Speakers

Low regularity dynamics are systems of PDEs with initial data and potentials or noises. We assume that both initial data and potentials are of low regularity. Then, one wants to prove well-posedness for this system and provide an approximation of the solution minimising the regularity of the initial data and the given potentials. It turns out that these two procedures rely on iterated integrals expansion of the solution where decorated trees are used for dealing with the complexity of such expansion. We will present two main examples one coming from singular SPDEs for well-posedness and the other which implements a resonance scheme for dispersive equations.

An introduction into the world of variational autoencoders and how they can be used in practice for volatility modelling and market simulation of option portfolios.

The first essential ingredient to build up Stein’s method for a continuous target distribution is to identify a so-called Stein operator, namely a linear differential operator with polynomial coefficients. In this paper, we introduce the notion of algebraic Stein operators , and provide a novel algebraic method to find all the algebraic Stein operators up to a given order and polynomial degree for a target random variable of the form Y = h(X), where X = (X_1 , . . . , X_d) has i.i.d. standard Gaussian components and h in K[X] is a polynomial with coefficients in the ring K. Our approach links the existence of an algebraic Stein operator with null controllability of a certain linear discrete system. This is the first paper that connects Stein’s method with computational algebra to find Stein operators for highly complex probability distributions, such as H_{20}(X 1)‚ where H_p is the p-th Hermite polynomial. Joint work with Ehsan Azmoodeh (Liverpool) and Rober Gaunt (Manchester).

In recent years, interest has grown for portfolio construction methods that do not rely on expected returns. Among risk-based methods, the most popular ones are minimum variance, maximum diversification, and risk budgeting (especially equal risk contribution, aka. ERC). Risk budgeting is particularly attractive because of its versatility: based on the Euler decomposition of positively homogeneous functions, it can be used with a large range of risk « measures », from volatility to expected shortfall/CVaR and beyond. The goal of this talk is to present new mathematical results about the construction of risk budgeting portfolios for a very wide spectrum of risk « measures » and to show that, in many cases, stochastic optimization techniques can be used to build risk budgeting portfolios.

I will present a (partly) new method for perfectly sampling from the stationary regime of point processes having an infinite number of interacting components. This method is based on what is called Kalikow decomposition of the stochastic intensity of the process, together with an exploration of the spatial-temporal past of the process called Clan-of-Ancestors method. Such decompositions have been introduced in the framework of discrete time processes with long memory and/or g-measures, by Kalikow in 1990. I will explain how this method can be generalized to the framework of point processes in continuous time and which are the limitations of this approach. Finally I will discuss several examples, in particular Hawkes processes with an infinite number of components, having a strict refractory period. My talk is based on joint work with Patricia Reynaud-Bouret and Tien Cuong Phi (Nice).

We consider a class of degenerate equations satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin-type diffusions.

We present a novel approach to the numerical resolution of McKean-Vlasov stochastic differential equations via stochastic gradient descent. The algorithm is presented for a class of mean-field diffusions with separable coefficients. After introducing the algorithm, we present a preliminary convergence result and some numerical tests to confirm the theoretical findings. We finally discuss further ongoing investigations about the performance of the algorithm and possible generalizations. This is a work in progress with Ankush Agarwal and Gonçalo dos Reis.

Energy solutions to the stochastic Burgers equation were introduced by Goncalves-Jara to derive fluctuations for weakly asymmetric particle systems. They give a probabilistically weak formulation of this singular SPDE and thus provide a complementary point of view to the pathwise perspective of regularity structures. As observed by Gubinelli-Jara, the construction of energy solutions extends to many parabolic stochastic PDEs with Gaussian invariant measures and quadratic nonlinearities. More recently, in a joint work with Gubinelli we were able to prove weak uniqueness for energy solutions of the fractional multi-component stochastic Burgers equation, and in follow-up works by Gubinelli-Turra and Luo-Zhu this was extended to certain 2d fluid dynamic models. By taking a more abstract, functional analytic point of view, Lukas Gräfner was able to greatly simplify the proof of weak uniqueness and to turn it into a general result which applies to a wide range of equations and even proves well-posedness of some scaling critical equations. In my talk I will discuss Gräfner's functional analytic approach and present some examples including d-dimensional hyperviscous stochastic Navier-Stokes equations with white noise invariant measures.

