2020
Activity report
Project-Team
MATHRISK
RNSR: 201221215M
Research center
In partnership with:
CNRS, Ecole des Ponts ParisTech, Université Paris-Est Marne-la-Vallée
Team name:
Mathematical Risk handling
In collaboration with:
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS)
Domain
Applied Mathematics, Computation and Simulation
Theme
Stochastic approaches
Creation of the Team: 2012 January 01, updated into Project-Team: 2013 January 01

# Keywords

• A6. Modeling, simulation and control
• A6.1. Methods in mathematical modeling
• A6.1.2. Stochastic Modeling
• A6.2.1. Numerical analysis of PDE and ODE
• A6.2.2. Numerical probability
• A6.2.3. Probabilistic methods
• A6.4.2. Stochastic control
• B3.1. Sustainable development
• B9.6.3. Economy, Finance
• B9.11. Risk management

# 1 Team members, visitors, external collaborators

## Research Scientists

• Agnès Bialobroda Sulem [Team leader, Inria, Senior Researcher, HDR]
• Aurélien Alfonsi [École Nationale des Ponts et Chaussées, Senior Researcher, HDR]
• Benjamin Jourdain [École Nationale des Ponts et Chaussées, HDR]
• Bernard Lapeyre [École Nationale des Ponts et Chaussées, Senior Researcher, HDR]

## Faculty Members

• Vlad Bally [Univ Paris-Est Marne La Vallée, Professor, HDR]
• Damien Lamberton [Univ Paris-Est Marne La Vallée, Professor, HDR]

## PhD Students

• Nerea Vadillo Fernandez [AXA Climate, CIFRE, From November 2020, ENPC]
• Edoardo Lombardo [ENPC, From November 2020, International PhD student]
• Oumaima Bencheikh [École Nationale des Ponts et Chaussées, until Oct 2020]
• Zhongyuan Cao [Inria, from Oct 2020]
• Adel Cherchali [École Nationale des Ponts et Chaussées, until November 2020]
• Rafael Coyaud [École Polytechnique, until December 2020]
• Ezechiel Kahn [École Nationale des Ponts et Chaussées]
• Hachem Madmoun [Bramham Gardens, CIFRE, ENPC]
• William Margheriti [École Nationale des Ponts et Chaussées]
• Yfen Qin [Université Gustave Eiffel, From October 2020]

## Technical Staff

• Pierre-Guillaume Raverdy [INRIA, Engineer]

## Interns and Apprentices

• Zhongyuan Cao [Inria, from Apr 2020 until Jul 2020]
• Melek Chaabane [Inria, from May 2020 until Aug 2020]
• Claire Zeng [Inria, from Mar 2020 until Jul 2020]

• Derya Gök [Inria]

## Visiting Scientist

• Antonino Zanette [Université d'Udine - Italie, until Mar 2020]

## External Collaborators

• Ahmed Kebaier [Univ Paris-Nord]
• Céline Labart [Univ Savoie Mont-Blanc]
• Jérôme Lelong [ENSIMAG, HDR]
• Antonino Zanette [Université d'Udine - Italie, from Apr 2020]

# 2 Overall objectives

The Inria project team MathRisk team was created in 2013. It is the follow-up of the MathFi project team founded in 2000. MathFi was focused on financial mathematics, in particular on computational methods for pricing and hedging increasingly complex financial products. The 2007 global financial crisis and its “aftermath crisis” has abruptly highlighted the critical importance of a better understanding and management of risk. The project MathRisk has been reoriented towards mathematical handling of risk, and addresses broad research topics embracing risk measurement and risk management, modeling and optimization in quantitative finance, but also in other related domains where risk control is paramount. The project team MathRisk aims both at producing mathematical tools and models in these domains, and developing collaborations with various institutions involved in risk control. Quantitative finance remains for the project an important source of mathematical problems and applications. Indeed, the pressure of new legislation leads to a massive reorientation of research priorities, and the interest of analysts shifted to risk control preoccupation.

