# Programme

Schedule

Overview for the week (PDF) and Book of Abstracts (PDF)

Detailed Schedule:

## Plenary Talks

### Francois Delarue - McKean Vlasov FBSDEs with a common noise

**Abstract:** Motivated by the theory of mean field games, I will address the solvability and the smoothness of the flow of fully coupled McKean Vlasov forward-backward SDEs with a common noise. Part of this presentation is taken from a joint work with Chassagneux and Crisan

#### Alison Etheridge - Modelling populations under fluctuating selection

**Abstract: **It has been recognised for a very long time that natural selection is not necessarily a constant force acting on a population; for example the genetic types favoured in a wet year may be different from those favoured in a dry year. As a result, fluctuating environmental conditions can maintain a balance between the different genotypes over extended periods. If, for example, we suppose that selection is acting on a single gene, then it is straightforward to write down a stochastic (ordinary) differential equation that captures the evolution of the frequencies of the different types of that gene (alleles) in the population. Crucially, such models can also capture genetic drift - the randomness due to reproduction in a finite population. What is much less studied is the evolution of allele frequencies in a population that is spatially structured. Here we discuss one such model, based on the so-called spatial Lambda-Fleming-Viot process, that can capture something of the interplay between fluctuating selection and genetic drift. We shall see that when viewed over sufficiently large spatial and temporal scales, in at least two spatial dimensions, allele frequencies are dominated by the fluctuations due to the environment and can be captured by an spde. Ideally, one would be able to capture family struture in the population. As time permits we shall explain some partial results in this direction.

#### Jianfeng Zhang - Some new types of path dependent PDEs

**Abstract:** Path dependent PDE is a PDE where the value (or solution) depends on the path of the state process, and is a convenient tool for studying path dependent (and non-Markovian) problems. In the standard literature, the PPDE is typically due to the (exogenous) path dependence imposed on the coefficients of the problem. In this talk, we present two examples where state dependent problems lead naturally to (new types of) PPDEs. The first example is that the state process is a fractional Brownian motion $B^H$ (or more generally solution to a Volterra SDE). In this case, the value function of the backward problem will take the form $u(t, (B^H_s)_{0\le s<t} \otimes_t (E_t[B^H_s])_{t\le s\le T})$. In particular, instead of being flat after $t$ as in standard literature, in this case an $\cF_t$-measurable path is involved over $[t, T]$. The second example is the stochastic control problem with information delay, namely the control $\alpha_t$ at time $t$ depends on $\cF_{t-\delta}$ for some delay parameter $\delta$. This problem leads naturally to a path dependent master equation, and under the optimal control (if it exists), the state process solves some McKean Vlasov SDE. We remark that in the latter example, the original control problem is in a standard setting without involvement of the laws. We also note that both examples involve some type of time inconsistency issue. The talk is based on two joint works, one with Frederi Viens and the other with Yuri Saporito.

## Invited Talks

#### Rainer Buckdahn - Representation formulas for long run avering stochastic optimal control problems

**Abstract:** We study a stochastic control problem with long run averaging cost for both the Cesaro and also the Abel means. The asymptotic behaviour of the associated value functions is a classical and meanwhile well studied problem in ergodic stochastic control. But unlike ergodic control problems we don't make assumptions here which lead to a limit value function which is a constant independent of the initial condition; our limit value functions can depend on the intial condition. While we have shown in previous work under the so-called non expansion condition the existence of a limit value function (together with Dan Goreac and Marc Quincampoix for classical cost functionals, and recently also Juan Li with Hui Min for cost functionals defined through controlled backward SDEs with infinite time horizon ), we study now representation formulas for accumulation points of the value functions - in the sense of the uniform convergence - when the time horizon of the Cesaro mean tends to infinity and the discount factor of the Abel mean converges to zero. The representation formulas are given by using occupational measures on the state spaces and invariant measures on the state path spaces, respectively. Their independence of the converging subsequence of value functions has as consequence that the limit of the value functions defined through the Cesaro or Abel means exists as soon as these value functions are equicontinuous. This is the case, for instance, under the non expansion condition. The talk is based on joint work with Juan Li (Shandong University, Weihai, P.R.China), Marc Quincampoix (Universite de Bretagne Occidentale, Brest, France) er Jerome Renault (Toulouse School of Economics, Toulouse, France)

