Past Stochastic Analysis & Mathematical Finance Seminars

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.

In this talk, we will present a loop expansion for lattice gauge theories and its application to prove ultraviolet stability in the Abelian Higgs model. We will first describe this loop expansion and how it relates to earlier works of Brydges-Frohlich-Seiler. We will then show how the expansion leads to a quantitative diamagnetic inequality, which in turn implies moment estimates, uniform in the lattice spacing, on the Holder-Besov norm of the gauge field marginal of the Abelian Higgs lattice model. Based on Gauge field marginal of an Abelian Higgs model, which is joint work with Ajay Chandra.

We consider a model of network of interacting  neurons based on jump processes. Briefly, the membrane potential $V^i_t$ of each individual neuron evolves according to a one-dimensional ODE. Neuron $i$ spikes at rate which only depends on its membrane potential, $f(V^i_t)$. After a spike, $V^i_t$ is reset to a fixed value $V^{\mathrm{rest}}$. Simultaneously, the membrane potentials of any (post-synaptic) neuron $j$ connected to the neuron $i$ receives a kick of value $J^{i,j}$.

We study the limit (mean-field) equation obtained where the number of neurons goes to infinity. In this talk, we describe the long time behaviour of the solution. Depending on the intensity of the interactions, we observe convergence of the distribution to a unique invariant measure (small interactions) or we characterize the occurrence of spontaneous oscillations for  interactions in the neighbourhood of critical values.

We prove convergence results of the simulation of the density solution to the McKean-Vlasov equation, when the measure variable is in the drift. Our method builds upon adaptive nonparametric results in statistics that enable us to obtain a data-driven selection of the smoothing parameter in a kernel-type estimator. In particular, we give a generalised Bernstein inequality for Euler schemes with interacting particles and obtain sharp deviation inequalities for the estimated classical solution. We complete our theoretical results with a systematic numerical study and gather empirical evidence of the benefit of using high-order kernels and data-driven smoothing parameters. This is a joint work with M. Hoffmann.

Generating high-fidelity time series data using generative adversarial networks (GANs) remains a challenging task, as it is difficult to capture the temporal dependence of joint probability distributions induced by time-series data. To this end, a key step is the development of an effective discriminator to distinguish between time series distributions. In this talk, I will introduce the so-called PCF-GAN, a novel GAN that incorporates the path characteristic function (PCF) as the principled representation of time series distribution into the discriminator to enhance its generative performance.  On the one hand, we establish theoretical foundations of the PCF distance by proving its characteristicity, boundedness, differentiability with respect to generator parameters, and weak continuity, which ensure the stability and feasibility of training the PCF-GAN. On the other hand, we design efficient initialisation and optimisation schemes for PCFs to strengthen the discriminative power and accelerate training efficiency. To further boost the capabilities of complex time series generation, we integrate the auto-encoder structure via sequential embedding into the PCF-GAN, which provides additional reconstruction functionality. Extensive numerical experiments on various datasets demonstrate the consistently superior performance of PCF-GAN over state-of-the-art baselines, in both generation and reconstruction quality. Joint work with Dr. Siran Li (Shanghai Jiao Tong Uni) and Hang Lou (UCL). Paper: [https://arxiv.org/pdf/2305.12511.pdf].

Neural SDEs are continuous-time generative models for sequential data. State-of-the-art performance for irregular time series generation has been previously obtained by training these models adversarially as GANs. However, as typical for GAN architectures, training is notoriously unstable, often suffers from mode collapse, and requires specialised techniques such as weight clipping and gradient penalty to mitigate these issues. In this talk, I will introduce a novel class of scoring rules on path space based on signature kernels and use them as an objective for training Neural SDEs non-adversarially. The strict properness of such kernel scores and the consistency of the corresponding estimators, provide existence and uniqueness guarantees for the minimiser. With this formulation, evaluating the generator-discriminator pair amounts to solving a system of linear path-dependent PDEs which allows for memory-efficient adjoint-based backpropagation. Moreover, because the proposed kernel scores are well-defined for paths with values in infinite-dimensional spaces of functions, this framework can be easily extended to generate spatiotemporal data. This procedure permits conditioning on a rich variety of market conditions and significantly outperforms alternative ways of training Neural SDEs on a variety of tasks including the simulation of rough volatility models, the conditional probabilistic forecasts of real-world forex pairs where the conditioning variable is an observed past trajectory, and the mesh-free generation of limit order book dynamics.

We describe the compact scaling limits of uniformly random quadrangulations with boundaries on a surface of arbitrary fixed genus. These limits, called Brownian surfaces, are homeomorphic to the surface of the given genus with or without boundaries depending on the scaling regime of the boundary perimeters of the quadrangulation. They are constructed by appropriate gluings of pieces derived from Brownian geometrical objects (the Brownian plane and half-plane). In this talk, I will review their definition and discuss possible alternative constructions. This is based on joint work with Jérémie Bettinelli.