The scientific issues related to quantitative finance we consider include systemic risk and contagion modeling, robust finance, market frictions, counterparty and liquidity risk, assets dependence modeling, market micro-structure modeling and price impact. In this context, models must take into account the multidimensional feature and various market imperfections. They are much more demanding mathematically and numerically, and require the development of risk measures taking into account incompleteness issues, model uncertainties, interplay between information and performance and various defaults.

Besides, financial institutions, submitted to more stringent regulatory legislations such as FRTB or XVA computation, are facing practical implementation challenges which still need to be solved. Research focused on numerical efficiency remains strongly needed in this context, renewing the interest for the numerical platform Premia (http://www.premia.fr) that Mathrisk is developing in collaboration with a consortium of financial institutions.

While these themes arise naturally in the world of quantitative finance, a number of these issues and mathematical tools are also relevant to the treatment of risk in other areas as economy, social insurance and sustainable development, of fundamental importance in today's society. In these contexts, the management of risk appears at different time scales, from high frequency data to long term life insurance management, raising challenging renewed modeling and numerical issues.

The MathRisk project is strongly involved in the development of new mathematical methods and numerical algorithms. Mathematical tools include stochastic modeling, stochastic analysis, in particular stochastic (partial) differential equations and various aspects of stochastic control and optimal stopping of these equations, nonlinear expectations, Malliavin calculus, stochastic optimization, dynamic game theory, random graphs, martingale optimal transport (especially in relation to numerical considerations), long time behavior of Markov processes (with applications to Monte-Carlo methods) and generally advanced numerical methods for effective solutions.

# 3 Research program

## 3.1 Risk management: modeling and optimization

### 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 Perspectives in Stochastic Analysis

### 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 Numerical Probability

Our project team is very much involved in numerical probability, aiming at pushing numerical methods towards the effective implementation. This numerical orientation is supported by a mathematical expertise which permits a rigorous analysis of the algorithms and provides theoretical support for the study of rates of convergence and the introduction of new tools for the improvement of numerical methods. This activity in the MathRisk team is strongly related to the development of the Premia software.

### 3.3.1 Simulation of stochastic differential equations

#### - Weak convergence of the Euler scheme in optimal transport distances.

With A. Kohatsu-Higa, A. Alfonsi and B. Jourdain 4 have proved using optimal transport tools that the Wasserstein distance between the time marginals of an elliptic SDE and its Euler discretization with $N$ steps is not larger than $C\sqrt{log\left(N\right)}/N$. The logarithmic factor may is removed when the uniform time-grid is replaced by a grid still counting $N$ points but refined near the origin of times.

#### - Strong convergence properties of the Ninomiya Victoir scheme and multilevel Monte-Carlo estimators.

With their former PhD student, A. Al Gerbi, E. Clément and B. Jourdain 1 have proved strong convergence with order $1/2$ of the Ninomiya-Victoir scheme which is known to exhibit order 2 of weak convergence 88. This study was aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator : indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used in the coarse and fine grids at each level. In 40, they proved that the order of strong convergence of the crude Ninomiya Victoir scheme is improved to 1 when the vector fields corresponding to each Brownian coordinate in the SDE commute, and in 41, they studied the error introduced by discretizing the ordinary differential equations involved in the Ninomiya-Victoir scheme.

#### - Non-asymptotic error bounds for the multilevel Monte Carlo Euler method.

A. Kebaier and B. Jourdain are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, they dealt in 81 with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. They obtained Gaussian-type concentration. To do so, they used the Clark-Ocone representation formula and derived bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives. The estimation of such differences is much more complicated than the one of a single Euler scheme contribution and explains why they suppose the diffusion coefficient to be constant. This assumption ensures boundedness of the Malliavin derivatives of both the SDE and its Euler scheme.

