Contributed Speakers

Mathias Beiglböck

Ofelia Bonesini

Andrea Bovo

Jodi Dianetti

Berenice Anne Neumann

Benjamin Robinson

Emel Savku

Stefan Schrott

Mathias Beiglböck

Title: The Wasserstein space of stochastic processes

Abstract: Wasserstein distance induces a natural Riemannian structure for the probabilities on the Euclidean space. This insight of classical transport theory is fundamental for tremendous applications in various fields of pure and applied mathematics.

We believe that an appropriate probabilistic variant, the adapted Wasserstein distance AW, can play a similar role for the class FP of filtered processes, i.e. stochastic processes together with a filtration. In contrast to other topologies for stochastic processes, probabilistic operations such as the Doob-decomposition, optimal stopping and stochastic control are continuous w.r.t. AW. We also show that (FP,AW) is a geodesic space, isometric to a classical Wasserstein space, and that martingales form a closed geodesically convex subspace.

Ofelia Bonesini

Title: A McKean-Vlasov game of commodity production, consumption and trading

Abstract: We propose a model where a producer and a consumer can affect the price dynamics of some commodity controlling drift and volatility of, respectively, the production rate and the consumption rate. We assume that the producer has a short position in a forward contract on λ units of the underlying at a fixed price F, while the consumer has the corresponding long position. Moreover, both players are risk-averse with respect to their financial position and their risk aversions are modelled through an integrated-variance penalization. We study the impact of risk aversion on the interaction between the producer and the consumer as well as on the derivative price. In mathematical terms, we are dealing with a two-player linear-quadratic McKean-Vlasov stochastic differential game. Using methods based on the martingale optimality principle and BSDEs, we find a Nash equilibrium and characterize the corresponding strategies and payoffs in semi-explicit form. Furthermore, we compute the two indifference prices (one for the producer and one for the consumer) induced by that equilibrium and we determine the quantity λ such that the players agree on the price. Finally, we illustrate our results with some numerics. In particular, we focus on how the risk aversions and the volatility control costs of the players affect the derivative price.

Andrea Bovo

Title: Variational inequalities on unbounded domains for zero-sum singular-controller vs. stopper games

Abstract: We study a class of zero-sum games between a singular-controller and a stopper over finite-time horizon. The underlying process is a multi-dimensional (locally non-degenerate) controlled stochastic differential equation (SDE) evolving in an unbounded domain. We prove that such games admit a value and provide an optimal strategy for the stopper. The value of the game is shown to be the maximal solution, in a suitable Sobolev class, of a variational inequality of `min-max' type with obstacle constraint and gradient constraint. Although the variational inequality and the game are solved on an unbounded domain we do not require boundedness of either the coefficients of the controlled SDE or of the cost functions in the game.

Jodi Dianetti

Title: Submodular mean field games: Existence and approximation of solutions

Abstract: We study mean field games with scalar Itô-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. Firstly, it allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Secondly, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist a minimal and a maximal solution. Thirdly, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. Our approach also allows to treat submodular mean field games with common noise, as well as mean field games with singular controls, optimal stopping and reflecting boundary conditions.

This talks is based on some joint works together with Giorgio Ferrari, Markus Fischer and Max Nendel.

Matteo Garbelli

Title: Deep Learning and Mean Field Optimal Control from a Stochastic Optimal Control perspective

Abstract: We provide a concrete proposal for the mathematical formulation of Deep Learning (DL) based approaches within the contexts of both Stochastic Optimal Control (SOC) and Mean-Field Control (MFC). Following a dynamical system perspective, we conduct an in-depth analysis of the Supervised Learning (SL) procedures characterizing Neural Networks (NNs) based models, also showing how to translate a SL task into an optimal (stochastic) MFC one. Moreover, we derive two methods within such a Mean Field setting: the first is obtained considering the Hamilton–Jacobi–Bellman (HJB) approach in the Wasserstein space of measures, while the second is based on the MF-Pontryagin maximum principle, providing necessary, weaker, condition the solution must satisfy. We conclude sketching future research directions we are considering.

Maddalena Ghio

Title: Mean-field games of finite-fuel capacity expansion with singular controls: analytical and numerical construction of solutions

Abstract: Joint work with: Tiziano De Angelis (Università degli Studi di Torino), Luciano Campi (Università degli Studi di Milano) and Giulia Livieri (Scuola Normale Superiore).

We study symmetric N-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart.

We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions.

We then show that solutions of the MFG of capacity expansion induce approximate Nash equilibria for the N -player games with approximation error ε going to zero as N tends to infinity.

The constructive proof of the existence result suggests a numerical method to approximate the MFG solution by solving (via Picard iterations) an integral equation for the boundary between the action and the inaction region. We implement the numerical scheme in a relevant example and observe convergence.

Our theoretical analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework.

The added value of our approach is that it results both in an analytical characterization of the solution of the MFG and in a numerical method to actually construct these solutions.

References

Campi, L., De Angelis, T., Ghio, M., & Livieri, G. (2020). Mean-field games of finite-fuel capacity expansion with singular controls. arXiv preprint arXiv:2006.02074.

