3.1.1 Contagion modeling and systemic risk

After the recent financial crisis, systemic risk has emerged as one of the major research topics in mathematical finance. Interconnected systems are subject to contagion in time of distress. The scope is to understand and model how the bankruptcy of a bank (or a large company) may or not induce other bankruptcies. By contrast with the traditional approach in risk management, the focus is no longer on modeling the risks faced by a single financial institution, but on modeling the complex interrelations between financial institutions and the mechanisms of distress propagation among these.

The mathematical modeling of default contagion, by which an economic shock causing initial losses and default of a few institutions is amplified due to complex linkages, leading to large scale defaults, can be addressed by various techniques, such as network approaches (see in particular R. Cont et al. 48 and A. Minca 86) or mean field interaction models (Garnier-Papanicolaou-Yang 77).

We have contributed in the last years to the research on the control of contagion in financial systems in the framework of random graph models : In 50, 87, 5, A. Sulem with A. Minca and H. Amini consider a financial network described as a weighted directed graph, in which nodes represent financial institutions and edges the exposures between them. The distress propagation is modeled as an epidemics on this graph. They study the optimal intervention of a lender of last resort who seeks to make equity infusions in a banking system prone to insolvency and to bank runs, under complete and incomplete information of the failure cluster, in order to minimize the contagion effects. The paper 5 provides in particular important insight on the relation between the value of a financial system, connectivity and optimal intervention.

The results show that up to a certain connectivity, the value of the financial system increases with connectivity. However, this is no longer the case if connectivity becomes too large. The natural question remains how to create incentives for the banks to attain an optimal level of connectivity. This is studied in 62, where network formation for a large set of financial institutions represented as nodes is investigated. Linkages are source of income, and at the same time they bear the risk of contagion, which is endogeneous and depends on the strategies of all nodes in the system. The optimal connectivity of the nodes results from a game. Existence of an equilibrium in the system and stability properties is studied. The results suggest that financial stability is best described in terms of the mechanism of network formation than in terms of simple statistics of the network topology like the average connectivity.

3.1.2 Liquidity risk and Market Microstructure

Liquidity risk is the risk arising from the difficulty of selling (or buying) an asset. Usually, assets are quoted on a market with a Limit Order Book (LOB) that registers all the waiting limit buy and sell orders for this asset. The bid (resp. ask) price is the most expensive (resp. cheapest) waiting buy or sell order. If a trader wants to sell a single asset, he will sell it at the bid price, but if he wants to sell a large quantity of assets, he will have to sell them at a lower price in order to match further waiting buy orders. This creates an extra cost, and raises important issues. From a short-term perspective (from few minutes to some days), it may be interesting to split the selling order and to focus on finding optimal selling strategies. This requires to model the market microstructure, i.e. how the market reacts in a short time-scale to execution orders. From a long-term perspective (typically, one month or more), one has to understand how this cost modifies portfolio managing strategies (especially delta-hedging or optimal investment strategies). At this time-scale, there is no need to model precisely the market microstructure, but one has to specify how the liquidity costs aggregate.

For rather liquid assets, liquidity risk is usually taken into account via price impact models which describe how a (large) trader influences the asset prices. Then, one is typically interested in the optimal execution problem: how to buy/sell a given amount of assets optimally within a given deadline. This issue is directly related to the existence of statistical arbitrage or Price Manipulation Strategies (PMS). Most of price impact models deal with single assets. A. Alfonsi, F. Klöck and A. Schied  47 have proposed a multi-assets price impact model that extends previous works. Price impact models are usually relevant when trading at an intermediary frequency (say every hour). At a lower frequency, price impact is usually ignored while at a high frequency (every minute or second), one has to take into account the other traders and the price jumps, tick by tick. Midpoint price models are thus usually preferred at this time scale. With P. Blanc, Alfonsi 3 has proposed a model that makes a bridge between these two types of model: they have considered an (Obizhaeva and Wang) price impact model, in which the flow of market orders generated by the other traders is given by an exogeneous process. They have shown that Price Manipulation Strategies exist when the flow of order is a compound Poisson process. However, modeling this flow by a mutually exciting Hawkes process with a particular parametrization allows them to exclude these PMS. Besides, the optimal execution strategy is explicit in this model. A practical implementation is given in 42.

3.1.3 Dependence modeling

- Calibration of stochastic and local volatility models. The volatility is a key concept in modern mathematical finance, and an indicator of market stability. Risk management and associated instruments depend strongly on the volatility, and volatility modeling is a crucial issue in the finance industry. Of particular importance is the assets dependence modeling.

By Gyongy's theorem, a local and stochastic volatility model is calibrated to the market prices of all call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented by Guyon and Henry-Labordère 79, provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no existence result is available for the SDE nonlinear in the sense of McKean. In the particular case when the local volatility function is equal to the inverse of the root conditional mean square of the stochastic volatility factor multiplied by the spot value given this value and the interest rate is zero, the solution to the SDE is a fake Brownian motion. When the stochastic volatility factor is a constant (over time) random variable taking finitely many values and the range of its square is not too large, B. Jourdain and A. Zhou proved existence to the associated Fokker-Planck equation 22. Thanks to results obtained by Figalli in 73, they deduced existence of a new class of fake Brownian motions. They extended these results to the special case of the LSV model called Regime Switching Local Volatility, when the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level.

- Interest rates modeling. Affine term structure models have been popularized by Dai and Singleton 63, Duffie, Filipovic and Schachermayer 64. They consider vector affine diffusions (the coordinates are usually called factors) and assume that the short interest rate is a linear combination of these factors. A model of this kind is the Linear Gaussian Model (LGM) that considers a vector Ornstein-Uhlenbeck diffusions for the factors, see El Karoui and Lacoste  72. A. Alfonsi et al.  39 have proposed an extension of this model, when the instantaneous covariation between the factors is given by a Wishart process. Doing so, the model keeps its affine structure and tractability while generating smiles for option prices. A price expansion around the LGM is obtained for Caplet and Swaption prices.

