The price impact of large orders si wel known ot be a concave function of trade size. We discuss how ot extend models consistent with this "square-root law" to multivariate setings with cross impact, where trading each asset also impacts the prices of the others. nI this context, we derive consistency conditions that rule out price manipulation. These minimal conditions make risk-neutral trading problems tractable and also naturally lead ot parsimonious specifications that can be calibrated ot historical data. We ilustrate this with a case study using proprietary CFM meta order data.

(Joint work ni progress with Natascha Hey and lacopo Mastromateo)

## Concave Cross Impact

## Portfolio optimization under transaction costs with recursive preferences

The solution to the investment-consumption problem ni a frictionless Black-Scholes market for an investor with additive CRRA preferences is to keep a constant fraction of wealth ni the risky asset. But this requires continuous adjustment of the portfolio and as soon as transaction costs are added, any attempt to folow the frictionless strategy wil lead to immediate bankruptcy. Instead as many authors have proposed the optimal solution si to keep the pair (cash, value of risky assets) ni a no-transaction (NT) wedge.

We return ot this problem ot see what we can say about: When si the problem well-posed? Where does the NT wedge lie? How do the results change fi we use recursive preferences? We introduce the shadow fraction of wealth and show how we can make significant progress towards the solution yb focussing on this quantity. Indeed many of the qualitative features of the solution can described by looking at a quadratic whose parameters depend on the parameters of the problem.

This is joint work with Martin Herdegen and Alex Tse.

## Lifetime Investment and Consumption with Epstein-Zin Stochastic Differential Utility

The Merton problem about how to invest and consume optimally over the infi- nite horizon is a classical problem in both finance and stochastic control. But, the conclusions do not always match observed behaviour and this has led economists to generalise the set-up. One such generalisation is to assume preferences are described by stochastic differential utility (SDU).

The problem under SDU can be recast as a problem about a Backward Stochas- tic Differential Equation over the infinite horizon. So we ask, when does this formulation make sense? When does there exist a solution to the BSDE? When is the solution unique? Interestingly, the answer to these questions is not always "Yes", and in the "No" cases we have to decide how to proceed.

In the talk I will discuss some of these issues and suggest how to resolve them. Joint work with Martin Herdegen and Joe Jerome.

## 7th Berlin Workshop on Mathematical Finance for Young Researchers

The 7th Berlin Workshop on Mathematical Finance for Young Researchers provides a forum for PhD students, postdoctoral researchers, and young faculty members from all over the world to discuss their research in an informal atmosphere. Keynote lectures will be given by

- Sara Biagini (Rome)
- Luciano Campi (Milano)
- Giorgio Ferrari (Bielefeld)
- Mete H. Soner (Princeton)
- Luitgard Veraart (London)

We also invite up to 20 contributed talks from young researchers. The deadline for abstract submission is May 20. Notification of acceptance will be sent by May 31. Accommodation for speakers will be arranged.

Limited support for travel expenses may be available upon request. Here a link to the workshop webpage.

## Risk Mitigation - Climate, Energy and Finance

This workshop brings together mathematicians and economists to discuss recent developments in the field of Risk Mitigation with a particular focus on applications to climate, energy and financial risk. The workshop is part of a conference series initiated by the Editors-in-Chief of the Springer published journal Mathematics and Financial Economics to promote the interaction between mathematics, economics and finance. Previous workshops focussed on Knightian Uncertain in Financial Markets, Mathematics of Behavioral Economics, and Many-Player Games and Applications.

The workshop will be held on September 2nd and 3rd, 2024 at the Humboldt University Berlin. It is sponsored by Springer Verlag, the Collaborative Research Center 1283(Bielefeld), the Collaborative Research Center 190 (Berlin, Munich) and the Berlin-Oxford International Research Training Group IRTG 2544. The workshop is jointly organized by the chair Applied Financial Mathematics at Humboldt University Berlin and the Center for Mathematical Economics at Bielefeld University.

## Consensus-based optimization for equilibrium points of games

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.

## 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.