#### Peter Imkeller - Reflected BSDE with irregular obstacles and optimal stopping

**Abstract:** Reflected backward stochastic differential equations (RBSDE) are a well known tool for solving American and game option problems. The case of reflection at continuous adapted or cadlag lower barriers is well understood. We develop tools to deal with obstacles that are optional but have essentially less regularity. Among the tools is Mertens' decomposition of optional strong supermartingales. The corresponding RBSDE are applied to an optimal stopping problem in which the risk of a financial position is assessed by an nonlinear conditional expectation. We characterize the value function, and formulate conditions under which optimal stopping times exist. This is joint work with Miryana Grigorova, Marie-Claire Quenez, Elias Offen and Youssef Ouknine.

#### Monique Jeanblanc - BSDE and enlargement of filtration

**Abstract:** We present two problems where one considers BSDE in various filtrations. The first one corresponds to an optimisation problem, when the measurability of the strategies varies. The second one is a comparison between two BSDEs written in different filtrations.

#### Juan Li - Mean-field forward and backward SDEs with jumps. Associated nonlocal quasi-linear integral-PDEs

**Abstract:** In this talk we consider a decoupled mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. The existence and the uniqueness of the solution $(Y^{t,x,P_\xi},Y^{t,\xi})$ of the decoupled equation are proved. We prove that under our assumptions the value function $V(t,x,P_\xi):=Y_t^{t,x,P_\xi}$ is regular, and it is the unique classical solution of the related quasi-linear integral-partial differential equation of mean-field type with the help of a new It\^{o} formula.

#### Jin Ma - Time-Consistent Approaches in Time-inconsistent Problems

**Abstract:** Time-inconsistency appears naturally and frequently in economics and finance, and has been noticed for more than half a century. Many attempts have been made in recent years to break through the major barrier in this problem: the failure of the dynamic programming principle, and various methods have been proposed to and optimal strategies for these problems that are “time-inconsistent" by nature. In this talk we discuss possible dynamic approaches to study time-inconsistent optimization problems that are time-consistent. Originated on the idea of dynamic programming, these approaches will focus more on the value of the optimization problem rather than on the optimal control, which may not even exit. In particular, we shall present a newly discovered fact that a commonly known time-inconsistent nonlinear expectation under probability distortion can actually be turned into a non-sub-additive, but time-consistent (or filtration-consistent) one, with a careful localization of the distortion impact. More interestingly, we show it has many features that resembles Peng's theory of non-linear expectation, and we shall derive the corresponding PDE and BSDE representing such a nonlinear expectation. We hope that such an observation will lead to some new developments in the study of time-inconsistent problems. This is based on joint works with Chandrasekhar Karnam, Ting-Kam Leonard Wong, and Jianfeng Zhang.

#### Anis Matoussi - Reflected second order BSDEs under weak regularity conditions

**Abstract:** We present a wellposdness solutions of reflected second order BSDEs in a general filtration and under weak regularity conditions on the terminal value, the generator and the obstacle (lower and upper cases). We introduce also a wellposdness Skorokhod-solution to the second order BSDEs.

#### Annie Millet - On generalized KdV equation subject to some random perturbation

**Abstract:** We will describe global well-posedness and orbital stability for the generalized KdV equation in the subcritical case, subject to some additive random perturbation. The critical and super-critical cases will also be discussed. This is joint work with S. Roudenko.

#### Etienne Pardoux - Nonlinear filtering with degenerating noise

**Abstract:** We consider a new class of filtering problem, where the observation lives in a bounded set, and the observation noise vanishes when the observation visits the boundary of the set where it lives. We show that an adaptation of the classical arguments yields a version of the traditional filtering equations. In the case where the observation is scalar and the observation and signal noises are independent, we obtain a robust version of the Zakai equation, which has a unique measure valued solution, thanks to a duality argument. We extend the uniqueness result via duality between a forward and a backward SPDE, beyond the "robust case".