3.1.4 Robust finance

- Numerical Methods for Martingale Optimal Transport problems.

The Martingale Optimal Transport (MOT) problem introduced in 61 has received a recent attention in finance since it gives model-free hedges and bounds on the prices of exotic options. The market prices of liquid call and put options give the marginal distributions of the underlying asset at each traded maturity. Under the simplifying assumption that the risk-free rate is zero, these probability measures are in increasing convex order, since by Strassen's theorem this property is equivalent to the existence of a martingale measure with the right marginal distributions. For an exotic payoff function of the values of the underlying on the time-grid given by these maturities, the model-free upper-bound (resp. lower-bound) for the price consistent with these marginal distributions is given by the following martingale optimal transport problem : maximize (resp. minimize) the integral of the payoff with respect to the martingale measure over all martingale measures with the right marginal distributions. Super-hedging (resp. sub-hedging) strategies are obtained by solving the dual problem. With J. Corbetta, A. Alfonsi and B. Jourdain 13 have studied sampling methods preserving the convex order for two probability measures $\mu$ and $\nu$ on ${𝐑}^d$, with $\nu$ dominating $\mu$.

Their method is the first generic approach to tackle the martingale optimal transport problem numerically and can also be applied to several marginals.

- Robust option pricing in financial markets with imperfections.

A. Sulem, M.C. Quenez and R. Dumitrescu have studied robust pricing in an imperfect financial market with default. The market imperfections are taken into account via the nonlinearity of the wealth dynamics. In this setting, the pricing system is expressed as a nonlinear g-expectation ${ℰ}^g$ induced by a nonlinear BSDE with nonlinear driver $g$ and default jump (see 65). A large class of imperfect market models can fit in this framework, including imperfections coming from different borrowing and lending interest rates, taxes on profits from risky investments, or from the trading impact of a large investor seller on the market prices and the default probability. Pricing and superhedging issues for American and game options in this context and their links with optimal stopping problems and Dynkin games with nonlinear expectation have been studied. These issues have also been addressed in the case of model uncertainty, in particular uncertainty on the default probability. The seller's robust price of a game option has been characterized as the value function of a Dynkin game under ${ℰ}^g$ expectation as well as the solution of a nonlinear doubly reflected BSDE in 9. Existence of robust superhedging strategies has been studied. The buyer's point of view and arbitrage issues have also been studied in this context.

In a Markovian framework, the results of the paper 8 on combined optimal stopping/stochastic control with ${ℰ}^g$ expectation allows us to address American nonlinear option pricing when the payoff function is only Borelian and when there is ambiguity both on the drift and the volatility of the underlying asset price process. Robust optimal stopping of dynamic risk measures induced by BSDEs with jumps with model ambiguity is studied in 89.

3.2.1 Optimal transport and longtime behavior of Markov processes

The dissipation of general convex entropies for continuous time Markov processes can be described in terms of backward martingales with respect to the tail filtration. The relative entropy is the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, J. Fontbona and B. Jourdain 75 used Girsanov theory to explicit the Doob-Meyer decomposition of this submartingale. They deduced a stochastic analogue of the well known entropy dissipation formula, which is valid for general convex entropies, including the total variation distance. Under additional regularity assumptions, and using Itô's calculus and ideas of Arnold, Carlen and Ju 51, they obtained a new Bakry-Emery criterion which ensures exponential convergence of the entropy to 0. This criterion is non-intrinsic since it depends on the square root of the diffusion matrix, and cannot be written only in terms of the diffusion matrix itself. They provided examples where the classic Bakry Emery criterion fails, but their non-intrinsic criterion applies without modifying the law of the diffusion process.

With J. Corbetta, A. Alfonsi and B. Jourdain have studied the time derivative of the Wasserstein distance between the marginals of two Markov processes 44. The Kantorovich duality leads to a natural candidate for this derivative. Up to the sign, it is the sum of the integrals with respect to each of the two marginals of the corresponding generator applied to the corresponding Kantorovich potential. For pure jump processes with bounded intensity of jumps, J. Corbetta, A. Alfonsi and B. Jourdain 43 proved that the evolution of the Wasserstein distance is actually given by this candidate. In dimension one, they showed that this remains true for Piecewise Deterministic Markov Processes. They applied the formula to estimate the exponential decrease rate of the Wasserstein distance between the marginals of two birth and death processes with the same generator in terms of the Wasserstein curvature.

3.2.2 Mean-field systems: modeling and control

- Mean-field limits of systems of interacting particles. In 82, B. Jourdain and his former PhD student J. Reygner have studied a mean-field version of rank-based models of equity markets such as the Atlas model introduced by Fernholz in the framework of Stochastic Portfolio Theory. They obtained an asymptotic description of the market when the number of companies grows to infinity. Then, they discussed the long-term capital distribution, recovering the Pareto-like shape of capital distribution curves usually derived from empirical studies, and providing a new description of the phase transition phenomenon observed by Chatterjee and Pal. They have also studied multitype sticky particle systems which can be obtained as vanishing noise limits of multitype rank-based diffusions (see 84). Under a uniform strict hyperbolicity assumption on the characteristic fields, they constructed a multitype version of the sticky particle dynamics. In 83, they obtain the optimal rate of convergence as the number of particles grows to infinity of the approximate solutions to the diagonal hyperbolic system based on multitype sticky particles and on easy to compute time discretizations of these dynamics.

