Schedule for Spring 2023

1/26/23

Emma Hubert (Princeton)

Title: Large-scale principal-agent problems

Abstract: In this talk, we introduce two problems of contract theory, in continuous-time, with a multitude of agents. First, we will study a model of optimal contracting in a hierarchy, which generalises the one-period framework of Sung (2015). The hierarchy is modelled by a series of interlinked principal-agent problems, leading to a sequence of Stackelberg equilibria. More precisely, the principal (she) can contract with a manager (he), to incentivise him to act in her best interest, despite only observing the net benefits of the total hierarchy. The manager in turn subcontracts the agents below him. We will see through a simple example that, while the agents only control the drift of their outcome, the manager controls the volatility of the Agents’ continuation utility. Therefore, even this relatively simple introductory example justifies the use of recent results on optimal contracting for drift and volatility control, and therefore the theory on 2BSDEs.

This will lead us to introduce the second problem, namely optimal contracting for demand- response management, which consists in extending the model by Aid, Possamai, and Touzi (2022)  to a mean-field of consumers. More precisely, the principal (an electricity producer, or provider) contracts with a continuum of agents (the consumers), to incentivise them to decrease the mean and the volatility of their energy consumption during high peak demand. In addition, we introduce a common noise, impacting all consumption processes, to take into account the impact of weather conditions on the agents’ electricity consumption. This mean- field framework with common noise leads us to consider a more extensive class of contracts. In particular, we prove that these results can be improved by indexing the contracts on the consumption of one agent and aggregate consumption statistics from the distribution of the entire population of consumers.

2/2/23

Renyuan Xu (USC)

Title: Asymptotic Analysis of Deep Residual Networks and Global Convergence of Gradient Descent Methods

Abstract: Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, we prove the existence of an alternative ODE limit, a stochastic differential equation, or neither of these. For each case, we also derive the limit of the backpropagation dynamics and address its adaptiveness issue. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

When the gradient descent method is applied to the training of ResNets, we prove that it converges linearly to a global minimum if the network is sufficiently deep and the initialization is sufficiently small. In addition, the global minimum found by the gradient descent method has finite quadratic variation without using any regularization in the training. This confirms existing empirical results that the gradient descent method enjoys an implicit regularization property and is capable of generalizing to unseen data.

This is based on several papers with Rama Cont (Oxford), Alain Rossier (Oxford), and Alain-Sam Cohen (InstaDeep).

2/9/23

Samuel Cohen (Oxford)

Title: Neural Q-learning solutions to elliptic PDEs

Abstract: Solving high-dimensional partial differential equations (PDEs) is a major challenge in scientific computing. We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning. Using a neural tangent kernel (NTK) approach, we prove that the neural network approximator for the PDE solution, trained with the Q-PDE algorithm, converges to the trajectory of an infinite-dimensional ordinary differential equation (ODE) as the number of hidden units becomes infinite. For monotone PDE (i.e. those given by monotone operators), despite the lack of a spectral gap in the NTK, we then prove that the limit neural network, which satisfies the infinite-dimensional ODE, converges in $L^2$ to the PDE solution as the training time increases. The numerical performance of the Q-PDE algorithm is studied for several elliptic PDEs. Based on joint work with Deqing Jiang and Justin Sirignano.

2/16/23

Ulrich Horst (HU Berlin)

Title: Mean-Field Liquidation Games with Market Drop-out

Abstract: We consider a novel class of portfolio liquidation games with market drop-out (``absorption''). More precisely, we consider mean-field and finite player liquidation games where a player drops out of the market once his position hits zero. In particular round-trips are not admissible. This can be viewed as a no statistical arbitrage condition on trading. In a model with only sellers we prove that the absorption condition is equivalent to a short selling constraint. We prove that equilibria (both in the mean-field and the finite player game) are given as solutions to integro differential equations with endogenous terminal conditions. We prove the existence of a unique solution to the differential equation from which we obtain the existence of an equilibrium in the MFG and the existence of a unique equilibrium in the $N$-player game. We establish the convergence of the equilbria in the finite player games to the obtained mean-field equilibrium and illustrate the impact of the drop-out constraint on equilibrium trading rates. 

The talk is based on joint work with Guanxing Fu and Paul Hager.

2/23/23

3/2/23

Melih Iseri (USC)

Title: Set Valued HJB Equations

Abstract:
In this talk, we introduce a notion of set valued PDEs. The set values have been introduced for many applications, such as time inconsistent stochastic optimization problems, multivariate dynamic risk measures, and nonzero sum games with multiple equilibria. One crucial property they enjoy is the dynamic programming principle (DPP). Together with the set valued Itô formula, which is a key component, DPP induces the PDE. In the context of multivariate optimization problems, we introduce the set valued Hamilton-Jacobi-Bellman equations and established its wellposedness. In the standard scalar case, our set valued PDE reduces back to the standard HJB equation. 

3/9/23

Per Mykland (U Chicago)

Title: Nonparametric Observed Standard Errors for High Frequency Data

Abstract: High frequency financial data has become an essential component of the digital world, giving rise to an increasing number of estimators. However, it is hard to reliably assess the uncertainty of such estimators. The Observed Asymptotic Variance (observed AVAR) is a non-parametric (squared) standard error for high- frequency-based estimators. We have earlier developed such an AVAR with time-discretization and two tuning parameters (per dimension). The current paper shows that these two parameters are confounded, and one can move to a single tuning parameter. This is shown by passing to continuous time (which is natural since observations are usually irregularly spaced). We show that the new time-continuous observed AVAR is a limit of the original observed AVAR. We also obtain a central limit theory for the new time-continuous observed AVAR, and the latter permits a sharper definition of our standard error. The device is related to observed information in likelihood theory, but in this case it is non-parametric and uses the high-frequency data structure. [With Lan Zhang, University of Illinois at Chicago.]

3/16/23

3/23/23

4/6/23

4/13/23

4/20/23

4/27/23