# Titres et résumés pour la rencontre WS3

• Philippe Briand : Simulation of BSDEs and Wiener chaos expansions

• René Carmona : Weak formulation for Mean Field Games

We present a simple model for high frequency market making which reduces to the search for Nash equilibria in (stochastic differential) Mean Field Games for which the interactions between the players is through their controls. We present a set of existence results using the weak formulation approach, and show how these abstract results can be applied to the market making model and a few models with non-standard interactions.

• Patrick Cheridito : BSDEs with regular terminal conditions

BSDEs with regular terminal conditions can be shown to have solutions even if the driver violates the standard Lipschitz assumptions. The following cases will be discussed: terminal conditions that are Lipschitz continuous, uniformly continuous or lower semicontinuous in the paths of the underlying Brownian motion and terminal conditions with bounded Malliavin derivative.

• Samuel Cohen

• François Delarue : Large population stochastic control and BSDEs.

I will present several results about large population stochastic control and related backward SDEs of the McKean-Vlasov type. Precisely, I will discuss two types of optimization problem about the control of a large population of players interacting one with the others through their statistical distribution. The first type of problem corresponds to the mean-field game framework as considered by Lasry and Lions on the one hand and by Huang, Caines and Malhame on the other hand. In this case, players choose their strategy in order to maximize their own wealth. When the population of players is infinite, Nash-equilibria can be described by a suitable version of the stochastic maximum principle by means of fully coupled forward-backward equations of the McKean-Vlasov type, and, under suitable assumptions, a sort propagation of chaos applies in the sense that the corresponding optimal controls provide nearly optimal controls for games driven by a large but finite population of players. In the second problem, players are assumed to obey a common policy in order to optimize a common wealth. When the population is infinite, equilibria can be also described by fully coupled forward-backward equations of the McKean-Vlasov, but of a different form than in the first case. Under suitable assumptions, propagation of chaos also holds. I will insist on the difference between the two types of problems and present some solvability results for both of them.

• Marco Fuhrman : BSDEs and point processes

We formulate and solve a class of BSDEs driven by the compensated random measure associated to a given marked point process on a general state space. We present basic well-posedness results, then we address applications to optimal control of marked point processes, where the solution of the BSDE allows to identify the value function and the optimal control. We also address a (backward) stochastic version of the Hamilton-Jacobi-Bellman equation in this nonmarkovian situation. Specific results are then given in the markovian case, i.e. when the point process is a pure jump Markov process.
This is work in progress in collaboration with Fulvia Confortola.

• Stefan Geiss : Decoupling on the Wiener space and applications to BSDEs

We use a decoupling technique on the Wiener space to introduce an-isotropic spaces of fractional smoothness and use these spaces to obtain estimates for the $L_p$-variation of the solution of a backward stochastic differential equation. Our decoupling method is based on a general approach not restricted to Gaussian random variables, the Wiener space is based on.

• Emmanuel Gobet : Recent advances in empirical regression schemes for BSDEs

We will review recent results regarding the resolution of BSDEs using empirical regression schemes: it will address the quadratic case, the high-dimensional case, under data with low regularity.
Joint work with P. Turkedjiev.

• Giuseppina Guatteri : Operator Valued BSDEs: a mild formulation

We study backward stochastic Lyapunov equation arising in optimal LQ control with stochastic differential state equation taking values in an Hilbert space H. Such equation is a BSDE taking values in the
non-Hilbertian space L(H). In this talk we show the well posedness
of the equation in its mild formulation. To this purpose we introduce
a suitable Hilbert space that involves fractional powers of the driving operator. Moreover we show that the solution exhibits the typical parabolic regularity at the interior of the domain [0, T).

• Saïd Hamadène : Viscosity Solutions of Systems of Variational Inequalities with
Interconnected Bilateral Obstacles

The talk is related to existence and uniqueness of a solution in viscosity sense of a system of PDEs with bilateral inter-connected obstacles. This problem is connected with two players zero-sum switching game. Mainly, under the non-free loop property on the switcing costs and monotonicity of the generators, we show existence and uniqueness of a solution for the system. Our method combines probabilistic tools and analytical ones as well. It is based on systems of reflected backward stochastic dierential equations and Perron's method.

• Peter Imkeller : On the Skorokhod embedding problem and FBSDE

A link between martingale representation and solutions of the Skorokhod embedding
problem has been established by R. Bass. A generalization of his approach to FBSDE leads
us to solutions of the Skorokhod embedding problem for certain diffusion processes.

• Monique Jeanblanc : Robust utility maximization in a discontinuous filtration

We study a problem of utility maximization under model uncertainty with information including jumps. We prove first that the value process of the robust stochastic control problem is described by the solution of a quadratic-exponential backward stochastic differential equation with jumps. Then, we establish a dynamic maximum principle for the optimal control of the maximization problem. The characterization of the optimal model and the optimal control (consumption-investment) is given via a forward-backward system which generalizes the result of Duffie and Skiadas and El Karoui et al. in the case of maximization of recursive utilities including model with jumps. Joint work with A. Matoussi and A. Ngoupeyou
• Hanqing Jin : Backward Stochastic Differential Equations with Law Invariance

Motivated by portfolio selection with law-invariant objective, we tried to studied the BSDE whose terminal condition is not specified by a random variable, but a distribution. We understand the problem is extreme hard for us, hence we focus on those BSDEs whose solution only depend on the (conditional)distribution of the terminal condition. We will give some equivalent condition for the law invariance under some regularity condition.