In 76, N. Fournier and B. Jourdain are interested in the two-dimensional Keller-Segel partial differential equation. This equation is a model for chemotaxis (and for Newtonian gravitational interaction).

- Mean field control and Stochastic Differential Games (SDGs). To handle situations where controls are chosen by several agents who interact in various ways, one may use the theory of Stochastic Differential Games (SDGs). Forward–Backward SDG and stochastic control under Model Uncertainty are studied in 91 by A. Sulem and B. Øksendal. Also of interest are large population games, where each player interacts with the average effect of the others and individually has negligible effect on the overall population. Such an interaction pattern may be modeled by mean field coupling and this leads to the study of mean-field stochastic control and related SDGs. A. Sulem, Y. Hu and B. Øksendal have studied singular mean field control problems and singular mean field two-players stochastic differential games 80. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions, the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem. Applications to optimal irreversible investments under uncertainty have been investigated. Predictive mean-field equations as a model for prices influenced by beliefs about the future are studied in 92.

3.2.3 Stochastic control and optimal stopping (games) under nonlinear expectation

M.C. Quenez and A. Sulem have studied optimal stopping with nonlinear expectation ${ℰ}^g$ induced by a BSDE with jumps with nonlinear driver $g$ and irregular obstacle/payoff (see 89). In particular, they characterize the value function as the solution of a reflected BSDE. This property is used in 69 to address American option pricing in markets with imperfections. The Markovian case is treated in 71 when the payoff function is continuous.

In 8, M.C. Quenez, A. Sulem and R. Dumitrescu study a combined optimal control/stopping problem under nonlinear expectation ${ℰ}^g$ in a Markovian framework when the terminal reward function is only Borelian. In this case, the value function $u$ associated with this problem is irregular in general. They establish a weak dynamic programming principle (DPP), from which they derive that the upper and lower semi-continuous envelopes of $u$ are the sub- and super- viscosity solution of an associated nonlinear Hamilton-Jacobi-Bellman variational inequality.

The problem of a generalized Dynkin game problem with nonlinear expectation ${ℰ}^g$ is addressed in 70. Under Mokobodzki's condition, we establish the existence of a value function for this game, and characterize this value as the solution of a doubly reflected BSDE. The results of this work are used in 9 to solve the problem of game option pricing in markets with imperfections.

A generalized mixed game problem when the players have two actions: continuous control and stopping is studied in a Markovian framework in 68. In this work, dynamic programming principles (DPP) are established: a strong DPP is proved in the case of a regular obstacle and a weak one in the irregular case. Using these DPPs, links with parabolic partial integro-differential Hamilton-Jacobi- Bellman variational inequalities with two obstacles are obtained.

With B. Øksendal and C. Fontana, A. Sulem has contributed on the issues of robust utility maximization 90, 92, and relations between information and performance 74.

3.2.4 Generalized Malliavin calculus

Vlad Bally has extended the stochastic differential calculus built by P. Malliavin which allows one to obtain integration by parts and associated regularity probability laws. In collaboration with L. Caramellino (Tor Vegata University, Roma), V. Bally has developed an abstract version of Malliavin calculus based on a splitting method (see 53). It concerns random variables with law locally lower bounded by the Lebesgue measure (the so-called Doeblin's condition). Such random variables may be represented as a sum of a "smooth" random variable plus a rest. Based on this smooth part, he achieves a stochastic calculus which is inspired from Malliavin calculus 6. An interesting application of such a calculus is to prove convergence for irregular test functions (total variation distance and more generally, distribution distance) in some more or less classical frameworks as the Central Limit Theorem, local versions of the CLT and moreover, general stochastic polynomials 55. An exciting application concerns the number of roots of trigonometric polynomials with random coefficients 56. Using Kac Rice lemma in this framework one comes back to a multidimensional CLT and employs Edgeworth expansions of order three for irregular test functions in order to study the mean and the variance of the number of roots. Another application concerns U statistics associated to polynomial functions. The techniques of generalized Malliavin calculus developed in 53 are applied in for the approximation of Markov processes (see 60 and 59). On the other hand, using the classical Malliavin calculus, V. Bally in collaboration with L. Caramellino and P. Pigato studied some subtle phenomena related to diffusion processes, as short time behavior and estimates of tubes probabilities (see 54, 57).

3.3.1 Simulation of stochastic differential equations

3.3.2 Estimation of the parameters of a Wishart process

A. Alfonsi, A. Kebaier and C. Rey  46 have computed the Maximum Likelihood Estimator for the Wishart process and studied its convergence in the ergodic and in some non ergodic cases. In the ergodic case, which is the most relevant for applications, they obtain the standard square-root convergence. In the non ergodic case, the analysis rely on refined results for the Laplace transform of Wishart processes, which are of independent interest.

3.3.3 Optimal stopping and American options

In joint work with A. Bouselmi, D. Lamberton studied the asymptotic behavior of the exercise boundary near maturity for American put options in exponential Lévy models. In 7, they deal with jump-diffusion models, and establish that, in some cases, the behavior differs from the classical Black and Scholes setting. D. Lamberton has also worked on the binomial approximation of the American put. The conjectured rate of convergence is $O(1/n)$ where $n$ is the number of time periods. He was able to derive a $O({(lnn)}^{\alpha }/n)$ bound, where the exponent $\alpha$ is related to the asymptotic behavior of the exercise boundary near maturity.

