## 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

## Local Volatility Models for Commodity Forwards

We present a dynamic model for forward curves in commodity markets, which is defined as the solution to a stochastic partial differential equation (SPDE) with state-dependent coefficients, taking values in a Hilbert space H of real valued functions. The model can be seen as an infinite dimensional counterpart of the classical local volatility model frequently used in equity markets. We first investigate a class of point-wise operators on H, which we then use to define the coefficients of the SPDE. Next, we derive growth and Lipchitz conditions for coefficients resulting from this class of operators to establish existence and uniqueness of solutions. We also derive conditions that ensure positivity of the entire forward curve. Finally, we study the existence of an equivalent measure under which related traded, 1-dimensional projections of the forward curve are martingales.

Our approach encompasses a wide range of specifications, including a Hilbert-space valued counterpart of a constant elasticity of variance (CEV) model, an exponential model, and a spline specification which can resemble the S shaped local volatility function that well reproduces the volatility smile in equity markets. A particularly pleasant property of our model class is that the one-dimensional projections of the curve can be expressed as one dimensional stochastic differential equation. This provides a link to models for forwards with a fixed delivery time for which formulas and numerical techniques exist. In a first numerical case study we observe that a spline based, S shaped local volatility function can calibrate the volatility surface in electricity markets.

Joint work with Silvia Lavagnini (BI Norwegian Business School)

## A path-dependent PDE solver based on signature kernels

We develop a provably convergent kernel-based solver for path-dependent PDEs (PPDEs). Our numerical scheme leverages signature kernels, a recently introduced class of kernels on path-space. Specifically, we solve an optimal recovery problem by approximating the solution of a PPDE with an element of minimal norm in the signature reproducing kernel Hilbert space (RKHS) constrained to satisfy the PPDE at a finite collection of collocation paths. In the linear case, we show that the optimisation has a unique closed-form solution expressed in terms of signature kernel evaluations at the collocation paths. We prove consistency of the proposed scheme, guaranteeing convergence to the PPDE solution as the number of collocation points increases. Finally, several numerical examples are presented, in particular in the context of option pricing under rough volatility. Our numerical scheme constitutes a valid alternative to the ubiquitous Monte Carlo methods. Joint work with Cristopher Salvi (Imperial College London).

## Polynomial Volterra processes

Recent studies have extended the theory of affine processes to the stochastic Volterra equati- ons framework. In this talk, I will describe how the theory of polynomial processes extends to the Volterra setting. In particular, I will explain the moment formula and an interesting stochastic invariance result in this context. This is joint work with Eduardo Abi Jaber, Christa Cuchiero, Luca Pelizzari and Sara Svaluto-Ferro.

## Geometric properties of some rough curves via dynamical systems: SBR measure, local time and Rademacher chaos

We investigate geometric properties of graphs of Takagi type functions, repre- sented by series based on smooth functions. They are Hölder continuous, and can be embedded into smooth dynamical systems, where their graphs emerge as pull- back attractors. It turns out that occupation measures and Sinai-Bowen-Ruelle (SBR) measures on their stable manifolds are dual by ’time’ reversal.

A suitable version of approximate self-similarity for deterministic functions al- lows us to ’telescope’ small-scale properties from macroscopic ones. As one conse- quence, absolute continuity of the SBR measure is seen to be dual to the existence of local time. The investigation of questions of smoothness both for SBR as for oc- cupation measures surprisingly leads us to the Rademacher version of Malliavin’s calculus, Bernoulli convolutions, and into probabilistic number theory. The link between the rough curves considered and smooth dynamical systems can be gen- eralized in various ways. For instance, Gaussian randomizations of Takagi curves just reproduce the trajectories of Brownian motion. Applications to regularization of singular ODE by rough signals are on our agenda.

## Optimal consumption with labor income and borrowing constraints for recursive preferences

In this talk, we present an optimal consumption and investment problem for an investor with liquidity constraints who has isoelastic recursive Epstein-Zin utility preferences and re- ceives a stochastic stream of income. We characterize the optimal consumption strategy as well as the terminal wealth for recursive utility under dynamic liquidity constraints, which pre- vent the investor to borrow against his stochastic future income. Using duality and backward SDE methods in a possibly non-Markovian diffusion model for the financial market, this gives rise to an interplay of singular control and optimal stopping problems. Our analysis extends to more general liquidity constraints. (Joint work with Dirk Becherer and Olivier Menoukeu Pamen)

## On two Formulations of McKean–Vlasov Control with Killing

We study a McKean–Vlasov control problem with killing and common noise. The particles in this control model live on the real line and are killed at a positive intensity whenever they are in the negative half-line. Accordingly, the interaction between particles occurs through the subprobability distribution of the living particles. We establish the existence of an optimal semiclosed-loop control that only depends on the particles’ location and not their cumulative intensity. This problem cannot be addressed through classical mimicking arguments, because the particles’ subprobability distribution cannot be reconstructed from their location alone. Instead, we represent optimal controls in terms of the solutions to semilinear BSPDEs and show those solutions do not depend on the intensity variable.