Hugo Lavenant

Title: Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion

Abstract: Links between optimal transport and Brownian motion are diverse and have turned out to be fruitful. This can be seen in martingale optimal transport, but an other important connection is via the Schrödinger problem which corresponds to entropy minimization with respect to the law of the Brownian motion. In this presentation, I will I will explain what happens in the Schrödinger problem when we replace Brownian motion by branching Brownian motion (that is, when particles may also split or die at random instants): the optimal transport counterpart becomes regularized unbalanced optimal transport, enabling to match distributions of unequal mass.

This is joint work with Aymeric Baradat (Arxiv preprint 2111.01666).

Alessandro Milazzo

Title: A resource extraction problem with possible fraud

Abstract: We consider a resource extraction problem which extends the classical de Finetti dividend problem to include the case when a competitor, who is equipped with the possibility to extract all remaining resources in one piece, may exist; we interpret this unknown competition as the agent being subject to possible fraud. This situation is modelled as a controller-stopper stochastic non-zero-sum game with incomplete information. In order to allow the fraudster to hide his/her existence, we consider strategies where his/her action time is randomised. Under these conditions, we provide a Nash equilibrium which is fully described in terms of the corresponding single-player resource extraction problem. In this equilibrium, the agent and the fraudster use singular strategies in such a way that a two- dimensional process, which represents available resources and the filtering estimate of active competition, reflects at a certain boundary.

This work is based on a joint collaboration with Erik Ekström (Uppsala University) and Marcus Olofsson (Umeå University).

Max Nendel

Title: Wasserstein perturbations of Markovian transition semigroups

Abstract: We deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modeled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup, which solves a Hamilton-Jacobi-Bellman equation in a viscosity sense. As a consequence, in standard situations, the nonlinear transition operators arising from Wasserstein uncertainty coincide with the value function of an optimal control problem. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The talk is based on joint work with Jonas Blessing, Robert Denk, Sven Fuhrmann, and Michael Kupper.

Berenice Anne Neumann

Title: Finite State Mean Field Games with Common Shocks

Abstract: We present a new framework for mean field games with finite state space and common noise, where the common noise is given through shocks that occur at random times. We first analyse the game for up to n shock times in which case we are able to characterize mean field equilibria through a family of parametrized and coupled forward-backward systems and prove existence of solutions to these systems for a small time horizon. Thereafter, we show that for the case of an unbounded number of shocks the equilibria of the game restricted to n shocks are approximate mean field equilibria.

The talk is based on joint work with Frank Seifried.

Benjamin Robinson

Title: A regularized Kellerer theorem in arbitrary dimensions

Abstract: The problem of finding a Markov martingale with prescribed marginals goes back to Strassen in 1965 and Kellerer in 1972. Strassen showed that for a discrete-time peacock (processus croissant pour l’ordre convex) there exists a Markov martingale whose marginals coincide with the given measures. We call such a process a mimicking martingale. Kellerer was the first to prove existence of Markovian mimicking martingales in continuous time in one dimension. The later work of Lowther strengthened Kellerer’s result, showing that there exists a unique strong Markov mimicking martingale for any given peacock in dimension one.

We provide the first known extension of Kellerer’s theorem to arbitrary dimensions. We show that, for a continuous-time peacock, after some Gaussian regularization, there exists a strong Markov mimicking martingale, which is an Itô diffusion. Our construction makes use of the structure of standard stretched Brownian motion in arbitrary dimensions, and the Gaussian regularization allows us to construct diffusions with coefficients that are bounded from above and below. We also provide a counter-example to show that such a mimicking martingale may not be unique in higher dimensions. This is joint work with Gudmund Pammer and Walter Schachermayer.

Emel Savku

Title: A constrained stochastic differential game application: Bancassurance

Abstract: We develop an approach for two player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different type of constraints, which lead us to find real valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for an example of cooperation between a bank and an insurance company, which is a popular, well-known business agreement type, called Bancassurance. By using stochastic maximum principle, we investigated optimal dividend strategy for the company as a best response according to the optimal mean rate of return choice of a bank for its own cash flow and vice versa. We found out a Nash equilibrium for this game and solved the adjoint equations explicitly for each state. It is well known that the timing and the amount of dividend payments are strategic decisions for companies. The announcement of a dividend payment may reduce or increase the stock prices of a company. From the side of the bank, it is clear that creating a cash flow with high returns would be the main goal. Hence, in our formulation, we provide an insight to both of the bank and the insurance company about their best moves in a bancassurance commitment under specified technical conditions.

Reference: E. Savku, A stochastic control approach for constrained stochastic differential games with jump and regimes. Submitted 2022.

Stefan Schrott

Title: The space of stochastic processes in continuous time

Abstract: Researchers from different areas have independently defined extensions of the usual weak topology between laws of stochastic processes. This includes Aldous' extended weak convergence, Hellwig's information topology and convergence in adapted distribution in the sense of Hoover-Keisler. We show that on the set of continuous processes with canonical filtration these topologies coincide and are metrized by a suitable adapted Wasserstein distance AW. Moreover, we show that the resulting topology is the weakest topology that guarantees continuity of optimal stopping.

While the set of processes with natural filtration is not complete, we establish that its completion consists precisely in the space processes with filtration FP.

We also observe that (FP, AW) exhibts several desirable properties. Specifically, (FP, AW) is Polish, Martingales form a closed subset and approximation results like Donsker's theorem extend to AW.

This talk is based on joint work with Daniel Bartl, Mathias Beiglböck, Gudmund Pammer and Xin Zhang.

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