5.1.1 PREMIA

• Keywords: Financial products, Computational finance, Option pricing
• Scientific Description: Premia is a numerical platform for computational finance. It is designed for option pricing, hedging and financial model calibration. Premia is developed by the MathRisk project team in collaboration with a consortium of financial institutions. The Premia project keeps track of the most recent advances in the field of computational finance in a well-documented way. It focuses on the implementation of numerical analysis techniques for both probabilistic and deterministic numerical methods. An important feature of the platform Premia is the detailed documentation which provides extended references in option pricing. Premia contains various numerical algorithms: deterministic methods (Finite difference and finite element algorithms for partial differential equations, wavelets, Galerkin, sparse grids ...), stochastic algorithms (Monte-Carlo simulations, quantization methods, Malliavin calculus based methods), tree methods, approximation methods (Laplace transforms, Fast Fourier transforms...) These algorithms are implemented for the evaluation of vanilla and exotic options on equities, interest rate, credit, energy and insurance products. Moreover Premia provides a calibration toolbox for Libor Market model and a toolbox for pricing Credit derivatives. The latest developments of the software address evaluation of financial derivative products, risk management and computations of risk measures required by new financial regulation. They include the implementation of advanced numerical algorithms taking into account model dependence, counterparty credit risk, hybrid features, rough volatility and various nonlinear effects. High-dimensional problems are addressed by deep learning techniques using neural network approximations.
• Functional Description: Premia is a software designed for option pricing, hedging and financial model calibration, developed by the MathRisk project team in collaboration with a consortium of financial institutions presently composed of Crédit Agricole CIB and NATIXIS. The Premia project keeps track of the most recent advances in computational finance in a well documented way. It focuses on the implementation of numerical techniques, be they probabilistic or deterministic, to solve financial problems. Premia is thus a powerful tool to assist Research and Development professional teams in their day-to-day duty. It is also a useful support for academics who wish to perform tests on new algorithms or pricing methods. Besides being a single entry point for accessible overviews and basic implementations of various numerical methods, the aim of the Premia project is: - to elaborate a powerful testing platform for comparing different numerical methods between each other, - to build a link between professional financial teams and academic researchers, - to provide a useful teaching support for Master and PhD students in mathematical finance. An important feature of the platform Premia is its detailed documentation which provides extended references in computational finance. The project Premia has started in 1999 and is now considered as a standard reference platform for quantitative finance among the academic mathematical finance community.
• Release Contributions: The latest developments of the software address evaluation of financial derivative products, risk management and computations of risk measures required by new financial regulation. They include the implementation of advanced numerical algorithms taking into account model dependence, counterparty credit risk, hybrid features, rough volatility and various nonlinear effects. High-dimensional problems are addressed by deep learning techniques using neural network approximations. We also develop our activity on insurance contracts, in particular on the computation of risk measures (Value at Risk, Condition Tail Expectation) of variable annuities contracts like GMWB (guaranteed minimum withdrawal benefit) including taxation and customers mortality modeling.
• News of the Year: Release 22 of the Premia software has been delivered to the Consortium in March 2020. It contains the following new implemented algorithms: (i) Machine Learning, Risk Management: - Solving high-dimensional partial differential equations using,Deep Learning, J. Han, A. Jentzen, E.Weinan, Proceedings of the National Academy of Sciences, 115(34),2018. - Longstaff-Schwartz algorithm with Neural Network. L.Goudenege, T.Sainrat - Neural network regression for Bermudan option pricing. B.Lapeyre J.Lelong - Conditional Monte Carlo Learning for Diffusions. L.A. Abbas-Turki, B.Diallo, G.Pagès. - Machine Learning for Pricing American Options in High-Dimensional Markovian and non-Markovian models. L.Goudenege, A.Molent A.Zanette, Quantitative Finance 2020. - Variance Reduction applied to Machine Learning for Pricing Bermudan/American Options in High Dimension, L. Goudenege, A.Molent A.Zanette. - PDE models and numerical methods for total value adjustment in,European and American options with counterparty risk. I. Arregui, B. Salvador, C. Vazquez, Applied Mathematics and Computation. 308, 2017. - Efficient risk estimation via nested sequential simulation. M.Broadie C.Moallemi, Management Science 2011 (ii) Equity Derivatives - Pricing American Options by Exercise Rate Optimization. C.Bayer, R.Tempone, S. Wolfer. Quantitative Finance, 20(11), 2020 -Robust pricing of European Options with Wavelets and the Characteristic Function. L. O. Gracia, C.W.Oosterlee, SIAM J. Sci. Comput Vol. 35, No. 5, 2013 - American option pricing under the double Heston model based on asymptotic expansion. S. M. Zhang, Y. Feng, Quantitative Finance 2018 - American and exotic option pricing with jump diffusions and other Lévy processes. J. L. Kirkby, The Journal of Computational Finance, 22(3), 2018. - An Efficient Transform Method For Asian Option Pricing. J. L. Kirkby,SIAM J. Financial Math., 2016 - Approximate Wiener-Hopf factorization and the Monte Carlo methods for Lévy processes. O. Kudryavtsev,Theory of Probability and its Applications 64–2, 2019
• URL: http://www.premia.fr
• Publications: hal-03013603, hal-03013606, hal-01873346
• Authors: Antonino Zanette, Benjamin Jourdain, Jérôme Lelong, Agnes Sulem, Aurélien Alfonsi, Peter Tankov, Pierre Cohort, Emmanuel Temam, Ismail Laachir, Jacques Printems, David Pommier
• Contact: Agnes Sulem
• Participants: Agnes Sulem, Antonino Zanette, Aurélien Alfonsi, Benjamin Jourdain, Jérôme Lelong, Bernard Lapeyre
• Partners: Inria, Ecole des Ponts ParisTech, Université Paris-Est

