Past Stochastic Analysis & Mathematical Finance Seminars

In this talk, we consider Dirichlet forms related to stochastic quantisation for the \(exp(φ)_{2}\)-model on the torus. We show strong uniqueness of the corresponding Dirichlet operators by applying an idea of (singular) SPDEs. This talk is based on ongoing joint work with Hirotatsu Nagoji (Kyoto University).

When and how well can a high-dimensional system of stochastic differential equations (SDEs) be approximated by one with independent coordinates? This fundamental question is at the heart of the theory of mean field limits and the propagation of chaos phenomenon, which arise in the study of large (many-body) systems of interacting particles. This talk will present recent sharp quantitative answers to this question, both for classical mean field models and for more recently studied non-exchangeable models. Two high-level ideas underlie these answers. The first is a simple non-asymptotic construction, called the independent projection, which is a natural way to approximate a general SDE system by one with independent coordinates. The second is a "local" perspective, in which low-dimensional marginals are estimated iteratively by adding one coordinate at a time, leading to surprising improvements on prior results obtained by "global" arguments such as subadditivity inequalities. In the non-exchangeable setting, we exploit a surprising connection with first-passage percolation.

Spin glasses are models of statistical mechanics in which a large number of elementary units interact with each other in a disordered manner. In the simplest case, there are direct interactions between any two units in the system, and I will start by reviewing some of the key mathematical results in this context. For modelling purposes, it is also desirable to consider models with more structure, such as when the units are split into two groups, and the interactions only go from one group to the other one. I will then discuss some of the technical challenges that arise in this case, as well as recent progress.

We provide sharp bounds in the supremum- and Hölder norm for parameter-dependent stochastic integrals. As an application we obtain novel long-term bounds for stochastic partial differential equations as well as novel bounds on the space-time modulus of continuity of the stochastic heat equation. This concerns joint work with Joris van Winden (TU Delft).

I will discuss  the well-posedness of a class of stochastic second-order in time-damped evolution equations in Hilbert spaces, subject to the constraint that the solution lies on  the unit sphere. A specific example is provided by  the stochastic damped wave equation in a bounded domain of a $d$-dimensional Euclidean space, endowed with the Dirichlet boundary conditions, with the added constraint that the $L^2$-norm of the solution is equal to one. We introduce a small mass $\mu>0$ in front of the second-order derivative in time and examine the validity of the Smoluchowski-Kramers diffusion approximation. We demonstrate that, in the small mass limit, the solution converges to the solution of a stochastic parabolic equation subject to the same constraint. We further show that an extra noise-induced drift emerges, which  in fact does not account for the Stratonovich-to-It\^{o} correction term. This talk is based on joint research with S. Cerrai (Maryland), hopefully to be published in Comm Maths Phys.

Global well-posedness of 3D Navier-Stokes equations (NSEs) is one of the biggest open problems in modern mathematics. A long-standing conjecture in stochastic fluid dynamics suggests that physically motivated noise can prevent (potential) blow-up of solutions of the 3D NSEs. This phenomenon is often referred to as `regularization by noise'. In this talk, I will review recent developments on the topic and discuss the solution to this problem in the case of the 3D NSEs with small hyperviscosity, for which the global well-posedness in the deterministic setting remains as open as for the 3D NSEs. An extension of our techniques to the case without hyperviscosity poses new challenges at the intersection of harmonic and stochastic analysis, which, if time permits, will be discussed at the end of the talk.

The construction of the measure of the $\Phi^4_3$ model in the 1970s has been one of the major achievements of constructive quantum field theory. In the 1980s Parisi and Wu suggested an alternative way of constructing quantum field theory measures by viewing them as invariant measures of certain stochastic PDEs. However, the highly singular nature of these equations prevented their application in rigorous constructions until the breakthroughs in the area of singular stochastic PDEs in the past decade. After explaining the basic idea behind stochastic quantization proposed by Parisi and Wu I will show how to apply this technique to construct the measure of a certain quantum field theory model generalizing the $\Phi^4_3$ model called the fractional $\Phi^4$ model. The measure of this model is obtained as a perturbation of the Gaussian measure with covariance given by the inverse of a fractional Laplacian. Since the Gaussian measure is supported in the space of Schwartz distributions and the quartic interaction potential of the model involves pointwise products, to construct the measure it is necessary to solve the so-called renormalization problem. Based on joint work with M. Gubinelli and P. Rinaldi.

The talk will focus on recent developments regarding the (near-)critical behaviour of certain statistical physics models with long-range dependence in dimensions larger than 2, but smaller than 6, above which mean-field behaviour is known to set in. This “intermediate” regime remains a great challenge for mathematicians. The models revolve around a certain percolation phase transition that brings into play very natural probabilistic objects, such as random walk traces and the Gaussian free field. 

The Seventeenth Brooke Benjamin Lecture 2024

The Elusive Singularity

I will describe the open problems of singularity formation in incompressible fluids. I will discuss a list of related models, some results, and some more open problems.

Date: Monday, 11 November 2024 

Time: 5pm GMT

Location: Lecture Theatre 1, Mathematical Institute 

Speaker: Professor Peter Constantin        

More information about The Brooke Benjamin Lecture.

