Probability

We establish sharp upper and lower bounds for distortion riskmetrics under distributional uncertainty. The uncertainty sets are  characterized by four key features of the underlying distribution: mean, variance, unimodality, and Wasserstein distance to a reference distribution. We first examine very general distortion riskmetrics, assuming only finite variation for the underlying distortion function and without requiring continuity or monotonicity. This broad framework includes notable distortion riskmetrics such as range value-at-risk, glue value-at-risk, Gini deviation, mean-median deviation and inter-quantile difference. In this setting, when the uncertainty set is characterized by a fixed mean, variance and a Wasserstein distance, we determine both the worst- and best-case values of a given distortion risk metric and identify the corresponding extremal distribution. When the uncertainty set is further constrained by unimodality with a fixed reflection point, we establish for the case of absolutely continuous distortion functions the extremal values
along with their respective extremal distributions.

We apply our results to robust portfolio optimization and model risk assessment offering improved decision-making under model uncertainty.

(This talk is based on a joint work with Steven Vanduffel and Yi Xia).

We study how to construct a stochastic process on a finite interval with given `roughness'. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of Hölder regularity of a function in terms of its Schauder coefficients. Using this characterization, we provide a better (pathwise) estimator of Hölder exponent. Furthermore, we study the concept of (generalized) p-th variation of a real-valued continuous function along a sequence of partitions. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of ℓp-norm of the coefficients along a Schauder basis. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.

Finding a robust representation of the conditional distribution of a signal given a noisy observation is a classical problem in stochastic filtering. Such representations are of interest as they justify the use of discrete observation data and ensure robustness of the signal approximation to slight model misspecification.
When the signal and observation are correlated through their noise, Crisan, Diehl, Friz, and Oberhauser (2013) showed that such a robust representation typically cannot exist as a functional on the space of continuous paths, but must instead be formulated on the space of geometric rough paths.
In this talk, I will discuss how to extend these results to stochastic filtering problems involving correlated multidimensional jump diffusions, using the theory of rough stochastic differential equations (RSDEs) with jumps. Specifically, I will discuss the consistency of (randomised) RSDEs with their purely stochastic counterparts, as well as exponential moment bounds provided by a version of the John–Nirenberg inequality for BMO processes with jumps, as first introduced by Lê in 2022.
Building on these results, I will then address the existence of a robust representation of the conditional distribution in a filtering model with correlation in both the continuous and jump noise.
This is an ongoing work with Andrew Allan and Josef Teichmann.

Importance sampling (IS) is an elegant, theoretically sound, flexible, and simple-to-understand methodology for approximation of intractable integrals and probability distributions. The only requirement is the point-wise evaluation of the targeted distribution. The basic mechanism of IS consists of (a) drawing samples from simple proposal densities, (b) weighting the samples by accounting for the mismatch between the targeted and the proposal densities, and (c) approximating the moments of interest with the weighted samples. The performance of IS methods directly depends on the choice of the proposal functions. For that reason, the proposals have to be updated and improved with iterations so that samples are generated in regions of interest. In this talk, we will first introduce the basics of IS and multiple IS (MIS), motivating the need to use several proposal densities. Then, the focus will be on motivating the use of adaptive IS (AIS) algorithms, describing an encompassing framework of recent methods in the current literature. Finally, we review the problem of combining Monte Carlo estimators in the context of MIS and AIS.

We study the probability that an AR(1) Markov chain $X_{n+1}=aX_n+\xi_{n+1}$, where $a$ is a constant, stays non-negative for a long time. Assuming that the i.i.d. innovations $\xi_n$ take only two values $\pm 1$ and $a \le \frac23$, we find the exact asymptotics of this probability and the weak limit of $X_n$ conditioned to stay non-negative. This limiting distribution is quasi-stationary. It has no atoms and is singular with respect to the Lebesgue measure when $\frac12< a \le \frac23$, except for the case $a=\frac23$ and $P(\xi_n=1)=\frac12$, where this distribution is uniform on the interval $[0,3]$. These properties are similar to those of the Bernoulli convolutions. To solve our problem, we employ a dynamical system defined by a certain linear mod 1 transform. Such mappings are well studied due to their use in expansions of numbers in non-integer bases, the so-called generalised $\beta$-expansions. This is a joint work with V. Wachtel.

In this talk, I will discuss the ergodicity of the stochastic heat equation driven by centred Gaussian noise, which is white in time and coloured in space, satisfying the Dalang condition. I will also provide a sufficient condition for the ergodicity, and classify the invariant measures based on their expectations. Assuming the spatial correlation has a Riesz-type tail of the form $|x|^{- \gamma}$, a Gaussian fluctuation result under diffusive scaling was established. In the case of a heavy tail, specifically when $\gamma < d$, the diffusive scaling limit satisfies an Edwards–Wilkinson equation. This talk is based on joint work with Le Chen, Alex Dunlap, Cheng Ouyang, and Samy Tindel.

In this talk, I will show how the theory of rough stochastic differential equations (rough SDEs) — introduced by Friz, Hocquet, and Lê in 2021 — helps to establish the existence, uniqueness, or smoothness of solutions to certain rough partial differential equations (rough PDEs).

A key motivation comes from stochastic filtering, where the Zakai equation, an SPDE describing the unnormalized conditional density, can be reformulated as a rough PDE using rough path theory.

I will present results from [1], where we develop a solution theory for linear rough PDEs and derive a Feynman–Kac-type representation via rough SDEs. If time permits, I will briefly discuss how we extend Hörmander’s theory to the rough setting in [2] using Malliavin calculus.



[1] F.B., Peter K. Friz, Wilhelm Stannat, Parameter dependent rough SDEs with applications to rough PDEs, 2024 (arXiv:2409.11330)

[2] F.B., Michele Coghi, Torstein K. Nilssen,  Malliavin calculus for rough stochastic differential equations, 2024 (arXiv:2402.12056)

In recent years, many models in mathematical physics have been encoded into graphical models, which are more accessible through the lens of probability theory. These graphical models often exhibit a natural percolation structure. One such model is the Random Loop Model introduced by Daniel Ueltschi. Peter Mühlbacher showed that the loop threshold for the Random Loop Model with θ=1 is larger than the percolation threshold. This is due to so-called blocking events in graphs with uniformly bounded degree. The proof primarily relies on a coupling method.

In my talk, I will introduce the model and the basic proof techniques. Furthermore, I will discuss a recent result where we generalize the method to obtain new results for general trees.

I will explain why the tree case differs from the case of a general graph. If time permits, I will use the Galton-Watson case to illustrate how the coupling in the proof works.

This talk is based on joint work with V. Betz, M. Kraft, B. Lees and C. Mönch