• Michael Kupper : Existence and Duality of minimal Supersolutions under Model Uncertainty

We discuss the superhedging problem under model uncertainty based on existence
and duality results for minimal supersolutions of backward stochastic differential equations.
The talk is based on joint works with Samuel Drapeau, Gregor Heyne and Reinhard Schmidt.

• Anis Matoussi : Reflected second order BSDE's and Dynkin game under uncertainty

We first prove the existence and uniqueness of second-order reflected 2BSDEs to the case of upper obstacles. Then, under some regularity assumptions on one of the barrier, and when the two barriers are completely separated, we provide a complete wellposedness theory for doubly reflected second-order BSDEs. We also show that these objects are related to non-standard optimal stopping games, thus generalizing the connection between DRBSDEs and Dynkin games first proved by Cvitanić and Karatzas (2006). More precisely, we show that the second order DRBSDEs provide solutions of what we call uncertain Dynkin games and that they also allow us to obtain super and subhedging prices for American game options (also called Israeli options) in financial markets with volatility uncertainty. This talk is based on joint works with Lambert Piozin (Université du Mans), Dylan Possamaï (Université Dauphine) and Cha Zhou (National Singapor University).

• Shige Peng

• Huyên Pham : Backward SDEs with partially nonpositive jumps and Hamilton-Jacobi-Bellman IPDE

We aim to provide a Feynman-Kac type representation for Hamilton-Jacobi-Bellman equation, in terms of
Forward Backward Stochastic Differential Equation (FBSDE) with a simulatable forward process.
For this purpose, we introduce a class of BSDE where the jumps component of the solution is subject to a partial nonpositive constraint. Existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions. We then show how minimal solution to this BSDE class provides a new probabilistic representation for fully nonlinear integro-partial differential equations (IPDEs) of Hamilton-Jacobi-Bellman (HJB) type, when considering a regime switching forward SDE in a Markovian framework. This includes in particular equations in finance arising from option pricing under model uncertainty. In contrast with the recent theory of G-expectation and 2BSDEs, our representation is formulated under a single probability measure, thus avoiding quasi-sure analysis and nondominated measures. We also introduce a direct numerical scheme for BSDEs with constrained jumps, which gives an original probabilistic algorithm for solving HJB equations.

• Anthony Réveillac : BSDEs with weak terminal conditions and application to Finance

In this talk, we introduce a new class of BSDEs with so-called weak terminal condition. As we will see, this new type of equations naturally arises when studying so-called stochastic target problems with controlled loss (like for example the quantile hedging problem). In this talk we will present some existence result for these BSDEs and provide a characterization of the minimal initial value of supersolutions as well as some regularity properties. This talk is based on a joint work with Bruno Bouchard and Romuald Elie.

• Adrien Richou : Numerical simulation of BSDEs with drivers of quadratic growth with respect to Z

In this talk we will see how it is possible to adapt the classical linearization trick to time discretized BSDEs. Thanks to this representation we can use BMO Martingale tools to obtain a nice speed of convergence for time discretized quadratic BSDEs. We will also provide some numerical experiments.
This is a joint work with Jean-François Chassagneux (Imperial College).

• Gianmario Tessitore : Two cases of stochastic maximum principle in the optimal control of SPDEs

We consider two situations in which the Stochastic Maximum Principe for optimal control of stochastic partial differential equations requires new results on backward stochastic differential equation in infinite dimensions. First we consider the case of a stochastic heat equation in dimension one driven by a space time white noise and with convex set of admissible controls. In this case the BSDE corresponding to the adjoint of the first variation equation already poses serious new difficulties. We prove existence of a solution in a class of operators with finite trace.Then we consider a stochastic parabolic equation driven by a finite dimensional noise. In that case we allow non convex set of admissible controls and control depending diffusion coefficients. The difficulty is in this case represented by the characterization of the process representing the adjoint of the second variation.

• Hao Xing : Large time behavior of solutions to HJB equations and multivariate portfolio turnpikes

We study the large time limiting behavior of solutions to a Cauchy problem for semilinear parabolic equations with quadratic nonlinearity in gradients. The spatial domain for the equation can either be R^n or the space of positive definite matrice. When a Lyapunov function exists, as time tends to infinity, the solution (and its gradient) to the Cauchy problem converges to a solution (and its gradient) of the associated ergodic problem. Applications to long term portfolio choice problems and their turnpike properties will be discussed. When the investment opportunities are driven by a multivariate factor process with the state space R^n or the space of positive definite matrice, the convergence of solutions implies the optimal investment strategy for the power utility agent converges to its long run optimal analogue, when the investment horizon tends to infinity.
This is a joint work with Scott Robertson.