#### Shige Peng - Quantifying nonlinear expected value by BSDE and PDE

**Abstract:** Frank Knight challenged the feasibility of using probability to treat uncertainty. The term ``Knightian uncertainty" or ``ambiguity'' is now widely accepted, specially among economists, to refer situations where there is no objective probability, or distribution, available for making decisions. Nonlinear expectation (e.g. $g$(and $G$)-expectation) provided a new framework of to deal with this type of uncertainties. But a challenging problem is: how can we use real-world data to construct the corresponding nonlinear expectation hidden behind? We have introduced a new algorithm for computing the nonlinear expected value based on a given historical data under a much weaker assumption than the classical i.i.d. one. The method is fundamentally based on our framework of nonlinear expectation theory. It also reveals a deep relation between robust limit theory and statistics and quasilinear and fully nonlinear PDEs.

#### Michael Röckner - Nonlinear Fokker–Planck equations driven by Gaussian linear multiplicative noise

**Abstract:** Existence and uniqueness of a strong solution in H^{-}^{1}(R^{d}) is proved for the stochastic nonlinear Fokker-Planck equation

dX−div(DX)dt−△β(X)dt=X dW in (0,T)×R^{d}, X(0)=x,

via a corresponding random PDE. W is a Wiener process in H^{-1}(R^{d}); D ∈ C^{1}(R^{d},R^{d}) and β is a Lipschitz continuous monotonically increasing function. The solution is pathwise Lipschitz continuous with respect to initial data in H^{1}(R^{d}) and preserves positivity.

In case of initial conditions bounded below by a constant ρ > 0, the Lipschitz condition on β can be relaxed. Stochastic equations with nonlinear drift of the form dX−div(a(X))dt+△β(X)dt = XdW are also considered for Lipschitzian functions a : R → R^{d}. The main technique for the proof is a variant of the Crandall-Liggett approach applied to the ran- dom PDE, but with time dependent coefficients.

This is joint work with Viorel Barbu.

#### Saïd Hamadene - Existence and uniqueness of viscosity solutions for second order integro-differential equations without monotonicity condition

**Abstract: **In this talk, we discuss a new existence and uniqueness result of a continuous viscosity solution for integro-partial differential equation (IPDE in short). The novelty is that we relax the so-called monotonicity assumption on the driver which is classically assumed in the literature of viscosity solution of equation with a nonlocal term. Our method is based on the link of those IPDEs with backward stochastic differential equations (BSDEs in short) with jumps for which we already know that the solution exists and is unique.

#### Panagiotis Souganidis - Regularizing effects and long time behavior of stochastic viscosity solutions

**Abstract:** I will review briefly the notion of stochastic viscosity solution and discuss some regularizing effects as well as results about the long time behavior of the solutions. This is joint work with P. L. Lions.

#### Nizar Touzi - Continuous-time Principal-Agent problem: a stackelberg stochastic differential game

**Abstract:** We provide a systematic method for solving general Principal-Agent problems with possibly infinite horizon. Our main result reduces such Stackelberg stochastic differential games to a standard stochastic control problem, which may be addressed by the standard tools of control theory. Our proofs rely on the backward stochastic differential equations approach to non-Markovian stochastic control, and more specifically, on the recent extensions to the second order case. The infinite horizon setting requires an extension of second order BSDEs to the random horizon setting.

#### Thaleia Zariphopoulou - Mean field and n-agent games for optimal investment under relative performance criteria

**Abstract:** I will discuss the optimal portfolio management of a population of fund managers who trade in a common horizon [0,T], aiming at maximizing their expected utility but are also concerned about their relative performance. I will present the n-agent model as well as the mean field game under both CARA and CRRA risk preferences, construct the equilibria explicitly and provide conditions for their existence and uniqueness. (joint work with D. Lacker).

## Contributed Talk Titles

See the Boof of Abstracts (PDF).

## Special Sessions

See the section on Special Sessions.

## Mini Courses

See the section on Mini-Courses.