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
We study various formulation of zero-sum games between a singular-controller and a stopper with a finite-time horizon, where the underlying process is a multi-dimensional controlled stochastic differential equation evolving in an unbounded domain. We prove that such games admit a value and present an optimal strategy for the stopper. In some cases, we show the game's value is the maximal solution, in a suitable Sobolev class, of a variational inequality of 'min-max' type with both obstacle and gradient constraint. Under stricter assumptions, we provide an optimal strategy for the controller and establish a connection between the space derivative of the value function and the solution of an optimal stopping problem with absorption.