#### - Computation of sensibilities of integrals with respect to the invariant measure.

In 52, R. Assaraf, B. Jourdain, T. Lelièvre and R. Roux considered the solution to a stochastic differential equation with constant diffusion coefficient and with a drift function which depends smoothly on some real parameter $\lambda$, and admitting a unique invariant measure for any value of $\lambda$ around $\lambda =0$. Their aim was to compute the derivative with respect to $\lambda$ of averages with respect to the invariant measure, at $\lambda =0$. They analyzed a numerical method which consists in simulating the process at $\lambda =0$ together with its derivative with respect to $\lambda$ on a long time horizon. They gave sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to $\lambda$ of the mean of an observable through Monte Carlo simulations.

#### - Approximation of doubly reflected Backward stochastic differential equations.

R. Dumitrescu and C. Labart have studied the discrete time approximation scheme for the solution of a doubly reflected Backward Stochastic Differential Equation with jumps, driven by a Brownian motion and an independent compensated Poisson process 67, 66.

#### - Parametrix methods.

V. Bally and A. Kohatsu-Higa have recently proposed an unbiased estimator based on the parametrix method to compute expectations of functions of a given SDE ( 58). This method is very general, and A. Alfonsi, A. Kohastu-Higa and M. Hayashi  45 have applied it to the case of one-dimensional reflected diffusions. In this case, the estimator can be obtained explicitly by using the scheme of Lépingle  85 and is quite simple to implement. It is compared to other simulation methods for reflected SDEs.

### 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 $O\left(1/n\right)$ where $n$ is the number of time periods. He was able to derive a $O\left({\left(lnn\right)}^{\alpha }/n\right)$ bound, where the exponent $\alpha$ is related to the asymptotic behavior of the exercise boundary near maturity.

# 4 Application domains

## 4.1 Financial Mathematics, Insurance

The domains of application are quantitative finance and insurance with emphasis on risk modeling and control. In particular, Mathrisk focuses on dependence modeling, systemic risk, market microstructure modeling and risk measures.

# 5 New software and platforms

## 5.1 New software

### 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:
• Publications:
• 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 New results

## 6.1 Control of systemic risk in a dynamic framework

The goal of the project is to develop a model that captures the dynamics of a complex financial network and to provide methods for the control of default contagion, both by a regulator and by the institutions themselves. This introduces a new class of problems that are very challenging mathematically, as it relies on using mean field games and random graphs theory. Agnès Sulem, Andreea Minca (Cornell University), Hamed Amini (J. Mack Robinson College of Business, Georgia State University) have studied a Dynamic Contagion Risk Model With Recovery Features. They introduce threshold growth in the classical threshold contagion model, in which nodes have downward jumps when there is a failure of a neighboring node. Choosing the configuration model as underlying graph, they prove fluid limits for the baseline model, as well as extensions to the directed case, state-dependent inter-arrival times and the case of growth driven by upward jumps. They then allow nodes to choose their connectivity by trading off link benefits and contagion risk. They define a rational equilibrium concept in which nodes choose their connectivity according to an expected failure probability of any given link, and then impose condition that the expected failure probability coincides with the actual failure probability under the optimal connectivity. Existence of an asymptotic equilibrium is shown as well as convergence of the sequence of equilibria on the finite networks. In particular, these results show that systems with higher overall growth may have higher failure probability in equilibrium 49. Zhangyong Cao (DIM Math-Innov doctoral allocation) has started a PhD thesis under the direction of A. Sulem on dynamics and stability of complex financial networks (after an internship in Spring 2020). The first objective is to develop a ruin theory in random networks, in particular on the configuration model as underlying graph. Some limit results for default cascades in sparse heterogeneous financial networks have already been obtained.

## 6.2 Mean-field Backward Stochastic Differential Equations and systemic risk measures

Agnès Sulem, Rui Chen, Andreea Minca, Roxana Dumitrescu have studied mean-field BSDEs with a generalized mean-field operator which can capture system influence with higher order interactions. Convergence of finite approximations to the mean-field BSDE have been obtained. In the finite system, the mean-field term can incorporate for example an inhomogeneous graph model in which the intensity of bilateral interactions depends on the states of the end nodes by means of a kernel function. This opens the path towards using dynamic risk measures induced by mean-field BSDE as a complementary approach to systemic risk measurement.

