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.

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

## 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.

## General Equilibrium with Unhedgeable Fundamentals and Heterogeneous Agents

We examine the implications of unhedgeable fundamental risk, combined with agents’ hete- rogeneous preferences and wealth allocations, on dynamic asset pricing and portfolio choice. We solve in closed form a continuous-time general equilibrium model in which unhedgeable fundamental risk affects aggregate consumption dynamics, rendering the market incomplete. Several long-lived agents with hete- rogeneous risk-aversion and time-preference make consumption and investment decisions, trading risky assets and borrowing from and lending to each other. We find that a representative agent does not exist. Agents trade assets dynamically. Their consumption rates depend on the history of unhedgeable shocks. Consumption volatility is higher for agents with preferences and wealth allocations deviating more from the average. Unhedgeable risk reduces the equilibrium interest rate only through agents’ heterogeneity and proportionally to the cross-sectional variance of agents’ preferences and allocations.

## Space-time white noises in a nonlinear expectation space

Under the framework of nonlinear expectation, we introduce a new type of random fields, which contains a type of space-time white noise as a special case. Based on this result, we also introduce a space white noise. Different from the case of linear expectation, in which the probability measure is given and fixed.

Under the uncertainty of probability measures, space white noises are intrinsi- cally different from the space cases, which is generalized from G-Gaussian processes which are different from a G-Brownian motion (joint work with Xiaojun JI).

## Extreme value theory in the insurance sector

## Sample Duality

Heuristically, two processes are dual if one can find a function to study one process by using the other. Sampling duality is a duality which uses a duality function S(n,x) of the form "what is the probability that all the members of a sample of size n are of a certain type, given that the number (or frequency) of that type of individuals is x".

Implicitly, this technique can be traced back to the work of Blaise Pascal. Explic- itly, it was studied in a paper of Martin Möhle in 1999 in the context of population genetics. We will discuss examples for which this technique proves to be useful, including applications to the Simple Exclusion Process and (work in progress) a universality result for the FKPP equation.

## Reduced-form framework and affine processes with jumps under model uncertainty

We introduce a sublinear conditional operator with respect to a family of possibly non- dominated probability measures in presence of multiple ordered default times. In this way we generalize the results in [3] where a consistent reduced-form framework under model uncertainty for a single default is developed. Moreover, we present a probabilistic construction Rd-valued non-linear affine processes with jumps, which allows to model intensities in a reduced-form framework. This yields a tractable model for Knightian uncertainty for which the sublinear expectation of a Markovian functional can be calculated via a partial integro-differential equation. This talk is based on [1] and [2].

[1] Francesca Biagini, Georg Bollweg, and Katharina Oberpriller. Non-linear affine processes with jumps. Probability, Uncertainty and Quantitative Risk, 8(3):235–266, 2023.

[2] Francesca Biagini, Andrea Mazzon, and Katharina Oberpriller. Reduced-form framework for multiple default times under model uncertainty. Stochastic Processes and Their Applications, 156:1–43, 2023.

[3] Francesca Biagini and Yinglin Zhang. Reduced-form framework under model uncertainty. The Annals of Applied Probability, 29(4):2481–2522, 2019