6.3.1 Option pricing in a non-linear incomplete market model with default

Agnès Sulem has studied with Miryana Grigorova (University of Leeds) and Marie-Claire Quenez (Université Paris Denis Diderot) superhedging prices and the associated superhedging strategies for both European and American options (see 20 and 78 in a non-linear incomplete market model with default. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio processes follow non-linear dynamics with a non-linear driver $f$. . By using a dynamic programming approach, we provide a dual formulation of the seller's (superhedging) price for the European option involving a suitable set of equivalent probability measures, which we call $f$-martingale probability measures. We also establish a characterization of the seller's price as the initial value of the minimal supersolution of a constrained BSDE with default. Our results rely on first establishing a non-linear optional and a non-linear predictable decomposition for processes which are ${ℰ}^f$-strong supermartingales under $Q$, for all $Q\in 𝒬$. We then studied American options with irregular payoff in this market. Both points of view of the seller and of thave he buyer are analyzed. We give a dual representation of the seller's (superhedging) price in terms of the value of a non-linear mixed control/stopping problem, and provide two infinitesimal characterizations of the seller's price process in terms of the minimal supersolution of a constrained reflected BSDE and of an optional reflected BSDE. Under some regularity assumptions on the payoff, we also prove a duality result for the buyer's price in terms of the value of a non-linear control/stopping game problem.

6.3.2 Insurance products

6.3.3 Deep learning for large dimensional financial problems

Neural network regression for Bermudan option pricing. The pricing of Bermudan options amounts to solving a dynamic programming principle, in which the main difficulty, especially in high dimension, comes from the conditional expectation involved in the computation of the continuation value. These conditional expectations are classically computed by regression techniques on a finite dimensional vector space. In 38, Bernard Lapeyre and Jérôme Lelong study neural networks approximations of conditional expectations. They prove the convergence of the well-known Longstaff and Schwartz algorithm when the standard least-square regression is replaced by a neural network approximation. They illustrate the numerical efficiency of neural networks as an alternative to standard regression methods for approximating conditional expectations on several numerical examples.

Machine learning for pricing American options. In 19, L. Goudenège, A. Molent and A. Zanette develop techniques, called GPR Tree and GPR Exact Integration, both based on Machine Learning, to compute prices of American basket options in high-dimension. Both Markovian and non Markovian models are studied, in particular rough Bergomi model, which provides stochastic volatility with memory.

Big data techniques for portfolio optimization. With his PhD student Hachem Madmoun, Bernard Lapeyre analyses the trajectories of asset prices by Fourier transform, and big data techniques such as vae (variational autoencoder) and hmm (hidden Markov chains) for portfolio management.

6.4.1 Optimal transport and mean-field SDEs

In 36, Benjamin Jourdain and Alvin Tse propose a generalised version of the central limit theorem for nonlinear functionals of the empirical measure of i.i.d. random variables, provided that the functional satisfies some regularity assumptions for the associated linear functional derivatives of various orders. This generalisation can be applied to Monte-Carlo methods, even when there is a nonlinear dependence on the measure component. As a consequence of this result, they also analyse the convergence of fluctuation between the empirical measure of particles in an interacting particle system and their mean-field limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear. In 15, A. Alfonsi and B. Jourdain study the structure of optimal couplings for the squared quadratic Wasserstein distance of probability measures with finite second order moments.

In 28, Oumaima Bencheikh and Benjamin Jourdain study the approximation in Wasserstein distance with index $\rho \ge 1$ of a probability measure $\mu$ on the real line with finite moment of order $\rho$ by the empirical measure of $N$ deterministic points. The minimal error converges to 0 as $N\to +\infty$. Apart when $\mu$ is a Dirac mass and the error vanishes, the order of convergence is not larger than 1. They give a necessary condition and a sufficient condition for the order to be equal to this threshold 1 in terms of the density of the absolutely continuous with respect to the Lebesgue measure part of $\mu$. They also check that for the order to lie in the interval $\left(1/\rho ,1\right)$, the support of $\mu$ has to be a bounded interval, and that, when $\mu$ is compactly supported, the order is not smaller than $1/\rho$. Last, they give a necessary and sufficient condition in terms of the tails of $\mu$ for the order to be equal to some given value in the interval $\left(0,1/\rho \right)$.

With V. Ehrlacher and R. Coyaud, Aurelien Alfonsi is working on numerical methods based to approximate the optimal transport in the symmetric case in the multimarginal case. 14

6.4.2 Martingale optimal transport

It is known since 21 that two one-dimensional probability measures in the convex order admit a martingale coupling with respect to which the integral of $|x-y|$ is smaller than twice their Wasserstein distance ${𝒲}_1$ with index 1. Moreover, replacing $|x-y|$ and ${𝒲}_1$ respectively with ${|x-y|}^{\rho }$ and ${𝒲}_{\rho }^{\rho }$ does not lead to a finite multiplicative constant. In 32, B. Jourdain and W. Margheriti show that a finite constant is recovered when replacing ${𝒲}_{\rho }^{\rho }$ with the product of ${𝒲}_{\rho }$ times the centred $\rho$-th moment of the second marginal to the power $\rho -1$. Then they study the generalisation of this new stability inequality to higher dimension. In 27, with M. Beiglböck and G. Pammer they establish stability of martingale couplings in dimension one. If $\pi$ is a martingale coupling with marginals $\mu ,\nu$. Then, given approximating marginal measures $\tilde{\mu }\approx \mu$, $\tilde{\nu }\approx \nu$ in convex order, they show that there exists an approximating martingale coupling $\tilde{\pi }\approx \pi$ with marginals $\tilde{\mu },\tilde{\nu }$. In mathematical finance, prices of European call / put option yield information on the marginal measures of the arbitrage free pricing measures. The above result asserts that small variations of call / put prices lead only to small variations on the level of arbitrage free pricing measures. While these facts have been anticipated for some time they do not generalize to higher dimensions and the actual proof requires somewhat intricate stability results for the adapted Wasserstein distance. Notably the result has consequences for a several related problems. Specifically, it is relevant for numerical approximations, it leads to a new proof of the monotonicity principle of martingale optimal transport and it implies stability of weak martingale optimal transport as well as optimal Skorokhod embedding. On the mathematical finance side this yields continuity of the robust pricing problem for exotic options and VIX options with respect to market data. These applications will be detailed in two companion papers.