## 6.3 Risk management in finance and insurance

### 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

#### Solvency Capital Requirement in Insurance.

A. Alfonsi has obtained a grant from AXA Foundation on a Joint Research Initiative with a team of AXA France working on the strategic asset allocation. This team has to make recommendations on the investment over some assets classes as, for example, equity, real estate or bonds. In order to do that, each side of the balance sheet (assets and liabilities) is modeled in order to take into account their own dynamics but also their interactions. Given that the insurance products are long time contracts, the projections of the company's margins have to be done considering long maturities. When doing simulations to assess investment policies, it is necessary to take into account the SCR which is the amount of cash that has to be settled to manage the portfolio. Typically, the computation of the future values of the SCR involve expectations under conditional laws, which is greedy in computation time.

A. Alfonsi and his PhD student A. Cherchali have constructed a model of the ALM management of insurance companies that takes into account the regulatory constraints on life-insurance 12. They have developed Multilevel Monte-Carlo methods to approximate the SCR (Solvency Capital Requirement) at a future date and more generally, to calculate the worst of $P$ shocks 24.

#### Pricing and hedging variable annuities of GMWB type in advanced stochastic models.

Antonino Zanette with Ludovic Goudenège (Ecole Centrale de Paris) and Andrea Molent (University of Udine) study the valuation of GMWB variable annuity when both stochastic volatility and stochastic interest rate are considered in the Heston Hull-White model. 18.

### 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 Stochastic analysis and probabilistic numerical methods

### 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 $\stackrel{˜}{\mu }\approx \mu$, $\stackrel{˜}{\nu }\approx \nu$ in convex order, they show that there exists an approximating martingale coupling $\stackrel{˜}{\pi }\approx \pi$ with marginals $\stackrel{˜}{\mu },\stackrel{˜}{\nu }$. 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 $\left[0,1\right]$ 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

#### Euler-Maruyama discretization

In 29, Oumaima Bencheikh and Benjamin Jourdain are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order 1/2 in total variation distance. When the drift has a spatial divergence in the sense of distributions with $\rho$-th power integrable with respect to the Lebesgue measure in space uniformly in time for some $\rho \ge d$, the order of convergence at the terminal time improves to 1 up to some logarithmic factor. In dimension d=1, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. The theoretical analysis is confirmed by numerical experiments.

#### Approximation with rough paths.

A. Alfonsi and A. Kebaier are working on the approximation of some processes with rough paths.

#### Eigen values of Wishart processes.

B. Jourdain and his PhD student E. Kahn study stochastic differential equations coming from the eigenvalues of Wishart processes 31, 37.

### 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/).

# 7 Bilateral contracts and grants with industry

## 7.1 Bilateral contracts with industry

•   Consortium PREMIA, Natixis - INRIA
•   Consortium PREMIA, Crédit Agricole Corporate Investment Bank (CA - CIB ) - INRIA
•   AXA Joint Research Initiative on Numerical methods for the ALM, from September 2017 to August 2020. PhD grant of Adel Cherchali, Supervisor: A. Alfonsi.
•   CIFRE agreement Milliman company/Ecole des Ponts (http://fr.milliman.com),

PhD thesis of Sophian Mehalla (started November 2017) on "Interest rate risk modeling for insurance companies", Supervisor: Bernard Lapeyre.

•   CIFRE agreement Brahham gardens/Ecole des Ponts

PhD thesis of Hachem Madmoun: "Gestion de portefeuilles utilisant des techniques de (big) data" https://www.bramham-gardens.com

•   Collaboration with IRT Systemx

PhD grant of Adrien Touboul (started November 2017) on "Uncertainty computation in a graph of physical simulations", Supervisors: Bernard Lapeyre and Julien Reygner.

