In this talk, we will introduce Consensus-Based Optimization (CBO) for min-max problems, a novel multi-particle, derivative-free optimization method that can provably identify global equilibrium points. This paradigm facilitates the transition to the mean-field limit, making the method amenable to theoretical analysis and providing rigorous convergence guarantees under reasonable assumptions about the initialization and the objective function, including nonconvex-nonconcave objectives. Additionally, numerical evidence will be presented to demonstrate the algorithm's effectiveness. This talk is based on joint works with Giacomo Borghi, Enis Chenchene, Hui Huang, and Konstantin Riedl.

## Consensus-based optimization for equilibrium points of games

## Linear-quadratic stochastic control with state constraints on finite-time horizon

We obtain a probabilistic solution to linear-quadratic optimal control problems with state constraints. Given a closed set $\mathcal D\subseteq [0,T]\times\mathbb R^d$, a diffusion $X$ in $\mathbb R^d$ must be linearly controlled in order to keep the time-space process $(t,X_t)$ inside the set $\mathcal $mathcal C:=([0,T]\times\mathbb R^d)\setminus\mathcal D$, while at the same time minimising an expected cost that depends on the state $(t,X_t)$ and it is quadratic in the speed of the control exerted. We find an explicit probabilistic representation for the value function and the optimal control under a set of mild sufficient conditions concerning the coefficients of the underlying dynamics and the regularity of the set $\mathcal C$. Fully explicit formulae are presented in some relevant examples.

(Joint work with Erik Ekstr\"om, University of Uppsala, Sweden)

## Interacting Particle Systems for Optimization: from Particle Swarm Optimization to Consensus-based Optimization

In this talk, we delve into the application of metaheuristics via extensive systems of interacting particles to tackle complex optimization problems, starting from the Particle Swarm Optimization (PSO) method. This technique leverages collective intelligence, where individual particles adapt their trajectories based on their own success and the influence of their neighbors, directing the swarm toward the optimal solution. We will investigate the continuous model proposed by Grassi and Pareschi, providing evidence of its convergence to global minimizers and illustrating its relationship to Consensus-Based Optimization (CBO) in the limit of zero inertia. The talk is based on joint works with Cristina Cipriani and Hui Huang.

## Developments in Computational Finance and Stochastic Numerics

## Convexity propagation and convex ordering of one-dimensional stochastic differential equations

We consider driftless one-dimensional stochastic differential equations. We first recall how they propagate convexity at the level of single marginals. We show that some spatial convexity of the diffusion coefficient is needed to obtain more general convexity propagation and obtain functional convexity propagation under a slight reinforcement of this necessary condition. Such conditions are not needed for directional convexity. This is a joint work with Gilles Pages.

## Vulnerable European and American Options in a Market Model with Optional Hazard

We study the upper and lower bounds for prices of European and American style options with the possibility of an external termination, meaning that the contract may be terminated at some random time. Under the assumption that the underlying market model is incomplete and frictionless, we obtain duality results linking the upper price of a vulnerable European option with the price of an American option whose exercise times are constrained to times at which the external termination can happen with a non-zero probability. Similarly, the upper and lower prices for a vulnerable American option are linked to the price of an American option and a game option, respectively. In particular, the minimizer of the game option is only allowed to stop at times which the external termination may occur with a non-zero probability.

## An infinite-dimensional price impact model

In this talk, we introduce an infinite-dimensional price impact process as a kind of Markovian lift of non-Markovian 1-dimensional price impact processes with completely monotone decay kernels. In an additive price impact scenario, the related optimal control problem is extended and transformed into a linear-quadratic framework. The optimal strategy is characterized by an operator-valued Riccati equation and a linear backward stochastic evolution equation (BSEE). By incorporating stochastic in-flow, the BSEE is simplified into an infinite-dimensional ODE. With appropriate penalizations, the well-posedness of the Riccati equation is well-known.

This is a joint work with Prof. Dirk Becherer and Prof. Christoph Reisinger.

## Stochastic Fredholm equations: a passe-partout for propagator models with cross-impact, constraints and mean-field interactions.

We will provide explicit solutions to certain systems linear stochastic Fredholm equations. We will then show the versatility of these equations for solving various optimal trading problems with transient impact including: (i) cross-impact (multiple assets), (ii) constraints on the inventory and trading speeds, and (iii) N-player game and mean-field interactions (multiple traders).

Based on joint works with Nathan De Carvalho, Eyal Neuman, Huyˆen Pham, Sturmius Tuschmann, and Moritz Voss.

## Some path-dependent processes from signatures

We provide explicit series expansions to certain stochastic path-dependent in- tegral equations in terms of the path signature of the time augmented driving Brownian motion. Our framework encompasses a large class of stochastic linear Volterra and delay equations and in particular the fractional Brownian motion with a Hurst index H in (0, 1).

Our expressions allow to disentangle an infinite dimensional Markovian struc- ture. In addition they open the door to: (i) straightforward and simple approxima- tion schemes that we illustrate numerically, (ii) representations of certain Fourier- Laplace transforms in terms of a non-standard infinite dimensional Riccati equa- tion with important applications for pricing and hedging in quantitative finance.

Based on joint works with Louis-Amand Gérard and Yuxing Huang.

## Solving probability measure uncertainty by nonlinear expectations

In 1921, economist Frank Knight published his famous ”Uncertainty, Risk and Profit”in which his challenging is still largely open. In this talk we explain why nonlinear expectation theory provides a powerful and fundamentally important mathematical tool to this century problem.