Recently Linares-Otto-Tempelmayr have unveiled a very interesting algebraic structure which allows to define a new class of rough paths/regularity structures, with associated applications to stochastic PDEs or ODEs. This approach does not consider trees as combinatorial tools but their fertility, namely the function which associates to each integer k the number of vertices in the tree with exactly k children. In a joint work with J-D Jacques we have studied this algebraic structure and shown that it is related with a general and simple class of so-called post-Lie algebras. The construction has remarkable properties and I will try to present them in the simplest possible way.

We build a model of a financial market where a large number of firms determine their dynamic emission strategies under climate transition risk in the presence of both environmentally concerned and neutral investors. The firms aim to achieve a trade-off between financial and environmental performance, while interacting through the stochastic discount factor, determined in equilibrium by the investors' allocations. We formalize the problem in the setting of mean-field games and prove the existence and uniqueness of a Nash equilibrium for firms. We then present a convergent numerical algorithm for computing this equilibrium and illustrate the impact of climate transition risk and the presence of environmentally concerned investors on the market decarbonization dynamics and share prices. We show that uncertainty about future climate risks and policies leads to higher overall emissions and higher spreads between share prices of green and brown companies. This effect is partially reversed in the presence of environmentally concerned investors, whose impact on the cost of capital spurs companies to reduce emissions. However, if future climate policies are uncertain, even a large fraction of environmentally concerned investors is unable to bring down the emission curve: clear and predictable climate policies are an essential ingredient to allow green investors to decarbonize the economy.

Joint work with Pierre Lavigne

The Anderson operator is a perburbation of the Laplace-Beltrami operator by a space white noise potential. I will explain how to get a short self-contained functional analysis construction of the operator and how a sharp description of its heat kernel leads to useful quantitative estimates on its eigenvalues and eigenfunctions. One can associate to Anderson operator a (doubly) random field called the Anderson Gaussian free field. The law of its (random) partition function turns out to characterize the law of the spectrum of the operator. The square of the Anderson Gaussian free field turns out to be related to a probability measure on paths built from the operator, called the polymer measure.

Recently there is a rising interest in the research of mean-field optimization, in particular because of its role in analyzing the training of neural networks. In this talk, by adding the Fisher Information (in other word, the Schrodinger kinetic energy) as the regularizer, we relate the mean-field optimization problem with a so-called mean field Schrodinger (MFS) dynamics. We develop a free energy method to show that the marginal distributions of the MFS dynamics converge exponentially quickly towards the unique minimizer of the regularized optimization problem. We shall see that the MFS is a gradient flow on the probability measure space with respect to the relative entropy. Finally we propose a Monte Carlo method to sample the marginal distributions of the MFS dynamics. This is a joint work with Giovanni Conforti, Zhenjie Ren and Songbo Wang.

It was recently shown by Aidekon and Da Silva how to construct a growth fragmentation process from a planar Brownian excursion. I will explain how this same growth fragmentation process arises in another setting: when one decorates a certain “critical Liouville quantum gravity random surface” with a conformal loop ensemble of parameter 4. This talk is based on joint work with Juhan Aru, Nina Holden and Xin Sun. 
 

We consider an investor who is dynamically informed about the future evolution of one of the independent Brownian motions driving a stock's price fluctuations. The resulting rough semimartingale dynamics allow for strong arbitrage, but with linear temporary price impact the resulting optimal investment problem with exponential utility turns out to be well posed. The dynamically revealed Brownian path segment makes the problem infinite-dimensional, but by considering its convex-analytic dual problem, we show that it still can be solved explicitly and we give some financial-economic insights into the optimal investment strategy and the properties of maximum expected utility. (This is joint work with Yan Dolinsky, Hebrew University of Jerusalem).

Given a finite graph, the arboreal gas is the measure on forests (subgraphs without cycles) in which each edge is weighted by a parameter β greater than 0. Equivalently this model is bond percolation conditioned to be a forest, the independent sets of the graphic matroid, or the q→0 limit of the random cluster representation of the q-state Potts model. Our results rely on the fact that this model is also the graphical representation of the nonlinear sigma model with target space the fermionic hyperbolic plane H^{0|2}, whose symmetry group is the supergroup OSp(1|2).

The main question we are interested in is whether the arboreal gas percolates, i.e., whether for a given β the forest has a connected component that includes a positive fraction of the total edges of the graph. We show that in two dimensions a Mermin-Wagner theorem associated with the OSp(1|2) symmetry of the nonlinear sigma model implies that the arboreal gas does not percolate for any β greater than 0. On the other hand, in three and higher dimensions, we show that percolation occurs for large β by proving that the OSp(1|2) symmetry of the non-linear sigma model is spontaneously broken. We also show that the broken symmetry is accompanied by massless fluctuations (Goldstone mode). This result is achieved by a renormalisation group analysis combined with Ward identities from the internal symmetry of the sigma model.

Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total p-variation of trained weights for any p∈[1,3]. Unlike the C1-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [Cohen-Cont-Rossier-Xu, "Scaling Properties of Deep Residual Networks”, 2021]. Mathematically, we interpret residual neural network as solutions to (rough) difference equations, and analyse them based on recent results of discrete time signatures and rough path theory. Based on joint work with C. Bayer and P. K. Friz.