This is the page of the fall semester of 2021 of the United Seminar of the Department of Probability Theory of the Faculty of Mechanics and Mathematics of Moscow State University. The permanent website of the seminar is here. The seminar is a continuation of the research seminar of the Department of Probability Theory under the leadership of A.N. Kolmogorov and B.V. Gnedenko.

Previous reports

September 22, 16:45 MSK
E.A. Illarionov, D.D. Sokoloff, M.A. Listopad, Lomonosov Moscow State University

Dimensionality reduction to reveal properties of complex structures

Dimensionality reduction is often used for data compression. However, apart from this utilitarian purpose, one can apply dimensionality reduction to get an insight about the data and construct a parametric description. In the first part of the presentation we will give an overview of classical and recent models that help in data exploration. In the second part we will consider a problem of morphological description of sunspot groups. We will present a new approach based on deep embeddings leant by convolutional autoencoder models. The idea of autoencoder neural networks is to map input data (sunspot group images) into a latent space of lower dimension and preserve an inverse mapping. Proper architecture of the neural network and careful training process can provide a reasonable structure of the latent space and reveal many non-trivial relations. We investigate the structure of the latent space and discuss interpretation of latent features.

The recording of the report can be found at this link.

October 13, 16:45 MSK
V.A. Vatutin*, E.E. Dyakonova*, V.A. Topchii, *Steklov Mathematical Institute, **Omsk Branch of the Sobolev Institute of Mathematics

Critical Galton-Watson branching processes with countably infinitely many types and infinite second moments

An indecomposable branching Galton–Watson process with a countable set of types of particles is considered. Assuming that the process is critical and particles of some (or all) its types can have an infinite variance of the offspring number, we describe the asymptotic behavior of the survival probability of such a process and prove a Yaglom-type conditional limit theorem for the distribution of an infinite-dimensional vector of the number of particles of all types.

The recording of the report can be found at this link.

October 27, 16:45 MSK
Yana Belopolskaya, Sirius University, SPbGASU

Stochastic models of conservation and balance laws in systems with dissipations

Mathematical models for conservation and balance laws in systems with dissipation (reaction-diffusion systems, systems with cross-diffusion and so on) as a rule are stated in the form of systems of nonlinear parabolic equations. For scalar nonlinear parabolic equations which admit interpretation as forward or backward Kolmogorov equations probabilistic representations of the Cauchy problem solutions have been studied by a number of authors. Namely, a diffusion process with a nonlinear forward Kolmogorov equation was constructed by H.McKean (1966) and a diffusion process with a nonlinear backward Kolmogorov equation was constructed by M.Freidlin (1967). There exist a number of papers that extend both approaches in various directions. Nevertheless, there exist a few results of this type concerning systems of nonlinear parabolic equations. In the talk we consider two classes of nonlinear parabolic systems admitting a probabilistic interpretation. Systems of the first class admit (after a simple transformation) an interpretation as systems of backward Kolmogorov equations, while systems of the second class should be interpreted as systems of forward Kolmogorov equations. We reduce the Cauchy problem solution for both types of these systems to solution of corresponding stochastic problems and as a result construct probabilistic representations of the Cauchy problem solutions.

The recording of the report can be found at this link.

November 10, 16:45 MSK
Ernst Eberlein, University of Freiburg

Fourier Based Methods for the Management of Complex Life Insurance Products

This paper proposes a framework for the valuation and the management of complex life insurance contracts, whose design can be described by a portfolio of embedded options, which are activated according to one or more triggering events. These events are in general monitored discretely over the life of the policy, due to the contract terms. Similar designs can also be found in other contexts, such as counterparty credit risk for example. The framework is based on Fourier transform methods as they allow to derive convenient closed analytical formulas for a broad spectrum of underlying dynamics. Multidimensionality issues generated by the discrete monitoring of the triggering events are dealt with efficiently designed Monte Carlo integration strategies. We illustrate the tractability of the proposed approach by means of a detailed study of ratchet variable annuities, which can be considered a prototypical example of these complex structured products.

This is a joint project with Laura Ballotta, Thorsten Schmidt, and Raghid Zeineddine (doi link).

The recording of the report can be found at this link.

November 24, 16:45 MSK
Goran Peskir*, David Roodman**, *The University of Manchester (UK), *Open Philanthropy, (USA)

Sticky Feller Diffusions

We consider a Feller branching diffusion process 𝑋 with drift 𝑐 having 0 as a slowly reflecting (sticky) boundary point with a stickiness parameter 1/𝜇 ∈ (0, ∞). We show that (i) the process 𝑋 can be characterised as a unique weak solution to the SDE system

where 𝑏 ∈ 𝑅 and 0 < 𝑐 < 𝑎 are given and fixed, 𝐵 is a standard Brownian motion, and l0(𝑋) is a diffusion local time process of 𝑋 at 0, and (ii) the transition density function of 𝑋 can be expressed in the closed form by means of a convolution integral involving a new special function and a modified Bessel function of the second kind. The new special function embodies the stickiness of 𝑋 entirely and reduces to the Mittag-Leffler function when 𝑏 = 0. We determine a (sticky) boundary condition at zero that characterises the transition density function of 𝑋 as a unique solution to the Kolmogorov forward/backward equation of 𝑋. Letting 𝜇 ↓ 0 (absorption) and 𝜇 ↑ ∞ (instantaneous reflection) the closed-form expression for the transition density function of 𝑋 reduces to the ones found by Feller (1951) and Molchanov (1967) respectively. The results derived for sticky Feller diffusions translate over to yield closed-form expressions for the transition density functions of (a) sticky Cox-Ingersoll-Ross processes and (b) sticky reflecting Vasicek processes that can be used to model slowly reflecting interest rates.

The recording of the report can be found at this link.

December 08, 16:45 MSK
Evgeny Spodarev, Ulm University

Long memory of heavy-tailed random fields

Long range dependence is usually related to the asymptotic behaviour of (co)variance, spectral density, etc. of a stationary square integrable random function in infinity or at zero. For that, the second moment of the random function should be finite. We give a new definition of long range dependence of stationary random functions with an infinite second moment which is based on the asymptotic variance of the volume of their level sets. It follows that all mixing random functions are short range dependent. We also show how this definition can be applied to different heavy-tailed processes and fields such as subordinated Gaussian, random volatility, α- and max-stable. A connection to limit theorems for random volatility fields on Z^d is proven

The recording of the report can be found at this link.