In 33, B. Jourdain and W. Margheriti are interested in martingale rearrangement couplings. As introduced by Wiesel, in order to prove the stability of Martingale Optimal Transport problems, these are projections in adapted Wasserstein distance of couplings between two probability measures on the real line in the convex order onto the set of martingale couplings between these two marginals. In reason of the lack of relative compactness of the set of couplings with given marginals for the adapted Wasserstein topology, the existence of such a projection is not clear at all. Under a barycentre dispersion assumption on the original coupling which is in particular satisfied by the Hoeffding-Fréchet or comonotone coupling, Wiesel gives a clear algorithmic construction of a martingale rearrangement when the marginals are finitely supported and then gets rid of the finite support assumption by relying on a rather messy limiting procedure to overcome the lack of relative compactness. B. Jourdain and W. Margheriti give a direct general construction of a martingale rearrangement coupling under the barycentre dispersion assumption. This martingale rearrangement is obtained from the original coupling by an approach similar to the construction given in 21 of the inverse transform martingale coupling, a member of a family of martingale couplings close to the Hoeffding-Fréchet coupling, but for a slightly different injection in the set of extended couplings introduced by Beiglböck and Juillet and which involve the uniform distribution on $[0,1]$ in addition to the two marginals. They last discuss the stability in adapted Wassertein distance of the inverse transform martingale coupling with respect to the marginal distributions.

6.4.3 Quantization methods

In 34, Benjamin Jourdain and Gilles Pagès establish for dual quantization the counterpart of Kieffer's uniqueness result for compactly supported one dimensional probability distributions having a log-concave density (also called strongly unimodal): for such distributions, Lr-optimal dual quantizers are unique at each level N, the optimal grid being the unique critical point of the quantization error. An example of non-strongly unimodal distribution for which uniqueness of critical points fails is exhibited. In the quadratic r=2 case, they propose an algorithm which computes the unique optimal dual quantizer with geometric rate of convergence in the log-concave case. It provides a counterpart of Lloyd's method I algorithm in a Voronoi framework. Finally semi-closed forms of Lr-optimal dual quantizers are established for power distributions on compacts intervals and truncated exponential distributions.

Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly supported, it is smaller than any of its dual quantizations while the dominated original measure is greater than any of its stationary (and therefore any of its optimal) quadratic primal quantization. Moreover, the quantization errors then correspond to martingale couplings between each original probability measure and its quantization. This enables B. Jourdain and G. Pagès to prove in 35 that any martingale coupling between the original probability measures can be approximated by a martingale coupling between their quantizations in Wassertein distance with a rate given by the quantization errors but also in the much finer adapted Wassertein distance. As a consequence, while the stability of (Weak) Martingale Optimal Transport problems with respect to the marginal distributions has only been established in dimension 1 so far, their value function computed numerically for the quantized marginals converges in any dimension to the value for the original probability measures as the numbers of quantization points go to $\infty$.

6.4.4 Approximation of stochastic differential equations

6.4.5 Abstract Malliavin calculus and convergence in total variation

In collaboration with L. Caramellino (University Tor Vergata) and with G. Poly (University of Rennes), V. Bally has settled a Malliavin type calculus for a general class of random variables, which are not supposed to be Gaussian (as it is the case in the standard Malliavin calculus). This is an alternative to the $\Gamma$ calculus settled by Bakry, Gentile and Ledoux. The main application is the estimate in total variation distance of the error in general convergence theorems. This is done in 16.

6.4.6 Regularity of the low of the solution of jump type equations

In collaboration with L. Caramellino and A. Kohatsu Higa, V. Bally study the regularity of the solutions of jump type equations. A first result is obtained in 26.

6.4.7 Optimal stopping and free boundary problems

Damien Lamberton has revisited the results of Bensoussan and Lions on variational inequalities, using some semi-group theory. He has contributed to a winter school on "Theory and practice of optimal stopping and free boundary problems" (cf. https://conferences.leeds.ac.uk/osfbp/).

8.1.1 Inria associate team not involved in an IIL

8.1.2 Visits of international scientists

• Beatriz Salvador Mancho, CWI, Netherlands, (8-13 March 2020)

8.2.1 Competitivity Clusters

Pôle Finance Innovation

9.1.1 Scientific events: organisation

• A. Alfonsi:

  Co-organizer of the working group seminar of MathRisk “Méthodes stochastiques et finance”. http://cermics.enpc.fr/~alfonsi/GTMSF.html

• V. Bally

  Organizer of the seminar of the LAMA laboratory, Université Gustave Eiffel.