## 7.2 Bilateral grants with industry

Chair X-ENPC-SU-Société Générale "Financial Risks" of the Risk fondation : A. Alfonsi, B. Jourdain, B. Lapeyre

# 8 Partnerships and cooperations

## 8.1 International initiatives

### 8.1.1 Inria associate team not involved in an IIL

#### Informal international partners

• Georgia state university (Hamed Amini)
• Cornell University, ORIE department (Andreea Minca)
• Roma Tor Vergata University (Lucia Caramellino)
• Ritsumeikan University (A. Kohatsu-Higa).

### 8.1.2 Visits of international scientists

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

## 8.2 National initiatives

• Labex Bezout

### 8.2.1 Competitivity Clusters

Pôle Finance Innovation

# 9 Dissemination

## 9.1 Promoting scientific activities

### 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

#### Member of the editorial boards

• B. Jourdain

Associate editor of

• ESAIM : Proceedings and Surveys
• Stochastic Processes and their Applications (SPA)
• Stochastic and Partial Differential Equations : Analysis and Computations
• D. Lamberton

Associate editor of

• Mathematical Finance,
• Associate editor of ESAIM Probability & Statistics
• A. Sulem

Associate editor of

• Journal of Mathematical Analysis and Applications (JMAA)
• SIAM Journal on Financial Mathematics (SIFIN)

#### Reviewer - reviewing activities

• B. Jourdain : Reviewer for Mathematical Reviews
• A. Sulem: Reviewer for Mathematical Reviews

### 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)

• 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 Teaching - Supervision - Juries

### 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.

• "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.

# 10 Scientific production

## 10.1 Major publications

• 1 articleA. Al Gerbi, B. Jourdain and E. Clément. 'Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators'.Monte Carlo Method and Applications223https://arxiv.org/abs/1508.06492July 2016, 197-228
• 2 book A. Alfonsi. 'Affine Diffusions and Related Processes: Simulation, Theory and Applications'. 2015
• 3 article A. Alfonsi and P. Blanc. 'Dynamic optimal execution in a mixed-market-impact Hawkes price model'. Finance and Stochastics https://arxiv.org/abs/1404.0648 January 2016
• 4 article A. Alfonsi, B. Jourdain and A. Kohatsu-Higa. 'Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme'. Electronic Journal of Probability https://arxiv.org/abs/1405.7007 2015
• 5 articleH. Amini, A. Minca and A. Sulem. 'Control of interbank contagion under partial information'.SIAM Journal on Financial Mathematics61December 2015, 24
• 6 articleV. Bally and L. Caramellino. 'Convergence and regularity of probability laws by using an interpolation method'.Annals of Probability4522017, 1110--1159
• 7 articleA. Bouselmi and D. Lamberton. 'The critical price of the American put near maturity in the jump diffusion model'.SIAM Journal on Financial Mathematics71https://arxiv.org/abs/1406.6615May 2016, 236--272
• 8 articleR. Dumitrescu, M.-C. Quenez and A. Sulem. 'A Weak Dynamic Programming Principle for Combined Optimal Stopping/Stochastic Control with ${E}^{f}$-Expectations'.SIAM Journal on Control and Optimization5442016, 2090-2115
• 9 articleR. Dumitrescu, M.-C. Quenez and A. Sulem. 'Game Options in an Imperfect Market with Default'.SIAM Journal on Financial Mathematics81January 2017, 532 - 559
• 10 book B. Jourdain. 'Probabilités et statistique'. seconde édition Ellipses 2016
• 11 bookB. Øksendal and A. Sulem. 'Applied Stochastic Control of Jump Diffusions'.3rd editionSpringer, Universitext2019, 436

