Propagator models describe transient price impact by allowing execution prices to depend on the entire history of past trades through a Volterra kernel. While such models capture empirically observed features of market microstructure, they lead to infinite-dimensional and non-Markovian optimisation problems that fall outside the scope of classical arbitrage and duality theory. In this talk, I develop a comprehensive mathematical finance framework for propagator models, including a fundamental theorem of asset pricing, a superreplication theorem with liquidity-adjusted risk measures, and a convex-duality approach to utility maximisation. Despite non-linear preferences and path-dependent price impact, optimal strategies are obtained by solving a suitably constructed dual problem over a family of fictitious frictionless financial markets.

We introduce a categorical framework for higher-order probability theory based on quasi-M-spaces and Frölicher M-spaces. Besides Giry's and Forré's probability monads, we introduce our own hybrid pushforward probability monad generated by randomizations of a fixed reference measure. These constructions are canonically transported to saturated Frölicher M-spaces, yielding, in the measurable setting, Giry and quasi-probability monads on the corresponding Frölicher spaces. In the smooth setting, the resulting spaces of probability measures carry a natural smooth structure, which can be related to the continuity equation, Otto calculus and Lions' derivative.
This is joint work with Philipp Harms, Peter W. Michor and Josef Teichmann.

We formulate the design of a market stability reserve as a stationary mean-field singular control problem. The allowance balance follows a regime-switching diffusion, while the regulator injects or withdraws allowances through bounded- variation controls. The objective trades off the average allowance balance, its stationary dispersion, deviations from a long-run target, and intervention costs. The optimal policy is a regime-dependent band rule characterized by four reflecting thresholds. Its computation reduces to a scalar fixed-point problem combining a coupled variational inequality with the invariant distribution of the reflected process. A calibration to the EU ETS yields thresholds of the same order of magnitude as the existing Market Stability Reserve rule. Comparative statics show that volatility is the main determinant of the intervention band, whereas the frequency and duration of crisis regimes have limited effects. A simpler two- threshold policy based on average regime dynamics closely reproduces the optimal regime-dependent policy in terms of liquidity and intervention volumes. 

Joint work with Giorgio Ferrari and Federico Cannerozzi (Bielefeld University). 

We develop a general framework for optimization and stochastic control problems on large sparse graphs, where a control is assigned to each vertex and the objective aggregates local cost functionals depending on neighboring vertices. A central question is how such problems behave as the graph grows. It turns out that locally convergent sparse graphs can have divergent optimal values. Instead, we identify local-global convergence as the appropriate notion, with graphings serving as the limiting objects. For a broad class of graph optimization problems, we prove the convergence of optimal values to their graphing counterparts and derive first-order optimality conditions for the limiting problem. Furthermore, we extend the theory to dynamic decentralized stochastic control problems and establish a stochastic maximum principle for the corresponding graphing control problem. This talk is based on a joint work with Eyal Neuman and Sturmius Tuschmann.

Mean field type control problems lead to Hamilton–Jacobi–Bellman (HJB) equations on the space of probability measures. In this talk, we will present a probabilistic approach to constructing classical solutions through forward-backward stochastic differential equations (FBSDEs), allowing for degenerate and unbounded diffusion coefficients. Starting with linear dynamics, we will explain how convexity ensures the global well-posedness of the associated FBSDE system, and how variational equations establish the regularity of the value function needed for the HJB equation. We will then extend the framework to jump-diffusion models, where the associated HJB equation is an integro-partial differential equation. Finally, we will consider the case where a backward equation forms part of the state.

Ballistic deposition is a model of interface growth introduced by Vold in 1959,
which has remained largely mathematically intractable. It is defined as a process
of vertically falling blocks that stick to the top, the right or the left of growing
columns. We introduce a variant, the one-sided ballistic deposition model, in
which vertically falling blocks can only stick to the top or the upper right corner
of growing columns. We establish that in dimension d=2, strong KPZ universality
holds near the time axis, proving that the fluctuations of the height function there
are given by the Tracy-Widom GUE distribution. The proof is based on a graphical
construction of the process in terms of a last passage percolation model. This talk
is based on a joint work with Pablo Groisman, Santiago Saglietti and Sebastián
Zaninovich.

Mean field games (MFGs) and mean field control (MFC) provide a mathemat-
ical framework for modeling large controlled interacting particle systems, with
applications ranging from financial mathematics to biological neural networks. In
the first part of this talk, I will give an accessible introduction to MFGs and MFC.
I will then show how the infinite-dimensional case arises naturally in applications,
highlight its specific analytical challenges, and provide an overview of my recent
results.
In the second part of the talk, I will detail how we address these challenges in
recent joint work with A. Święch, where we investigate an MFG posed in a Hilbert
space and driven by degenerate noise. We study the associated MFG system
consisting of a Hamilton–Jacobi–Bellman (HJB) equation for the value function
coupled with a nonlinear Fokker–Planck (FP) equation for the distribution of the
particles, both governed by a degenerate Kolmogorov operator. This degeneracy
introduces significant analytical challenges. In particular, the HJB equation is
treated in the viscosity sense, while the FP equation is interpreted in a suitable
weak formulation. Proving the uniqueness of solutions to the FP equation becomes
particularly delicate. Under appropriate structural assumptions, we establish the
well-posedness of the MFG system.