Past Stochastic Analysis & Mathematical Finance Seminars

TBC

We will report on recent progress regarding the near-critical behavior of certain statistical physics models in dimension 3. Our results deal with the second-order phase transition associated to two percolation problems involving the Gaussian free field in 3D. In one case, they determine a unique ``fixed point'' corresponding to the transition, which is proved to obey one of several scaling relations. Such laws are classically conjectured to hold by physicists on the grounds of a corresponding scaling ansatz. 

Gibbs measures on spaces of functions or distributions play an important role in various contexts in mathematical physics.  They can, for example, be viewed as continuous counterparts of classical spin models such as the Ising model, they are an important stepping stone in the rigorous construction of Quantum Field Theories, and they are invariant under the 
flow of certain dispersive PDEs, permitting to develop a solution theory with random initial data, well below the deterministic regularity threshold. 

These measures have been constructed and studied, at least since the 60s, but over the last few years there has been renewed interest, partially due to new methods in stochastic analysis, including Hairer’s theory of regularity structures and Gubinelli-Imkeller-Perkowski’s theory of paracontrolled distributions. 

In this talk I will present two independent but complementary results that can be obtained with these new techniques. I will first show how to obtain estimates on samples from of the Euclidean $\phi^4_3$ measure, based on SPDE methods. In the second part, I will discuss a method to show the emergence of phase transitions in the $\phi^4_3$ theory.

Title: Measuring association with Wasserstein distances

Abstract: Let π ∈ Π(μ, ν) be a coupling between two probability measures μ and ν on a Polish space. In this talk we propose and study a class of nonparametric measures of association between μ and ν, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between ν and the disintegration of π with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures μ and ν. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.

Classical approaches to optimal portfolio selection problems are based 
on probabilistic models for the asset returns or prices. However, by 
now it is well observed that the performance of optimal portfolios are 
highly sensitive to model misspecifications. To account for various 
type of model risk, robust and model-free approaches have gained more 
and more importance in portfolio theory. Based on a rough path 
foundation, we develop a model-free approach to stochastic portfolio 
theory and Cover's universal portfolio. The use of rough path theory 
allows treating significantly more general portfolios in a model-free 
setting, compared to previous model-free approaches. Without the 
assumption of any underlying probabilistic model, we present pathwise 
Master formulae analogously to the classical ones in stochastic 
portfolio theory, describing the growth of wealth processes generated 
by pathwise portfolios relative to the wealth process of the market 
portfolio, and we show that the appropriately scaled asymptotic growth 
rate of Cover's universal portfolio is equal to the one of the best 
retrospectively chosen portfolio. The talk is based on joint work with 
Andrew Allan, Christa Cuchiero and Chong Liu.

 

 In this talk, we will present the tools of regularity structures to deal with singular stochastic PDEs that involve non-translation invariant differential operators. We describe in particular the renormalized equation for a very large class of spacetime dependent renormalization schemes. Our approach bypasses the previous approaches in the translation-invariant setting. This is joint work with Ismael Bailleul.

Given a point and a loop in the plane, one can define a relative integer which counts how many times the curve winds around the point. We will discuss how this winding function, defined for almost every points in the plane, allows to define some integrals along the loop. Then, we will investigate some properties of it when the loop is Brownian.
In particular, we will explain how to recover data such as the Lévy area of the curve and its occupation measure, based on the values of the winding of uniformly distributed points on the plane.

I will present recent results on the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We study the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained on the torus undergoes a phase transition, i.e., if it admits more than one steady state. A typical example of such a system on the torus is given by mean field plane rotator (XY, Heisenberg, O(2)) model. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature. This is joint work with Matias Delgadino (U Texas Austin) and Rishabh Gvalani (MPI Leipzig).