We discuss some recent results on Kinetic degenerate Kolmogorov SDEs with non-linear drift: we show two sided bounds and pointwise controls of its derivatives, under somehow minimal assumptions that guarantee that the equation is weakly well posed. These estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized flow.

This talk will present a joint work with B. de Bruyne and S. Redner on first-passage resetting, where the resetting of a random walk to a fixed position is triggered by a first-passage event of the walk itself. In an infinite domain, we calculate the resulting spatial probability distribution of the particle analytically, and also obtain this distribution by a path decomposition. In a finite interval, we define an optimization problem that is controlled by first- passage resetting: a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset-trigger point. This scenario is motivated by reliability theory. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost. Time permitting, I will also talk about an extension of first-passage resetting into a minimalist dynamical model of wealth evolution and wealth sharing among N agents as a platform to compare the relative merits of altruism and individualism. References: de Bruyne, B., R.-F., J., Redner, S. (2020). Phys. Rev. Letters, 125(5), 050602; de Bruyne, B., R.-F., J., Redner, S. (2021). J. Stat. Mech., 2021(10), 103405;

In this study we characterized the Ornstein-Uhlenbeck process having a symmetric normal tempered stable stationary law extending accordingly, the findings in Barndolff-Nielsen ([1] in the attached file). To this end, we represent the transition distribution in terms of the sum of independent laws and also write its relative background driving Lévy process in terms of the sum of two independent Lévy processes. Based on these facts we can design two alternate algorithms for the simulation of the skeleton of the Ornstein-Uhlenbeck process. The solution based on the transition law turns out to be faster since it is based on a lower number of computational steps; that is also confirmed by extensive numerical experiments. We also calculate the characteristic function of the transition density which is instrumental for the application of the FFT-based method of Carr and Madan ([2] in the attached file) to the pricing of a strip of call options written on markets whose price evolution is modeled by an Ornstein-Uhlenbeck dynamics. This approach is indeed common for spot prices in the energy and the commodity fields. Finally, we show how to extend the range of applications to future markets.

Exact simulation of NTS processes of OU type with applications

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We shall discuss aspects of a nonlocal SPDE with applications to the so-called Amari-type neural field model, which are mean-field models for neural activity in the cortex. In particular, under suitable assumptions on the coupling kernel, we show that via an infinite dimensional change of coordinates the equation admits a gradient flow structure, which has consequences for regularity, long-time behavior and uniqueness of invariant distributions. This is a joint work with Christian Kuehn, TU Munich.

In this presentation we study the limiting distribution for the joint-law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the so-called Lyapunov condition. Under an appropriated normalization, the joint-law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws. The latter implies that a normalized condition number converges in distribution to a Fréchet law as the dimension of the matrix increases. Roughly speaking, the condition number measures how much the output value of a linear system can change by a small perturbation in the input argument. The proof relies on the celebrated Einmahl--Komlós--Major--Tusnády coupling. This is a joint work with Paulo Manrique, National Polytechnic Institute, Mexico.

We introduce a novel sustainable capital instrument: the skin-in-the-game bond. With features inspired by contingent convertibles (CoCos), this bond is an alternative for the green, social, sustainability and sustainability-linked bonds available on the market. A skin-in-the-game bond is linked to the performance of a benchmark that relates to the broad concept of sustainability in at least one of its pillars, being the environment (E), society (S) or corporate governance (G). When the benchmark hits a preset trigger level, (part of) the bond’s face value is withheld and directed into a government-controlled fund by the issuer. The skin-in-the-game bond offers a higher yield to investors than a standard corporate bond, in order to compensate for the risk of losing out on (part of) the investment. Both issuer and investor have skin-in-the-game; the embedded financial penalty incentivizes the preservation of a favorable benchmark value. In this presentation, we elaborate on the general concept of a skin-in-the-game bond, as well as on a tailored valuation model, illustrated by two examples: the ESG and nuclear skin-in-the-game bonds.

Risk-neutral densities contain rich information about investors' risk averseness and expectations for the future. In this paper, we use the Bilateral Gamma model to fit the risk-neutral density because it provides more flexible tail behaviours than the widely used Variance Gamma model. A traditional model calibrating method uses numerical optimisation to minimize the sum of the distance between the market price and the model price. However, it suffers from local minima and is sensitive to the initial value used. Compared to numerical optimisation, the moment-matching method is more time-efficient but shows less accuracy than numerical optimisation. Motivated by this, we combine the moment-matching method and numerical optimisation to balance time-consuming and model performance.