## 10.2 Publications of the year

### International journals

• 12 article A. Alfonsi, A. Cherchali and J. 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
• 13 articleA. Alfonsi, J. Corbetta and B. Jourdain. 'Sampling of probability measures in the convex order by Wasserstein projection'.Annales de l'IHP - Probabilités et Statistiques5632020, 1706-1729
• 14 article A. Alfonsi, R. Coyaud, V. Ehrlacher and D. Lombardi. 'Approximation of Optimal Transport problems with marginal moments constraints'. Mathematics of Computation 2020
• 15 articleA. Alfonsi and B. Jourdain. 'Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability'.ESAIM: Probability and Statistics242020, 703-717
• 16 articleV. Bally, L. Caramellino and G. Poly. 'Regularization lemmas and convergence in total variation'.Electronic Journal of Probability2502020, paper no. 74, 20 pp
• 17 article L. Goudenège, A. Molent and A. Zanette. 'Computing Credit Valuation Adjustment solving coupled PIDEs in the Bates model'. Computational Management Science 17 2 April 2020
• 18 article L. Goudenège, A. Molent and A. Zanette. 'Gaussian process regression for pricing variable annuities with stochastic volatility and interest rate'. Decisions in Economics and Finance May 2020
• 19 articleL. Goudenège, A. Molent and A. Zanette. 'Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models'.Quantitative Finance204April 2020, 573-591
• 20 articleM. Grigorova, M.-C. Quenez and A. Sulem. 'European options in a non-linear incomplete market model with default'.SIAM Journal on Financial Mathematics113September 2020, 849–880
• 21 articleB. Jourdain and W. Margheriti. 'A new family of one dimensional martingale couplings'.Electronic Journal of Probability251362020, 1-50
• 22 articleB. Jourdain and A. Zhou. 'Existence of a calibrated Regime Switching Local Volatility model'.Mathematical Finance302April 2020, 501-546