• A. Sulem

  Co-organizer of the seminar INRIA-MathRisk /Université Paris Diderot LPSM “Numerical probability and mathematical finance”. https://www.lpsm.paris/mathfipronum/gt

9.1.2 Journal

9.1.3 Invited talks

• A. Alfonsi
  • 14th of January, 2020: "A generic construction for high order approximation schemes of semigroups using random grids", Advances in Financial Mathematics 2020, Paris.
  • 3rd of March, 2020: "Lifted and geometric differentiability of the squared quadratic Wasserstein distance", séminaire de probabilités du LPSM.
• B. Jourdain:
  • Oxford Stochastic Analysis and Mathematical Finance seminar, 20 January 2020 : The inverse transform martingale coupling.
• D. Lamberton
  • Variational inequalities in the Heston model, workshop on Stochastic Analysis, Control and Mathematical Finance, Leeds, January 2020.
• A. Sulem
  • Shandong University, Qingdao new campus, on September 7th-9th, 2020, dedicated to the 80th Anniversary of Professor Alain Bensoussan (postponed because of Covid)
  • Main Speaker, EPCO 2020 – “Portuguese Meeting on Optimal Control”, Faculty of Sciences and Technology, Universidade Nova de Lisboa, June, 2020 (postponed because of Covid)

9.1.4 Research administration

• A. Alfonsi
  • Director of CERMICS laboratory until August,
  • Deputy director of CERMICS since September.
  • In charge of the Master “Finance and Data” at the Ecole des Ponts.
• V. Bally
  • Member of the LAMA committee
  • Responsible of the Master 2, option finance.
• A. Sulem
  • Member of the Scientific Committee of AMIES
  • Corresponding member of the Operational Committee for the assesment of Legal and Ethical risks (COERLE) at INRIA Paris research center
  • Member of the Committee for INRIA international Chairs

9.2.1 Teaching

• Licence :
  • A. Alfonsi: "Probability theory”, first year course at Ecole des Ponts.
  • B. Jourdain : course "Mathematical tools for engineers", 1st year ENPC.

• Master :

  - Aurélien Alfonsi

  • “Données Haute Fréquence en finance”, lecture for the Master at UPEMLV.
  • “Mesures de risque”, Master course of UPEMLV and Sorbonne Université.
  • Professeur chargé de cours at Ecole Polytechnique.

  - Vlad Bally

  • "Taux d'Intêret", M2 Finance. Université Gustave Eiffel
  • "Calcul de Malliavin et applications en finance", M2 Finance, UGE
  • "Analyse du risque" M2 Actuariat, UGE
  • "Processus Stochastiques" M2 Recherche, UGE

  - Benjamin Jourdain

  • course "Mathematical finance", 2nd year ENPC
  • course "Monte-Carlo Markov chain methods and particle algorithms", Research Master Probabilités et Modèles Aléatoires, Sorbonne Université

  - B. Jourdain, B. Lapeyre

  • course "Monte-Carlo methods", 3rd year ENPC and Research Master Mathématiques et Application, University Gustave Eiffel

  - J.-F. Delmas, B.Jourdain

  • course "Jump processes with applications to energy markets", 3rd year ENPC and Research Master Mathématiques et Application, University Gustave Eiffel

  - D. Lamberton

  • "Calcul stochastique pour la finance", master 1 course, Université Gustave Eiffel
  • "Arbitrage, volatilité et gestion de portefeuille", master 2 course, Université Gustave Eiffel
  • "Méthodes mathématiques pour les probabilités", L3 course, Université Gustave Eiffel.
  • Variational inequalities for optimal stopping, seven hour course in Winter School on Theory and Practice of Optimal Stopping and Free Boundary Problems, Leeds, January 2020.

  - B. Lapeyre

  • Monte-Carlo methods in quantitative finance, Master of Mathematics, University of Luxembourg,

  - A. Sulem

  • "PDE methods in Finance", Master of Mathematics, University of Luxembourg, 22 h lectures and responsible of the module "Numerical Methods in Finance".

9.2.2 Supervision

• PhD defended :
  • Oumaima Bencheikh, "Analysis of the weak error of time and particles discretization of SDEs nonlinear in the sense of McKean", supervised by B.Jourdain, defended on October 22 2020, Université Paris Est, Cermics
  • William Margheriti "On the stability of the martingale optimal transport problem", supervised by B. Jourdain, defended on December 17 2020, Université Paris Est, Cermics
• PhD in progress :
  • Anas Bentaleb (started February 2018) : Mathematical techniques for expected exposure evaluation, Supervisor: B. Lapeyre.
  • Adel Cherchali, “Numerical methods for the ALM”, funded by Fondation AXA, started in September 2017, Supervisor: Aurélien Alfonsi
  • Zhonguyan Cao, "Dynamics and Stability of Complex Financial networks", Université Paris-Dauphine, Supervisor: Agnès Sulem, Dim Mathinnov doctoral allocation, started October 2020
  • Rafaël Coyaud, “Deterministic ans stochastic numerical methods for multimarginal and martingale constraint optimal transport problems”, started in October 2017, Supervisor: Aurélien Alfonsi
  • Ezechiel Kahn (started September 2018) "Functional inequalities for random matrices models", supervisors: B. Jourdain and D. Chafai
  • Hachem Madmoun (started September 2018) "Gestion de portefeuilles utilisant des techniques de (big) data", supervisor: B. Lapeyre.
  • Sophian Mehalla (started November 2017), CIFRE agreement Milliman company/Ecole des Ponts (http://fr.milliman.com, Supervisor: B. Lapeyre
  • Edoardo Lombardo (International PhD, co-advisor: Lucia Caramellino), “High order numerical approximation for some singular stochastic processes and related PDEs”, started in November 2020.
  • Nerea Vadillo Fernandez (CIFRE AXA Climate), “Risk valuation for weather derivatives in index-based insurance”, started in November 2020.
  • Yfen Qin, "Regularity properties for non linear problems", Dim-Mathinnov doctoral allocation, supervisor: V. Bally
• Internships
  • Claire Zeng (April to July 2020): Pricing in the rough Heston model. (supervisor : A. Alfonsi)
  • Zhonguyan Cao : (April to July 2020): Study of contagion models in random networks, (supervisor : A. Sulem)

9.2.3 Juries

• Aurélien Alfonsi
  • Member of the Committee for the recruitment of a Professor in applied mathematics, finance and numerical probability, Laboratoire de probabilités (LPSM), Université Paris-Diderot, Spring 2020.
  • Referee of the PhD thesis of Emilio Saïd on “Market Impact in Systematic Trading and Option Pricing”, Université Paris-Saclay, 23 June 2020.
• Benjamin Jourdain
  • PhD thesis of Thibaut Montes, defended on June 24 2020, Sorbonne University
• Agnès Sulem
  • PhD thesis of Johann Nicolle, Some contributions of Bayesian and computational learning methods to portfolio selection problems, LPSM, Université Paris-Diderot, 5 octobre 2020 (Chair of the Committee)
  • Member of the Committee for the recruitment of a Professor in applied mathematics, finance and numerical probability, Laboratoire de probabilités (LPSM), Université Paris-Diderot, Spring 2020.

