Past Stochastic Analysis & Mathematical Finance Seminars

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.

Many systems in the applied sciences are made of a large number of particles. One is often not interested in the detailed behaviour of each particle but rather in the collective behaviour of the group. An established methodology in statistical mechanics and kinetic theory allows one to study the limit as the number of particles in the system N tends to infinity and to obtain a (low dimensional) PDE for the evolution of the density of the particles. The limiting PDE is a non-linear equation, where the non-linearity has a specific structure and is called a McKean-Vlasov nonlinearity. Even if the particles evolve according to a stochastic differential equation, the limiting equation is deterministic, as long as the particles are subject to independent sources of noise. If the particles are subject to the same noise (common noise) then the limit is given by a Stochastic Partial Differential Equation (SPDE). In the latter case the limiting SPDE is substantially the McKean-Vlasov PDE + noise; noise is furthermore multiplicative and has gradient structure.  One may then ask the question about whether it is possible to obtain McKean-Vlasov SPDEs with additive noise from particle systems. We will explain how to address this question, by studying limits of weighted particle systems.  

This is a joint work with L. Angeli, J. Barre,  D. Crisan, M. Kolodziejzik.  

Rough path theory provides a framework for the study of nonlinear systems driven by highly oscillatory (deterministic) signals. The corresponding analysis is inherently distinct from that of classical stochastic calculus, and neither theory alone is able to satisfactorily handle hybrid systems driven by both rough and stochastic noise. The introduction of the stochastic sewing lemma (Khoa Lê, 2020) has paved the way for a theory which can efficiently handle such hybrid systems. In this talk, we will discuss how this can be done in a general setting which allows for jump discontinuities in both sources of noise.

This presentation focuses on the study of the regulartiy of random wavelet series. We first study their belonging to certain functional spaces and we compare these results with long-established results related to random Fourier series. Next, we show how the study of random wavelet series leads to precise pointwise regularity properties of processes like fractional Brownian motion. Additionally, we explore how these series helps create Gaussian processes  with random Hölder exponents.

One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and infinite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We discuss some dynamic games of this type, from static to two player to the associated mean field game, with some numerical and existence-uniqueness results.

Based on works with Andrew Ledvina and with Emre Parmaksiz.

Fluctuating hydrodynamics provides a framework for approximating density fluctuations in interacting particle systems by suitable SPDEs. The Dean-Kawasaki equation - a strongly singular SPDE - is perhaps the most basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N diffusing weakly interacting particles in the regime of large particle numbers N. The strongly singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification: Besides being non-renormalizable by approaches like regularity structures, it has recently been shown to not even admit nontrivial martingale solutions.

In this talk, we give an overview of recent quantitative results for the justification of fluctuating hydrodynamics models. In particular, we give an interpretation of the Dean-Kawasaki equation as a "recipe" for accurate and efficient numerical simulations of the density fluctuations for weakly interacting diffusing particles, allowing for an error that is of arbitarily high order in the inverse particle number. 

Based on joint works with Federico Cornalba, Jonas Ingmanns, and Claudia Raithel

We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence respecting  the temporal dynamics of the time series distribution. We  estimate the drift function from data samples by nonparametric, e.g. kernel regression methods,  and the simulation of the SB diffusion  yields new synthetic data samples of the time series. The performance of our generative model is evaluated through a series of numerical experiments.  First, we test with autoregressive models, a GARCH Model, and the example of fractional Brownian motion,  and measure the accuracy of our algorithm with marginal, temporal dependencies metrics, and predictive scores. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets. 

We consider stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter less than 1/2. The drift is a measurable function of time and space which belongs to a certain Lebesgue space. Under subcritical regime, we show that a strong solution exists and is unique in path-by-path sense. When the noise is formally replaced by a Brownian motion, our results correspond to the strong uniqueness result of Krylov and Roeckner (2005). Our methods forgo standard approaches in Markovian settings and utilize Lyons' rough path theory in conjunction with recently developed tools. Joint work with Toyomu Matsuda and Oleg Butkovsky.

Standard symmetric α-stable cylindrical processes in Hilbert spaces are the natural generalisation of the analogue processes in Euclidean spaces. However, like standard Brownian motions, standard symmetric α-stable processes in finite dimensions can only be generalised to infinite dimensional Hilbert spaces as cylindrical processes, i.e. processes in a generalised sense (of Gel’fand and Vilenkin (1964) or Segal (1954))  not attaining values in the underlying Hilbert space.

In this talk, we briefly introduce the theory of stochastic integrals with respect to standard symmetric α-stable cylindrical processes. As these processes exist only in the generalised sense, introducing a stochastic integral requires an approach different to the classical one by semi-martingale decomposition. The main result presented in this talk is the existence of a solution to an abstract evolution equation driven by a standard symmetric α-stable cylindrical process. The main tool for establishing this result is a Yosida approximation and an Itô formula for Hilbert space-valued semi-martingales where the martingale part is represented as an integral driven by cylindrical α-stable noise. While these tools are standard in stochastic analysis, due to the cylindrical nature of our noise, their application requires completely novel arguments and techniques.