### Doctoral dissertations and habilitation theses

• 23 thesis O. 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

### Reports & preprints

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

## 10.3 Cited publications

• 39 article A. Ahdida, A. Alfonsi and E. Palidda. 'Smile with the Gaussian term structure model'. Journal of Computational Finance https://arxiv.org/abs/1412.7412 2017
• 40 articleA. Al Gerbi, B. Jourdain and E. Clément. 'Asymptotics for the normalized error of the Ninomiya-Victoir scheme'.Stochastic Processes and their Applications12862018, 1889-1928
• 41 articleA. Al Gerbi, B. Jourdain and E. Clément. 'Ninomiya-Victoir scheme : Multilevel Monte-Carlo estimators and discretization of the involved Ordinary Differential Equations'.ESAIM: Proceedings and Surveys59https://arxiv.org/abs/1612.07017November 2017, 1-14
• 42 article A. Alfonsi and P. Blanc. 'Extension and calibration of a Hawkes-based optimal execution model'. Market microstructure and liquidity https://arxiv.org/abs/1506.08740 August 2016
• 43 articleA. Alfonsi, J. Corbetta and B. Jourdain. 'Evolution of the Wasserstein distance between the marginals of two Markov processes'.Bernoulli244A2018, 2461-2498
• 44 article A. Alfonsi, J. Corbetta and B. Jourdain. 'Sampling of one-dimensional probability measures in the convex order and computation of robust option price bounds'. International Journal of Theoretical and Applied Finance 22 3 This paper is an updated version of a part of the paper https://hal.archives-ouvertes.fr/hal-01589581 (or https://arxiv.org/pdf/1709.05287.pdf ) 2019
• 45 incollection A. Alfonsi, M. Hayashi and A. Kohatsu-Higa. 'Parametrix Methods for One-Dimensional Reflected SDEs'. Modern Problems of Stochastic Analysis and StatisticsSelected Contributions In Honor of Valentin Konakov Springer Proceedings in Mathematics & Statistics 208 Springer November 2017
• 46 article A. Alfonsi, A. Kebaier and C. Rey. 'Maximum Likelihood Estimation for Wishart processes'. Stochastic Processes and their Applications https://arxiv.org/abs/1508.03323 November 2016
• 47 article A. Alfonsi, A. Schied and F. Klöck. 'Multivariate transient price impact and matrix-valued positive definite functions'. Mathematics of Operations Research https://arxiv.org/abs/1310.4471 March 2016
• 48 articleH. Amini, R. Cont and A. Minca. 'Resilience to Contagion in Financial Networks'.Mathematical Finance2622016, 329-365
• 49 unpublishedH. Amini, A. Minca and A. Sulem. 'Dynamic Contagion Risk Model With Recovery Features'.2019,
• 50 articleH. Amini, A. Minca and A. Sulem. 'Optimal equity infusions in interbank networks'.Journal of Financial Stability31August 2017, 1 - 17
• 51 articleA. Arnold, E. Carlen and Q. Ju. 'Large-time behavior of non-symmetric Fokker-Planck type equations'.Communications on Stochastic Analysis212008, 153-175
• 52 articleR. Assaraf, B. Jourdain, T. Lelièvre and R. Roux. 'Computation of sensitivities for the invariant measure of a parameter dependent diffusion'.Stochastics and Partial Differential Equations: Analysis and Computationshttps://arxiv.org/abs/1509.01348October 2017, 1-59
• 53 articleV. Bally and L. Caramellino. 'Asymptotic development for the CLT in total variation distance'.Bernoulli222016, 2442-2485
• 54 articleV. Bally, L. Caramellino and P. Pigato. 'Tube estimates for diffusions under a local strong Hörmander condition'.Ann. Inst. H. Poincaré Probab. Statist.55411 2019, 2320--2369
• 55 articleV. Bally, L. Caramellino and G. Poly. 'Convergence in distribution norms in the CLT for non identical distributed random variables'.Electron. J. Probab.232018, 51 pp.
• 56 article V. Bally, L. Caramellino and G. Poly. 'Non universality for the variance of the number of real roots of random trigonometric polynomials'. Probability Theory and Related Fields 174 08 2019
• 57 articleV. Bally and L. Caramellino. 'Regularity of Wiener functionals under a Hörmander type condition of order one'.Annals of Probability4532017, 1488-1511
• 58 articleV. Bally and A. Kohatsu-Higa. 'A probabilistic interpretation of the parametrix method'.Annals of Applied Probability252015, 3095-3138
• 59 unpublishedV. Bally and V. Rabiet. 'Asymptotic behavior for multi-scale PDMP's'.April 2015, preprint
• 60 article V. Bally and C. Rey. 'Approximation of Markov semigroups in total variation distance'. Electronic Journal of Probability 21 12 2016
• 61 articleM. Beiglböck, P.-H. Labordère and F. Penkner. 'Model-independent bounds for option prices - a mass transport approach'.Finance Stoch.1732013, 477-501
• 62 articleR. Chen, A. Minca and A. Sulem. 'Optimal connectivity for a large financial network'.ESAIM: Proceedings and Surveys59Editors : B. Bouchard, E. Gobet and B. Jourdain2017, 43 - 55
• 63 articleQ. Dai and K. Singleton. 'Specification Analysis of Affine Term Structure Models'.The Journal of Finance5552000, 1943--1978
• 64 articleD. Duffie, D. Filipović and W. Schachermayer. 'Affine processes and applications in finance'.Ann. Appl. Probab.1332003, 984--1053
