Recent advances in computing power and the potential to make more realistic assumptions due to increased flexibility have led to the increased prevalence of simulation models in economics. While models of this class, and particularly agent-based models, are able to replicate a number of empirically-observed stylised facts not easily recovered by more traditional alternatives, such models remain notoriously difficult to estimate due to their lack of tractable likelihood functions. While the estimation literature continues to grow, existing attempts have approached the problem primarily from a frequentist perspective, with the Bayesian estimation literature remaining comparatively less developed. For this reason, we introduce a widely-applicable Bayesian estimation protocol that makes use of deep neural networks to construct an approximation to the likelihood, which we then benchmark against a prominent alternative from the existing literature.
 

TBC
 

In this talk, we will 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 modeled 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. Both agents and the manager each independently control a stochastic process representing their outcome. We will see through a simple example that even if the agents only control the drift of their outcome, the manager controls the volatility of the Agents’ continuation utility. Even this first simple example justifies the use of recent results on optimal contracting for drift and volatility control, and therefore the theory on 2BSDEs. We will also discuss some possible extensions of this model. In particular, one extension consists in the elaboration of more general contracts, indexing the compensation of one worker on the result of the others. This increase in the complexity of contracts is beneficial for the principal, and constitutes a first approach to even more complex contracts, in the case, for example, of a continuum of workers with mean–field interactions. This will lead us to introduce the second problem, namely optimal contracting for demand–response management, which consists in extending the model by Aïd, Possamaï, and Touzi (2019) to a mean–field of consumers. Finally, we will conclude by mentioning that this principal-agent approach with a multitude of agents can be used to address many situations, for example to model incentives for
lockdown in the current epidemic context.
 

We explore reinforcement learning methods for finding the optimal policy in the linear quadratic regulator (LQR) problem. In particular, we consider the convergence of policy gradient methods in the setting of known and unknown parameters. We are able to produce a global linear convergence guarantee for this approach in the setting of finite time horizon and stochastic state dynamics under weak assumptions. The convergence of a projected policy gradient method is also established in order to handle problems with constraints. We illustrate the performance of the algorithm with two examples. The first example is the optimal liquidation of a holding in an asset. We show results for the case where we assume a model for the underlying dynamics and where we apply the method to the data directly. The empirical evidence suggests that the policy gradient method can learn the global optimal solution for a larger class of stochastic systems containing the LQR framework and that it is more robust with respect to model mis-specification when compared to a model-based approach. The second example is an LQR system in a higher-dimensional setting with synthetic data.

It is well known that expected signatures can be used as the “moments” of the law of stochastic processes. Inspired by this fact, we introduced higher rank expected signatures to capture the essences of the weak topologies of adapted processes, and characterize the information evolution pattern associated with stochastic processes. This approach provides an alternative perspective on a recent important work by Backhoff–Veraguas, Bartl, Beiglbock and Eder regarding adapted topologies and causal Wasserstein metrics.

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The dimer model, a classical model of statistical mechanics, is the uniform distribution on perfect matchings of a graph. In two dimensions, one can define an associated height function which turns the model into a random surface (with specified boundary conditions). In the 1960s, Kasteleyn and Temperley/Fisher found an exact "solution" to the model, computing the correlations in terms of a matrix called the Kasteleyn matrix. This exact solvability was the starting point for the breakthrough work of Kenyon (2000) who proved that the centred height function converges to the Dirichlet (or zero boundary conditions) Gaussian free field. This was the first proof of conformal invariance in statistical mechanics.

In this talk, I will focus on a natural modification of the model where one allows the vertices on the boundary of the graph to remain unmatched: this is the so-called monomer-dimer model, or dimer model with free boundary conditions. The main result that we obtain is that the scaling limit of the height function of the monomer-dimer model in the upper half-plane is the Neumann (or free boundary conditions) Gaussian free field. Key to this result is a somewhat miraculous random walk representation for the inverse Kasteleyn matrix, which I hope to discuss.

Joint work with Marcin Lis (Vienna) and Wei Qian (Paris).

It is a basic fact of convexity that the volume of convex bodies is a polynomial, whose coefficients contain many familiar geometric parameters as special cases. A fundamental result of convex geometry, the Alexandrov-Fenchel inequality, states that these coefficients are log-concave. This proves to have striking connections with other areas of mathematics: for example, the appearance of log-concave sequences in many combinatorial problems may be understood as a consequence of the Alexandrov-Fenchel inequality and its algebraic analogues.

There is a long-standing problem surrounding the Alexandrov-Fenchel inequality that has remained open since the original works of Minkowski (1903) and Alexandrov (1937): in what cases is equality attained? In convexity, this question corresponds to the solution of certain unusual isoperimetric problems, whose extremal bodies turn out to be numerous and strikingly bizarre. In combinatorics, an answer to this question would provide nontrivial information on the type of log-concave sequences that can arise in combinatorial applications. In recent work with Y. Shenfeld, we succeeded to settle the equality cases completely in the setting of convex polytopes. I will aim to describe this result, and to illustrate its potential combinatorial implications through a question of Stanley on the combinatorics of partially ordered sets.