The seminal work of Cvitanic, Possamai and Touzi (2018) [1] introduced a general framework for continuous-time principal-agent problems using dynamic programming and second-order backward stochastic differential equations (2BSDEs). In this talk, we first propose an alternative formulation of the principal-agent problem that allows for a more direct resolution using standard BSDEs alone. Our approach is motivated by a key observation in [1]: when the principal observes the output process X continuously, she can compute its quadratic variation pathwise. While this information is incorporated into the contract in [1], we consider here a reformulation where the principal directly controls this process in a ‘first-best' setting. The resolution of this alternative problem follows the methodology known as Sannikov's trick [2] in continuous-time principal-agent problems. We then demonstrate that the solution to this ‘first-best' formulation coincides with the original problem's solution. More specifically, leveraging the contract form introduced in [1], we establish that the ‘first-best' outcome can be attained even when the principal lacks direct control over the quadratic variation. Crucially, our approach does not require the use of 2BSDEs to prove contract optimality, as optimality naturally follows from achieving the ‘first-best' scenario. We believe that this reformulation offers a more accessible approach to solving continuous-time principal-agent problems with volatility control, facilitating broader dissemination across various fields. In the second part of the talk, we will explore how this methodology extends to more complex settings, particularly multi-agent frameworks. Research partially supported by the NSF grant DMS-2307736.[-]

We review basic properties of the moment-LP and moment-SOS hierarchies for polynomial optimization and compare them. We also illustrate how to use such a methodology in two applications outside optimization. Namely :
- for approximating (as claosely as desired in a strong sens) set defined with quantifiers of the form
$R_1 =\{ x\in B : f(x,y)\leq 0 $ for all $y$ such that $(x,y) \in K \}$.
$D_1 =\{ x\in B : f(x,y)\leq 0 $ for some $y$ such that $(x,y) \in K \}$.
by a hierarchy of inner sublevel set approximations
$\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right \}\subset R_f$.
or outer sublevel set approximations
$\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right \}\supset D_f$.
for some polynomiales $(J_k)$ of increasing degree :
- for computing convex polynomial underestimators of a given polynomial $f$ on a box $B \subset R^n$.[-]

The question of using the available measurements to retrieve mathematical models characteristics (parameters, boundary conditions, initial conditions) is a key aspect of the modeling objective in biology or medicine. In a stochastic/statistical framework this question is seen as an estimation problems. From a deterministic point of view, we classical talk about inverse problems as we recover classical model inputs from outputs. When considering evolution problems,this question falls in the realm of data assimilation that can be seen from a deterministic of statistical point of view. Our objective in this course is to introduce the mathematical principles and numerical aspects behind data assimilation strategies with an emphasis on the deterministic formalism allowing to understand why data assimilation is a specific inverse problem. Our presentation will include considerations on finite dimensional problems but also on infinite dimensional problems such as the ones arising from PDE models. And we will illustrate the course with numerous examples coming from cardiovascular applications and biology.[-]

Processing signals presents many challenges by the quantity, structure, faults, heterogeneity of sensor data recorded over time. Supporting decisions using prediction or detection based on data streams naturally calls Machine Learning techniques (and theory!) for backup. The latter field has witnessed a tremendous development since the publication of Vladimir Vapnik's best-seller 'The Nature of Statistical Learning' and the invention of Support Vector Machines, Bagging, Boosting and Random Forests between 1995 and 1999 until the latest technological breakthroughs based on Deep Learning. However, most of its reference frameworks and methods consider vector observations which are essentially invariant up to a permutation of the indices of vector components. Beyond the obvious approach of featurization (or embedding) time series into vectors of characteristics (features), there are other more subtle interactions between the two fields of SP and ML but they first need to address some fundamental questions such as:
- how to monitor the lack of stationarity in time
- dependent data - how to supervise such data
- what is the objective of learning (prediction goal) in this context, and more generally what can be learned with signals
- how to account for additional structure in signals
- how Signal Processing as a field may benefit from modern optimization techniques

The purpose of this course is to offer an overview on some Signal Processing problems from the angle of Machine Learning philosophy and techniques in order to develop insights on the fundamental questions formulated above. In other terms, this is not a standard course on Signal Processing and we may skip some of the very fundamental concepts that would belong to such a course.

The topics presented in this doctoral course will include:
- local stationarity
- event detection methodology
- prediction problems with signals
- representation learning
- graph signal processing
In the practical sessions, a concrete example in the context of precision medicine will be developed. In particular, the central issues of segmentation, quantification, representation will be addressed with code.[-]

These lectures will consist in an overview of recent progresses made in contracting theory, using the so-called dynamic programming approach. The basic situation is that of a Principal wanting to hire an Agent to do a task on his behalf, and who has to be properly incentivized. We will show how this general framework allows to treat volatility control problems arising for instance in delegated portfolio management, or in electricity pricing. If time permit, we will also analyze the situation of a Principal hiring a finite number of Agents who can interact with each other, as well as the associated mean-field problem. The theory will be mostly illustrated by examples ranging from finance and insurance applications to regulation issues.[-]

We study the small-time local controllability (STLC) for scalar input control affine systems, in finite dimension. It is known that the entire information about STLC is contained in the evaluation at zero of the Lie brackets of the vector fields. In the 80's, several authors formulated necessary conditions for controllability (obstructions), relying on particular 'bad' brackets. In this talk, I will present a unified approach to determine and prove obstructions to STLC, that allows to recover known obstructions and prove new ones, in a relatively systematic way. This approach relies on a recent Magnus-type representation of the state, a new Hall basis of the free Lie algebra over 2 generators and interpolation inequalities. This is a joint work with Frédéric Marbach and Jérémy Leborgne.[-]