• 65 incollection R. Dumitrescu, M. Grigorova, M.-C. Quenez and A. Sulem. 'BSDEs with default jump'. Computation and Combinatorics in Dynamics, Stochastics and Control - The Abel Symposium, Rosendal, Norway August 2016 13 The Abel Symposia book series Springer 2018
• 66 articleR. Dumitrescu and C. Labart. 'Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles'.Journal of Mathematical Analysis and applications4421https://arxiv.org/abs/1406.3612October 2016, 206-243
• 67 articleR. Dumitrescu and C. Labart. 'Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles'.Journal of Computational and Applied Mathematics296https://arxiv.org/abs/1502.02888April 2016, 827-839
• 68 articleR. Dumitrescu, M.-C. Quenez and A. Sulem. ' Mixed generalized Dynkin game and stochastic control in a Markovian framework'.Stochastics8912016, 30
• 69 articleR. Dumitrescu, M.-C. Quenez and A. Sulem. 'American Options in an Imperfect Complete Market with Default'.ESAIM: Proceedings and Surveys2018, 93--110
• 70 article R. Dumitrescu, M.-C. Quenez and A. Sulem. 'Generalized Dynkin games and doubly reflected BSDEs with jumps'. Electronic Journal of Probability 2016
• 71 articleR. Dumitrescu, M.-C. Quenez and A. Sulem. 'Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems'.Journal of Optimization Theory and Applications16712015, 23
• 72 unpublishedN. El Karoui and V. Lacoste. 'Multifactor models of the term structure of interest rates'.1992, Preprint University of Paris 6
• 73 articleA. Figalli. 'Existence and uniqueness for martingale solutions of SDEs with rough or degenerate coefficients'.Journal of Functional Analysis2542008, 109--153
• 74 articleC. Fontana, B. \Oksendal and A. Sulem. 'Market viability and martingale measures under partial information'.Methodology and Computing in Applied Probability172015, 24
• 75 articleJ. Fontbona and B. Jourdain. 'A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations'.Annals of Probability441https://arxiv.org/abs/1107.3300February 2016, 131-170
• 76 articleN. Fournier and B. Jourdain. 'Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes'.The Annals of Applied Probability : an official journal of the institute of mathematical statistics275https://arxiv.org/abs/1507.01087November 2017, 2807-2861
• 77 articleJ. Garnier, G. Papanicolaou and T. Yang. 'Large deviations for a mean field model of systemic risk'.SIAM Journal on Financial Mathematics4112013, 151--184
• 78 unpublishedM. Grigorova, M.-C. Quenez and A. Sulem. 'American options in a non-linear incomplete market model with default'.February 2019, working paper or preprint
• 79 article J. Guyon and P. Henry-Labordère. 'Being particular about calibration'. Risk January 2012
• 80 articleY. Hu, B. \Oksendal and A. Sulem. 'Singular mean-field control games'.Stochastic Analysis and Applications355June 2017, 823 - 851
• 81 articleB. Jourdain and A. Kebaier. 'Non-asymptotic error bounds for The Multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient'.Electronic Journal of Probability24122019, 1-34
• 82 articleB. Jourdain and J. Reygner. 'Capital distribution and portfolio performance in the mean-field Atlas model'.Annals of Finance112https://arxiv.org/abs/1312.5660May 2015, 151-198
• 83 articleB. Jourdain and J. Reygner. 'Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data'.Discrete and Continuous Dynamical Systems - Series A369https://arxiv.org/abs/1507.01085September 2016, 4963-4996
• 84 articleB. Jourdain and J. Reygner. 'The small noise limit of order-based diffusion processes'.Electronic Journal of Probability1929https://arxiv.org/abs/1307.0490March 2014, 1-36
• 85 articleD. Lépingle. 'Euler scheme for reflected stochastic differential equations'.Math. Comput. Simulation381-3Probabilités numériques (Paris, 1992)1995, 119--126
• 86 phdthesis A. Minca. 'Modélisation mathématique de la contagion de défaut; Mathematical modeling of financial contagion'. Inria Université Pierre et Marie Curie Paris 6 September 5 2011
• 87 articleA. Minca and A. Sulem. 'Optimal Control of Interbank Contagion Under Complete Information'.Statistics and Risk Modeling3112014, 1001-1026
• 88 articleS. Ninomiya and N. Victoir. 'Weak approximation of stochastic differential equations and application to derivative pricing'.Applied Mathematical Finance152008, 107-121
• 89 articleM.-C. Quenez and A. Sulem. 'Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps'.Stochastic Processes and their Applications1249https://arxiv.org/abs/1212.6744September 2014, 23
• 90 articleB. \Oksendal and A. Sulem. 'Dynamic Robust Duality in Utility Maximization'.Applied Mathematics and Optimization2016, 1-31
• 91 articleB. \Oksendal and A. Sulem. 'Forward--Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty'.Journal of Optimization Theory and Applications1611April 2014, 22 - 55
• 92 incollectionB. \Oksendal and A. Sulem. 'Optimal control of predictive mean-field equations and applications to finance'.Springer Proceedings in Mathematics & Statistics138Stochastic of Environmental and Financial EconomicsSpringer Verlag2016, 301--320