Modelling joint dynamics of liquid vanilla options is crucial for arbitrage-free pricing of illiquid derivatives and managing risks of option trade books. This paper develops a nonparametric model for the European options book respecting underlying financial constraints and while being practically implementable. We derive a state space for prices which are free from static (or model-independent) arbitrage and study the inference problem where a model is learnt from discrete time series data of stock and option prices. We use neural networks as function approximators for the drift and diffusion of the modelled SDE system, and impose constraints on the neural nets such that no-arbitrage conditions are preserved. In particular, we give methods to calibrate neural SDE models which are guaranteed to satisfy a set of linear inequalities. We validate our approach with numerical experiments using data generated from a Heston stochastic local volatility model, and will discuss some initial results using real data.

Based on joint work with Christoph Reisinger and Sheng Wang

Abstract: In this talk we try to establish an analytic framework for studying Set-Valued Backward Stochastic Differential Equations (SVBSDE for short), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will be based on the notion of Hukuhara difference between sets, in order to compensate the lack of “inverse” operation of the traditional Minkowski addition, whence the vector space structure, in traditional set-valued analysis. We shall examine and establish a useful foundation of set-valued stochastic analysis under this algebraic framework, including some fundamental issues regarding Aumann-Itˆo integrals, especially when it is connected to the martingale representation theorem. We shall identify some fundamental challenges and propose some extensions of the existing theory that are necessary to study the SVBSDEs. This talk is based on the joint works with C¸ a˘gın Ararat and Wenqian Wu.

We present linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque, and Ichiba in a previous work. We solve explicitly for Nash equilibria with a finite number of players and we study more general finite-player games with a mixture of both directed chain interaction and mean field interaction. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. 

The limit is characterized by Catalan functions and the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain, with or without the presence of mean field interaction.

Joint work with Yichen Feng and Tomoyuki Ichiba.

Following the risk-taking model of Seel and Strack, n players decide when to stop privately observed Brownian motions with drift and absorption at zero. They are then ranked according to their level of stopping and paid a rank-dependent reward. We study the optimal reward design where a principal is interested in the average performance and the performance at a given rank. While the former can be related to reward inequality in the Lorenz sense, the latter can have a surprising shape. Next, I will present the mean-field version of this problem. A particular feature of this game is to be tractable without necessarily being smooth, which turns out to offer a cautionary tale. We show that the mean field equilibrium induces n-player ε-Nash equilibria for any continuous reward function— but not for discontinuous ones. We also analyze the quality of the mean field design (for maximizing the median performance) when used as a proxy for the optimizer in the n-player game. Surprisingly, the quality deteriorates dramatically as n grows. We explain this with an asymptotic singularity in the induced n-player equilibrium distributions. (Joint work with M. Nutz)

We consider a reaction-diffusion equation of the type ∂tψ=∂2xψ+V(ψ)+λσ(ψ)W˙on (0,∞)×?, subject to a "nice" initial value and periodic boundary, where ?=[−1,1] and W˙ denotes space-time white noise. The reaction term V:ℝ→ℝ belongs to a large family of functions that includes Fisher--KPP nonlinearities [V(x)=x(1−x)] as well as Allen-Cahn potentials [V(x)=x(1−x)(1+x)], the multiplicative nonlinearity σ:ℝ→ℝ is non random and Lipschitz continuous, and λ>0 is a non-random number that measures the strength of the effect of the noise W˙. The principal finding of this paper is that: (i) When λ is sufficiently large, the above equation has a unique invariant measure; and (ii) When λ is sufficiently small, the collection of all invariant measures is a non-trivial line segment, in particular infinite. This proves an earlier prediction of Zimmerman et al. (2000). Our methods also say a great deal about the structure of these invariant measures.

This is based on joint work with Carl Mueller (Univ. Rochester) and Kunwoo Kim (POSTECH, S. Korea).

In this talk, we will consider multidimensional fast-slow dynamical systems in discrete-time with random initial conditions but otherwise completely deterministic dynamics. The question we will investigate is whether the slow variable converges in law to a stochastic process under a suitable scaling limit. We will be particularly interested in the case when the limiting dynamic is superdiffusive, i.e. it coincides in law with the solution of a Marcus SDE driven by a discontinuous stable Lévy process. Under certain assumptions, we will show that generically convergence does not hold in any Skorokhod topology but does hold in a generalisation of the Skorokhod strong M1 topology which we define using so-called path functions. Our methods are based on a combination of ergodic theory and ideas arising from (but not using) rough paths. We will finally show that our assumptions are satisfied for a class of intermittent maps of Pomeau-Manneville type. 