We consider the problem of parametric and non-parametric statistical inference for systems of weakly interacting diffusions and of their mean field limit. We present several parametric inference methodologies, based on stochastic gradient descent in continuous time, spectral methods and the method of moments. We also show how one can perform fully nonparametric Bayesian inference for the mean field McKean-Vlasov PDE. The effect of non-uniqueness of stationary states of the mean field dynamics on the inference problem is elucidated.

The notion of Lip(gamma) Functions, for a parameter gamma > 0, introduced by Stein in the 1970s (building on earlier work of Whitney) is a notion of smoothness that is well-defined on arbitrary closed subsets (including, in particular, finite subsets) that is instrumental in the area of Rough Path Theory initiated by Lyons and central in recent works of Fefferman. Lip(gamma) functions provide a higher order notion of Lipschitz regularity that is well-defined on arbitrary closed subsets, and interacts well with the more classical notion of smoothness on open subsets. In this talk we will survey the historical development of Lip(gamma) functions and illustrate some fundamental properties that make them an attractive class of function to work with from a machine learning perspective. In particular, models learnt within the class of Lip(gamma) functions are well-suited for both inference on new unseen input data, and for allowing cost-effective inference via the use of sparse approximations found via interpolation-based reduction techniques. Parts of this talk will be based upon the works https://arxiv.org/abs/2404.06849 and https://arxiv.org/abs/2406.03232.

We develop a model based on mean-field games of competitive firms producing similar goods according to a standard AK model with a depreciation rate of capital generating pollution as a byproduct. Our analysis focuses on the widely-used cap-and-trade pollution regulation. Under this regulation, firms have the flexibility to respond by implementing pollution abatement, reducing output, and participating in emission trading, while a regulator dynamically allocates emission allowances to each firm. The resulting mean-field game is of linear quadratic type and equivalent to a mean-field type control problem, i.e., it is a potential game. We find explicit solutions to this problem through the solutions to differential equations of Riccati type. Further, we investigate the carbon emission equilibrium price that satisfies the market clearing condition and find a specific form of FBSDE of McKean-Vlasov type with common noise. The solution to this equation provides an approximate equilibrium price. Additionally, we demonstrate that the degree of competition is vital in determining the economic consequences of pollution regulation.

This is based on joint work with Gianmarco Del Sarto and Marta Leocata. 

https://arxiv.org/pdf/2407.12754

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In this talk I will describe a natural generalisation of the Hastings-Levitov family in which the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. In recent joint work with Norris and Silvestri, we show that when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. This phase transition in fluctuations can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks analogous to that present in the Hastings-Levitov family.

Optimal transport theory is a natural way to define both a distance and a geometry on the space of probability measures. In settings like graphical causal models (also called Bayes networks or belief networks), the space of probability measures is enriched by an information structure modeled by a directed graph. This talk introduces a variant of optimal transport including such a graphical information structure. The goal is to provide a concept of optimal transport whose topological and geometric properties are well suited for structural causal models. In this regard, we show that the resulting concept of Wasserstein distance can be used to control the difference between average treatment effects under different distributions, and is geometrically suitable to interpolate between different structural causal models.

In this talk we study wave propagation in random media using multiscale analysis.
We show that the wavefield can be described by a stochastic partial differential equation.
We can then address the following physical conjecture: for large propagation distances, the wavefield has Gaussian statistics, mean zero, and second-order moments determined by radiative transfer theory.
The results for the first two moments can be proved under general circumstances.
The Gaussian conjecture for the statistical distribution of the wavefield can be proved in some propagation regimes, but it turns out to be wrong in other regimes.

We consider the general framework of distributionally robust optimization under a martingale restriction. We provide explicit expressions for model risk sensitivities in this context by considering deviations in the Wasserstein distance and the corresponding adapted one. We also extend the dual formulation to this context.

We consider the sharp interface limit problem for 1D stochastic Allen-Cahn equation, and extend a classic result by Funaki to the full small noise regime. One interesting point is that the notion of "small noise" turns out to depend on the topology one uses. The main new idea in the proof is the construction of a series of functional correctors, which are designed to recursively cancel out potential divergences. At a technical level, in order to show these correctors are well behaved, we also develop a systematic decomposition of functional derivatives of the deterministic Allen-Cahn flow of all orders, which might have its own interest.
Based on a joint work with Wenhao Zhao (EPFL) and Shuhan Zhou (PKU).

We will introduce the Quintic Ornstein-Uhlenbeck model that jointly calibrates SPX-VIX options with a particular focus on its mathematical tractability namely for fast pricing SPX options using Fourier techniques. Then, we will consider the more general class of  stochastic volatility models where the dynamics of the volatility are given by a possibly infinite linear combination of the elements of the time extended signature of a Brownian motion. First, we show that the model is remarkably universal, as it includes, but is not limited to, the celebrated Stein-Stein, Bergomi, and Heston models, together with some path-dependent variants. Second, we derive the joint characteristic functional of the log-price and integrated variance provided that some infinite-dimensional extended tensor algebra valued Riccati equation admits a solution. This allows us to price and (quadratically) hedge certain European and path-dependent options using Fourier inversion techniques. We highlight the efficiency and accuracy of these Fourier techniques in a comprehensive numerical study.

Inspired by questions concerning the evolution of phase fields, we study the Allen-Cahn equation in dimension 2 with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint works with Simon Gabriel, Martin Hairer, Khoa Lê and Nikos Zygouras.