demonstrate that, given demonstrations of actions for the same reward under two distinct discount factors, or under sufficiently different environments, the unobserved reward can be recovered up to a constant. We also give general necessary and sufficient conditions for reconstruction of time-homogeneous rewards on finite horizons, and for action-independent rewards, generalizing recent results of Kim [2021] and Fu [2018].

Financial and Actuarial Mathematics, Numerical Probability
Thursday March 3, 2022, 5PM, Sophie Germain salle 1016
Jodi Dianetti (Bielefeld University) Submodular mean field games: Existence and approximation of solutions

We study mean field games with scalar Itô-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. Firstly, it allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Secondly, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist a minimal and a maximal solution. Thirdly, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. Our approach also allows to treat submodular mean field games with common noise, as well as mean field games with singular controls, optimal stopping and reflecting boundary conditions

Financial and Actuarial Mathematics, Numerical Probability
Thursday February 17, 2022, 4PM, Sophie Germain salle 1016
David Métivier (CMAP, Ecole Polytechnique) Interpretable hidden Markov model for stochastic weather generation and climate change analysis

The challenges raised by climate change force industrials to carefully analyze the resilience of their assets to anticipate future weather conditions. In particular, the estimation of future extreme hydrometeorological events, like the frequency of long-lasting dry spells, is critical for hydropower or nuclear generation. Stochastic Weather Generators (SWG) are essential tools to determine these future risks, as they can quickly sample climate statistics from models. They can be either trained on historical data or from simulated data like expert climate change scenarios. In our work, the SWG described and validated with France historical data is based on a spatial Hidden Markov Model (HMM). It generates correlated multisite rain occurrences and amounts, with special attention to the correct reproduction of the distribution of dry and wet spells. The hidden states are viewed as global climate states, e.g., dry all over France, rainy in the north, etc. The resulting model is fully interpretable. We describe how the model recovers large-scale structures that are compared with North Atlantic Oscillations. The model achieves good performances, specifically in terms of extremes, where for example, statistics of drought at the scale of France are well replicated. The model architecture allows easy integration of other weather variables like temperature. In a last part, we show how the model parameters change when trained on RCP climate scenarios.

Financial and Actuarial Mathematics, Numerical Probability
Thursday February 17, 2022, 5PM, Sophie Germain salle 1016
Sergio Pulido (LaMME, ENSIEE) The rough Heston model with self-exciting jumps

We introduce a novel affine Volterra stochastic volatility model by adding jumps with a self-exciting structure to the dynamics of the variance process and the log returns in the well-known rough Heston model. In particular, the variance process is given in terms of the solution of a stochastic affine Volterra equation of convolution type with jumps. Thanks to the affine structure of the model we can provide explicit formulas for the Laplace transforms of quantities like log-prices and their quadratic variation, variance swap rates, and integrated variance. These formulas can be exploited to price options using Fourier inversion techniques. The procedure to get the explicit expressions for these transforms essentially relies on the martingale property of complex-valued exponential processes and comparison results for (deterministic) Riccati-Volterra equations. In addition, when the Brownian component in the spot variance dynamics is equal to zero, we devise an exact and efficient conditional simulation scheme. We illustrate with numerical examples the behavior of the implied volatility smiles for options written on the underlying asset and on the VIX. Joint work with Alessandro Bondi, Giulia Livieri and Simone Scotti.

Financial and Actuarial Mathematics, Numerical Probability
Thursday February 3, 2022, 4PM, Jussieu, Salle Paul Lévy, 16-26 209
Peter Tankov (CREST, ENSAE) Optimal Exploration of an Exhaustible Resource with Stochastic Discoveries

The standard Hotelling model assumes that the stock of an exhaustible resource is known. &nbsp;We expand on the model by Arrow and Chang that introduced stochastic discoveries and for the first time completely solve such a model using impulse control. The model has two state variables: the ``proven reserves as well as a finite unexplored area available for exploration with constant marginal cost, resulting in a Poisson process of new discoveries. &nbsp;We prove &nbsp;that &nbsp;a frontier of critical levels of ``proven reserves exists, above which exploration is stopped, and below which it happens at infinite speed. This frontier is increasing in the explored area, and higher ``proven'' reserve levels along this critical threshold are indicative of more scarcity, not less. In our stochastic generalization of Hotelling's rule, price expectations conditional on the current state rise at the rate of interest across exploratory episodes. However, the state-dependent conditional &nbsp;expected &nbsp;path of prices realized prior to exhaustion of the exploratory area rises at a rate lower than the rate of interest, consistent with &nbsp;most &nbsp;empirical tests based on observed price histories.

Financial and Actuarial Mathematics, Numerical Probability
Thursday February 3, 2022, 5PM, Jussieu, Salle Paul Lévy, 16-26 209
Alexandre Pannier (Imperial College, Londres) Rough multi-factor volatility models for SPX and VIX

After a short review of VIX and rough volatility models, we provide explicit small-time formulae for the at-the-money implied volatility, skew and curvature in a large class of models. Our general setup encompasses both European options on a stock and VIX options, thereby providing new insights on their joint calibration. This framework also allows to consider rough volatility models and their multi-factor versions; in particular we develop a detailed analysis of the two-factor rough Bergomi model. The tools used are essentially based on Malliavin calculus for Gaussian processes. This is a joint work with A. Jacquier and A. Muguruza.

Financial and Actuarial Mathematics, Numerical Probability
Thursday January 20, 2022, 5PM, Jussieu, Salle Paul Lévy, 16-26 209
Philippe Bergault (Ecole Polytechnique) A mean field game of market making against strategic traders

We design a market making model à la Avellaneda–Stoikov in which the market takers act strategically, in the sense that they design their trading strategy based on a specific signal. The market maker chooses her quotes based on the average market takers' behaviour, modelled through a mean field interaction. We derive, up to the resolution of a coupled HJB–Fokker–Planck system, the optimal controls of the market maker and the representative market taker. This approach is flexible enough to incorporate different behaviours for the market takers and takes into account the impact of their strategies on the price process. Joint work with Bastien Baldacci and Dylan Possamai.