In my talk, I will introduce a family of human-machine interaction (HMI) models in optimal portfolio construction (robo-advising). Modeling difficulties stem from the limited ability to quantify the human’s risk preferences and describe their evolution, but also from the fact that the stochastic environment, in which the machine optimizes, adapts to real-time incoming information that is exogenous to the human. Furthermore, the human’s risk preferences and the machine’s states may evolve at different scales. This interaction creates an adaptive cooperative game with both asymmetric and incomplete information exchange between the two parties.

As a result, challenging questions arise on, among others, how frequently the two parties should communicate, what information can the machine accurately detect, infer and predict, how the human reacts to exogenous events, how to improve the inter-linked reliability between the human and the machine, and others. Such HMI models give rise to new, non-standard optimization problems that combine adaptive stochastic control, stochastic differential games, optimal stopping, multi-scales and learning.

We will discuss several versions of Ito’s formula in the case where the function is path dependent and only concave or C1 in the sense of Dupire. In particular, we will show that it can be used to solve (super) hedging problems, in the context of market impact or under volatility uncertainty. 

We discuss a novel backward Ito-Ventzell formula and an extension of the Aleeksev-Gröbner interpolating formula to stochastic flows. We also present some natural spectral conditions that yield direct and simple proofs of time uniform estimates of the difference between the two stochastic flows when their drift and diffusion functions are not the same, yielding what seems to be the first results of this type for this class of  anticipative models.

We illustrate the impact of these results in the context of diffusion perturbation theory, interacting diffusions and discrete time approximations.

Nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations This talk is about joint work with Viorel Barbu. We consider a class of nonlinear Fokker-Planck (- Kolmogorov) equations of type \begin{equation*} \frac{\partial}{\partial t} u(t,x) - \Delta_x\beta(u(t,x)) + \mathrm{div} \big(D(x)b(u(t,x))u(t,x)\big) = 0,\quad u(0,\cdot)=\mu, \end{equation*} where $(t,x) \in [0,\infty) \times \mathbb{R}^d$, $d \geq 3$ and $\mu$ is a signed Borel measure on $\mathbb{R}^d$ of bounded variation. In the first part of the talk we shall explain how to construct a solution to the above PDE based on classical nonlinear operator semigroup theory on $L^1(\mathbb{R}^d)$ and new results on $L^1- L^\infty$ regularization of the solution semigroups in our case. In the second part of the talk we shall present a general result about the correspondence of nonlinear Fokker-Planck equations (FPEs) and McKean-Vlasov type SDEs. In particular, it is shown that if one can solve the nonlinear FPE, then one can always construct a weak solution to the corresponding McKean-Vlasov SDE. We would like to emphasize that this, in particular, applies to the singular case, where the coefficients depend "Nemytski-type" on the time-marginal law of the solution process, hence the coefficients are not continuous in the measure-variable with respect to the weak topology on probability measures. This is in contrast to the literature in which the latter is standardly assumed. Hence we can cover nonlinear FPEs as the ones above, which are PDEs for the marginal law densities, realizing an old vision of McKean.

References V. Barbu, M. Röckner: From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE, Ann. Prob. 48 (2020), no. 4, 1902-1920. V. Barbu, M. Röckner: Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations, J. Funct. Anal. 280 (2021), no. 7, 108926.

Interacting particle systems have found diverse applications in mathematics and several related fields, including statistical physics, population dynamics, and machine learning.  We will focus, in particular, on the zero range process and the symmetric simple exclusion process.  The large-scale behavior of these systems is essentially deterministic, and is described in terms of a hydrodynamic limit.  However, the particle process does exhibit large fluctuations away from its mean.  Such deviations, though rare, can have significant consequences---such as a concentration of energy or the appearance of a vacuum---which make them important to understand and simulate.