International journals

• 12 article AurélienA. Alfonsi, AdelA. Cherchali and Jose Arturo InfanteJ. Acevedo. A full and synthetic model for Asset-Liability Management in life insurance, and analysis of the SCR with the standard formula European Actuarial Journal 2020
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• 13 articleAurélienA. Alfonsi, JacopoJ. Corbetta and BenjaminB. Jourdain. Sampling of probability measures in the convex order by Wasserstein projectionAnnales de l'IHP - Probabilités et Statistiques5632020, 1706-1729
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• 14 article AurélienA. Alfonsi, RafaëlR. Coyaud, VirginieV. Ehrlacher and DamianoD. Lombardi. Approximation of Optimal Transport problems with marginal moments constraints Mathematics of Computation 2020
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• 15 articleAurélienA. Alfonsi and BenjaminB. Jourdain. Squared quadratic Wasserstein distance: optimal couplings and Lions differentiabilityESAIM: Probability and Statistics242020, 703-717
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• 16 articleVladV. Bally, LuciaL. Caramellino and GuillaumeG. Poly. Regularization lemmas and convergence in total variationElectronic Journal of Probability2502020, paper no. 74, 20 pp
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• 17 article LudovicL. Goudenège, AndreaA. Molent and AntoninoA. Zanette. Computing Credit Valuation Adjustment solving coupled PIDEs in the Bates model Computational Management Science 17 2 April 2020
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• 18 article LudovicL. Goudenège, AndreaA. Molent and AntoninoA. Zanette. Gaussian process regression for pricing variable annuities with stochastic volatility and interest rate Decisions in Economics and Finance May 2020
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• 19 articleLudovicL. Goudenège, AndreaA. Molent and AntoninoA. Zanette. Machine learning for pricing American options in high-dimensional Markovian and non-Markovian modelsQuantitative Finance204April 2020, 573-591
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• 20 articleMiryanaM. Grigorova, Marie-ClaireM.-C. Quenez and AgnèsA. Sulem. European options in a non-linear incomplete market model with defaultSIAM Journal on Financial Mathematics113September 2020, 849–880
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• 21 articleBenjaminB. Jourdain and WilliamW. Margheriti. A new family of one dimensional martingale couplingsElectronic Journal of Probability251362020, 1-50
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• 22 articleBenjaminB. Jourdain and AlexandreA. Zhou. Existence of a calibrated Regime Switching Local Volatility modelMathematical Finance302April 2020, 501-546
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Doctoral dissertations and habilitation theses

• 23 thesis OumaimaO. Bencheikh. Weak error analysis for time and particle discretizations of some stochastic differential equations non linear in the sense of McKean Université Paris-Est Marne la vallée October 2020
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Reports & preprints

• 24 misc AurélienA. Alfonsi, AdelA. Cherchali and José ArturoJ. Infante Acevedo. Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests November 2020
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• 25 misc AurélienA. Alfonsi, RafaëlR. Coyaud and VirginieV. Ehrlacher. Constrained overdamped Langevin dynamics for symmetric multimarginal optimal transportation February 2021
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• 26 misc VladV. Bally and LuciaL. Caramellino. Transfer of regularity for Markov semigroups January 2020
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• 27 misc MathiasM. Beiglböck, BenjaminB. Jourdain, WilliamW. Margheriti and GudmundG. Pammer. Approximation of martingale couplings on the line in the weak adapted topology January 2021
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• 28 misc OumaimaO. Bencheikh and BenjaminB. Jourdain. Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures December 2020
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• 29 misc OumaimaO. Bencheikh and BenjaminB. Jourdain. Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise May 2020
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• 30 misc EricE. Cancès, RafaëlR. Coyaud and L. RidgwayL. Scott. Van der Waals interactions between two hydrogen atoms: The next orders July 2020
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• 31 misc BenjaminB. Jourdain and EzéchielE. Kahn. Strong solutions to a beta-Wishart particle system March 2020
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• 32 misc BenjaminB. Jourdain and WilliamW. Margheriti. Martingale Wasserstein inequality for probability measures in the convex order November 2020
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• 33 misc BenjaminB. Jourdain and WilliamW. Margheriti. One dimensional martingale rearrangement couplings February 2021
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• 34 misc BenjaminB. Jourdain and GillesG. Pagès. Optimal dual quantizers of 1D log-concave distributions: uniqueness and Lloyd like algorithm October 2020
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• 35 misc BenjaminB. Jourdain and GillesG. Pagès. Quantization and martingale couplings December 2020
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• 36 misc BenjaminB. Jourdain and AlvinA. Tse. Central limit theorem over non-linear functionals of empirical measures with applications to the mean-field fluctuation of interacting particle systems February 2020
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• 37 misc EzechielE. Kahn. About the eigenvalues of Wishart processes October 2020
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• 38 misc BernardB. Lapeyre and JérômeJ. Lelong. Neural network regression for Bermudan option pricing November 2020
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