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.[-]

In 1926 Statistician G. Udny Yule showed that for two independent standard random walks, the empirical correlation coefficient (Pearson's correlation) does not converge to 0, but rather appears to converge in distribution to a diffuse law supported by the entire interval (-1,1). This phenomenon, which has since been recognized for many highly non-stationary time series, is in sharp contrast with the classical result for two sequences of i.i.d. data, by which the same correlation converges to zero, a phenomenon which also extends to many stationary time series. Still, ignorance of this empirical fact for random walks and other non-stationary time series, known today as Yule's nonsense correlation, has lead practitioners to make dramatically ill-informed assertions about statistical associations. This improper use of methodology has occurred in recent times, particularly in environmental observational studies, e.g. for attribution in climate science. The mathematics behind the basic premise of Yule's nonsense correlation are a rather straightforward application of the classical Donsker's theorem; the Pearson correlation ρn of two random walks of length n converges in distribution to the law of a random variable ρ written explicitly as the ratio of two quadratic functionals of two Wiener processes on [0.1]. In this talk, we investigate the fluctuations around this convergence. We present elements of a new result by which n(ρ − ρn) has an asymptotic distribution in the so-called second Wiener chaos, whose characteristics are partly exogenous to the original data, as one would expect for a standard central limit theorem, and are partly conditional on the data. We will discuss the implications of this discovery in practical testing for independence and for attribution in environmental time series. We conjecture that the fluctuation scale, of order 1/n rather than 1/n1/2, is not accidentally related to the exotic convergence in law in the second Wiener haos.[-]

The tragedy of the commons (TOTC, introduced by Hardin, 1968) states that the individual incentives will result in overusing common pool resources which in turn may have detrimental future consequences that affect everyone involved negatively. However, in many real-life situations this does not happen and researchers such as the Nobel laureate Elinor Ostrom suggested mutual restraint by individuals can be the preventing factor. In mean field games (MFGs), since individuals are insignificant and fully non-cooperative, the TOTC is inevitable. This shows that MFG models should incorporate a mixture of self- ishness and altruism to capture real-life situations that include common pool resources. Motivated by this, we will discuss different equilibrium notions to capture the mixture of cooperative and non-cooperative behavior in the population. First, we will introduce mixed individual MFGs and mixed population MFGs where we also include the common pool resources. The former captures altruistic tendencies at the individual level and the latter models a population that is a mixture of fully cooperative and non-cooperative individuals. For both cases, we will briefly discuss definitions and characterization of equi- librium with the forward backward stochastic differential equations. Later, we will discuss a real-life inspired example of fishers where the fish stock is the common pool resource. We will analyze the existence and uniqueness results, and discuss the experimental results.[-]

In this talk, we will establish a primal-dual formulation for continuous-time mean field games (MFGs) and provide a complete analytical characterization of the set of all Nash equilibria (NEs). We first show that for any given mean field flow, the representative player's control problem with measurable coefficients is equivalent to a linear program over the space of occupation measures. We then establish the dual formulation of this linear program as a maximization problem over smooth subsolutions of the associated Hamilton-Jacobi-Bellman (HJB) equation, which plays a fundamental role in characterizing NEs of MFGs. Finally, a complete characterization of all NEs for MFGs is established by the strong duality between the linear program and its dual problem. This strong duality is obtained by studying the solvability of the dual problem, and in particular through analyzing the regularity of the associated HJB equation. its NE characterization do not require the convexity of the associated Hamiltonianor the uniqueness of its optimizer, and remain applicable when the HJB equation lacks classical or even continuous solutions.[-]

The propagation of chaos phenomenon states roughly that a large system of weakly interacting particles will remain approximately independent for all times if initialized as such. This can be quantified in terms of the distance between low-dimensional marginal distributions and suitably chosen product measures. This talk will discuss some recent sharp quantitative results of this nature, both for classical mean field diffusions and for more recently studied non-exchangeable models. These results are driven by a new "local" relative entropy method, in which low-dimensional marginals are estimated iteratively by adding one coordinate at a time, leading to surprising improvements on prior results obtained by "global" arguments such as subadditivity inequalities. In the non-exchangeable setting, we exploit a surprising connection with first-passage percolation.[-]

The celebrated mean-curvature flow describes the evolution of the interface between two domains which moves so that its orthogonal velocity at each point is proportional to its mean curvature, pointing in the direction of decreasing the curvature. In the second order mean-curvature flow, it is the derivative of the orthogonal velocity (i.e., the acceleration) that is proportional to the mean curvature. Both flows can be described through partial differential equations (PDEs) for the associated arrival time functions. However, unlike the PDE for the classical mean-curvature flow, the equation for its second order version -- which we refer to as the ``cascade equation" -- is hyperbolic and does not enjoy the comparison principle. For this reason, and due to other challenges, the standard PDE tools are not sufficient to develop a well-posedness theory for the cascade equation directly. Nevertheless, it turns out that solutions to the cascade PDE can be identified with minimal elements of a set of value functions in a family of mean-field games. As a result, the existence of a solution to the cascade equation can be shown by proving the compactness of the aforementioned set of value functions, which we accomplish by employing the tools from Geometric Measure Theory. Joint work with Y. Guo and M. Shkolnikov.[-]