In this talk, which is based on joint work with Benjamin Gess, I will introduce a continuum model for simulating rare events in the zero range and symmetric simple exclusion process.  The model is based on an approximating sequence of stochastic partial differential equations with nonlinear, conservative noise.  The solutions capture to first-order the central limit fluctuations of the particle system, and they correctly simulate rare events in terms of a large deviations principle.

We consider a chain of oscillators with one particle in contact with a thermostat at temperature T. The thermostat is modeled by a Langevin dynamics or a renewal of the velocity with a gaussian random variable with variance T. The dynamics of the oscillators is perturbed by a random exchange on velocities between nearest neighbor particles.
The (thermal) energy has a macroscopic superdiffusive behavior governed by a fractional heat equation (i.e. with a fractional Laplacian). The microscopic thermostat impose a particular boundary condition to the fractional Laplacian, corresponding to certain probabilities of transmission/reflection/absorption/creation for the corresponding superdiffusive Levy process.
This is from a series of works in collaboration with Tomazs Komorowski, Lenya Ryzhik, Herbert Spohn.

The goal of sequential learning is to draw inference from data that is gathered gradually through time. This is a typical situation in many applications, including finance. A sequential inference procedure is `anytime-valid’ if the decision to stop or continue an experiment can depend on anything that has been observed so far, without compromising statistical error guarantees. A recent approach to anytime-valid inference views a test statistic as a bet against the null hypothesis. These bets are constrained to be supermartingales - hence unprofitable - under the null, but designed to be profitable under the relevant alternative hypotheses. This perspective opens the door to tools from financial mathematics. In this talk I will discuss how notions such as supermartingale measures, log-optimality, and the optional decomposition theorem shed new light on anytime-valid sequential learning. (This talk is based on joint work with Wouter Koolen (CWI), Aaditya Ramdas (CMU) and Johannes Ruf (LSE).)
 

The CLE_4 Conformal Loop Ensemble in a 2D simply connected domain is a random countable collection of fractal Jordan curves that satisfies a statistical conformal invariance and appears, or is conjectured to appear, as a scaling limit of interfaces in various statistical physics models in 2D, for instance in the double dimer model. The CLE_4   is also related to the 2D Gaussian free field. Given a simply connected domain D and a point z in D, we consider the CLE_4 loop that surrounds z and study the extremal distance between the loop and the boundary of the domain, and the conformal radius of the interior surrounded by the loop seen from z. Because of the confomal invariance, the joint law of this two quantities does not depend (up to a scale factor) on the choice of the domain D and the point z in D. The law of the conformal radius alone has been known since the works of Schramm, Sheffield and Wilson. We complement their result by deriving the joint law of (extremal distance, conformal radius). Both quantities can be read on the same 1D Brownian path, by tacking a last passage time and a first hitting time. This joint law, together with some distortion bounds, provides some exponents related to the CLE_4. This is a joint work with Juhan Aru and Avelio Sepulveda.

An open market is a subset of a larger equity market, composed of a certain fixed number of top‐capitalization stocks. Though the number of stocks in the open market is fixed, their composition changes over time, as each company's rank by market capitalization fluctuates. When one is allowed to invest also in a money market, an open market resembles the entire “closed” equity market in the sense that the market viability (lack of arbitrage) is equivalent to the existence of a numéraire portfolio (which cannot be outperformed). When access to the money market is prohibited, the class of portfolios shrinks significantly in open markets; in such a setting, we discuss how to construct functionally generated stock portfolios and the concept of the universal portfolio.

This talk is based on joint work with Ioannis Karatzas.

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Mean field games (MFG) and mean field control problems (MFC) are frameworks to study Nash equilibria or social optima in games with a continuum of agents. These problems can be used to approximate competitive or cooperative situations with a large finite number of agents. They have found a broad range of applications, from economics to crowd motion, energy production and risk management. Scalable numerical methods are a key step towards concrete applications. In this talk, we propose several numerical methods for MFG and MFC. These methods are based on machine learning tools such as function approximation via neural networks and stochastic optimization. We provide numerical results and we investigate the numerical analysis of these methods by proving bounds on the approximation scheme. If time permits, we will also discuss model-free methods based on extensions of the traditional reinforcement learning setting to the mean-field regime.