Financial markets demand models that are both theoretically sound and operationally viable. This talk offers a practitioner’s perspective on the evolution and application of risk factor modeling—starting fromsimple flat volatility assumptions and progressing toward sophisticated jump-diffusion and pure jump models. We explore how these models are used across asset classes such as equities, FX, commodities, and electricity, where phenomena like fat tails, pricejumps, and regime shifts require careful treatment. Drawing on real-world experience, we examine the challenges of model calibration—highlighting issues such as parameter instability and overfitting. On the methodological front, we discuss numerical techniquesincluding Monte Carlo and Quasi Monte Carlo simulation, as well as methods based on partial (integro-)differential equations. Each comes with trade-offs in terms of performance, stability, and suitability for specific problem classes. Finally we address applicationsin valuation, scenario analysis and risk management.

In this talk, we study the existence and structure of monetary risk measures consistent with fractional stochastic orders, a refinement of second-order stochastic dominance defined via a threshold utility function $v$, whose absolute risk aversion determines the set of test utilities. Motivated by applications in different risk sharing setups, we derive representation results for this class. However, the deeper issue is whether such risk measures exist at all. Our results indicate that they are either highly constrained or non-existent, except when $v$ is given as exponential utility function. Imposing further properties such as convexity or positive homogeneity makes existence even less likely. In addition, we outline implications for practical risk management.

In this talk, we present a novel approach to pricing American options using piecewise diffusion Markov processes (PDifMPs), a generalised stochastic hybrid system combining continuous dynamics with discrete jumps. Standard models often assume constant drift and volatility, limiting their ability to capture the erratic nature of financial markets. Our method leverages PDifMPs to incorporate sudden market fluctuations, offering a more realistic asset price model. We benchmark our approach against the Longstaff-Schwartz algorithm, including its modified version with PDifMP-based asset price trajectories. Numerical simulations show that the PDifMP method reflects market behaviour more accurately while enhancing computational efficiency. The results suggest that PDifMPs significantly improve American options pricing by aligning with real-world stochastic volatility and market jumps, providing a more predictive and efficient approach.

We consider an insurance company whose surplus follows an Ornstein-Uhlenbeck (OU) process driven by a standard Brownian motion. The company pays dividends to its shareholders and seeks to maximise the expected value of the future discounted dividends. Late dividend payments are penalised/rewarded not only through the usual discounting, but through an additional exponential factor. We find the optimal strategy for the case of mean-reverting and non-mean-reverting OU processes and illustrate our findings by a numerical example. References: [1] S. Asmussen and M. Taksar: Controlled diffusion models for optimal dividend pay-out, Insurance: Math- ematics and Economics, (1997), 20(1):1–15. [2] B. Avanzi and B. Wong: On a mean reverting dividend strategy with Brownian motion, Insurance: Mathematics and Economics, (2012), 51(2):229–238. [3] A. N. Borodin and P. Salminen: Handbook of Brownian motion-facts and formulae, Birkh¨auser, (2012). [4] J. Eisenberg: Unrestricted consumption under a deterministic wealth and an Ornstein–Uhlenbeck process as a discount rate, Stochastic models, (2018), 34(2):139–153. [5] F. Locas and J.-F. Renaud: De Finetti’s control problem with a concave bound on the control rate, Journal of Applied Probability, (2024), pages 1–17. [6] S. E. Shreve, J. P. Lehoczky, and D. P. Gaver: Optimal consumption for general diffusions with absorbing and reflecting barriers, SIAM Journal on Control and Optimization, (1984), 22(1):55–75.

The asymmetric simple exclusion process and stochastic six vertex model are classical interacting particle systems. Their multi-class generalizations are natural generalizations stemming from a shared property called attractivity. We show that all extremal stationary measures of the multi-class stochastic six vertex model (with shift) or of ASEP are given either by a projections of the ASEP speed process or the q-Mallows measure.

We introduce a new framework for the construction, simulation, and statistical analysis of ranked, unlabelled trees—combinatorial structures whose state-space grows over-exponentially (following the Euler zigzag numbers). Building on the F-matrix representation of Salmyak & Palacios (2024) [1], we recast each tree as a sequence of column-vectors that inherit the Markov property, thereby defining a much smaller, tractable state space. Specifically, for a tree with n leaves, the number of distinct column states is Fib(n), and the corresponding transition matrix is a (relatively) sparse Fib(n)×Fib(n) array that admits highly efficient storage and computation. Within this column-state framework, we derive explicit algorithms for a Markov chain that sequentially builds ranked unlabelled trees. Once assembled, these trees support a variety of inferential procedures. We develop hypothesis tests to assess whether observed transition probabilities align with those of the Kingman coalescent, or come from a more complex model, like the Blum-Francois [2]. Moreover, we discuss tree-appropriate summary statistics — namely, the Fr´echet mean and variance on tree space — that succinctly characterize collections of trees. Finally, we present Vitreebi, a dynamic-programming algorithm that computes Frechet means under our metric, providing a principled summary of sample heterogeneity. Our approach dramatically reduces computational complexity, opening the door to scalable simulation and inference on large tree-shaped data without sacrificing mathematical rigor or probabilistic fidelity. References: [1] Samyak, Rajanala, and Julia A. Palacios. ”Statistical summaries of unlabelled evolutionary trees.” Biometrika 111.1 (2024): 171-193. [2] Sainudiin, Raazesh, and Amandine V´eber. ”A Beta-splitting model for evolutionary trees.” Royal Society open science 3.5 (2016): 160016.

Every first year probability student knows that one may shed light on a joint probability law by computing its marginals, but that this does not determine the joint law uniquely. Conversely, one may gain insight about a given probability law by understanding the joint laws that have it as a marginal. Such arguments are generally referred to as coupling methods, and they are of vital importance in the study of various statistical mechanics models such as random walks and the Ising model. As such, it is of inherent interest to simply understand criteria under which couplings exist. In this talk, we will discuss classical couplings between the Ising model and its graphical representations as well as try to give a flavour of the arguments that they help one accomplish. Finally, we will discuss a theorem which has all the classical couplings of the Ising model as special cases, and which yields at least three novel couplings. References: [1] U. T. Hansen, J. Jiang and F. R. Klausen: A General Coupling For Ising Models and Beyond, arXiv preprint arXiv:2506.10765, 2025. 1

Causal transport, together with the associated adapted Wasserstein distance, induces a natural geodesic structure on the space of stochastic processes. In this talk, we present recent advances in the study of the resulting geometry. In particular, we provide a characterization of absolutely continuous curves w.r.t.\ the adapted Wasserstein distance and further elucidate the geometric structure of Gaussian processes in discrete time. This talk is based on joint works with B. Acciaio, D. Bartl, M. Beiglböck, S. Hou, D. Krsek, M. Rodriguez, and S. Schrott.

The notion of spherical $t$-designs, introduced in the seminal paper [1], has attracted interest in various areas of mathematics over the past decades. In this work, we consider spherical $t$-designs as the set of sampling points in a fixed design non-parametric regression on spheres of arbitrary dimension. We show that the fixed design regression experiments defined this way are asymptotically equivalent, in the sense of Le Cam, to a sequence of Gaussian white noise experiments as the sample size tends to infinity. More precisely, asymptotic equivalence is established for Sobolev and Besov function classes on the sphere. These results provide further support for the use of spherical $t$-designs as sampling points in non-parametric regression with spherical regressors. References [1] P. Delsarte, J. M. Goethals, and J. J. Seidel: Spherical codes and designs, \emph{Geometriae Dedicata}, 6 (1977), 3, 363--388.