This is the page 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.

Upcoming reports

May 13, 16:45 msk

Artur Sidorenko, Lomonosov MSU, Russia

Stability of stochastic optimal control problems arising in modern finance

This talk is devoted to the stability of stochastic optimal control problems arising in mathematical finance. We consider a singular stochastic optimal control problem with conic constraints, formulated in the geometric framework of a market with proportional transaction costs. In this framework an arbitrary finite number of risky assets is allowed; transaction costs are described by solvency cones, while admissible strategies are processes of finite variation whose increments satisfy conic constraints. We establish the applicability of Skorokhod's representation theorem in this setting. To this end, we distinguish between the market model — understood as the joint distribution of the price process and unobservable parameters — and a particular realization of the model on a filtered probability space. We prove invariance of the value function (the Bellman function) of the control problem with respect to additional randomization of the model and with respect to the choice of its equivalent realization. Using the Skorokhod representation and the Meyer–Zheng topology, we obtain sufficient conditions for convergence of Bellman functions under weak convergence of the joint distributions of the corresponding price processes and unobservable parameters. We also prove the existence of a limiting optimal strategy for a sequence of asymptotically optimal strategies. The final part of the talk takes a step toward a quantitative analysis of the rate of such convergence in optimal control problems: we present regularity estimates for the diffusion semigroup in weighted function spaces without assuming uniform ellipticity, together with uniform estimates for the rate of convergence of discrete approximations of fractional diffusion models with coefficients of linear growth.

April 29, 16:45 msk

Elena Filichkina, Lomonosov MSU, Russia

Limit theorems for branching random walks with various configurations of absorbing sources

The report is devoted to branching random walks (BRW) on d-dimensional lattices with two types of sources located at lattice points. In sources of the first type the reproduction and death of particles occurs, as in a branching process, while in sources of the second type particles can only be absorbed. These assumptions lead to new effects that depend on the process parameters and source configuration. If absorption sources are located at every point and there is a single "branching" center, then additional phase transitions arise in the limit behavior of the process. Special attention is paid to the BRW on a one-dimensional lattice with a finite number of absorbing sources symmetrically located around the “branching” center. For this model a necessary and sufficient condition for the exponential growth of particle numbers at lattice points is obtained. For a random walk in which only a finite number of absorbing sources without branching are assumed, limit theorems on the behavior of particle numbers are proved for each dimension d of the integer lattice.

The recording: YouTube

April 15, 16:45 msk

Platon Promyslov, Lomonosov MSU, Russia

Ruin Probability in Insurance Models with Investments: Asymptotics and Integro-Differential Equations

The talk considers the problem of estimating the ruin probability for insurance models in which the company’s capital is partially or fully invested in a risky asset. The presence of investments fundamentally changes the dynamics of the process: the exponential decay of the ruin probability is replaced by a power-law decay, and the mathematical analysis becomes significantly more complicated due to the singularity of the resulting equations. We will discuss the method of implicit renewal theory, which allows one to find the rate of decay of the ruin probability using affine distributional equations, as well as new analytical results describing the survival function as a solution to second-order integro-differential equations.

The recording: YouTube

April 1, 16:45 msk

Lomonosov readings

I.D. Stepanov, D.A. Shabanov, Lomonosov MSU, MIPT, Russia

Algorithmic threshold probabilities for random discrete structures

The phenomenon of threshold probabilities in random discrete structures allows us to propose a reasonable probabilistic criterion for determining the presence of a certain desired property. However, it does not allow us to quickly find the object under study, but only indicates a high probability of its presence in the structure. It turns out that there may be obstacles to finding fast algorithms for finding such objects due to the specific structure of their set. In this paper, we will explore this effect through several specific examples in the theory of random graphs and hypergraphs. 

Alexander Veretennikov, Lomonosov MSU, Russia

Efficient estimations of convergence rate for Markov processes

Efficient approaches for evaluating convergence rate in total variation for finite and general Markov chains will be discussed in the talk. The motivation for studying convergence rate in this metric is its usefulness in various limit theorems. For homogeneous Markov chains the goal is to compare several different methods: (1) the second eigenvalue for the transition matrix method (the ''method no. 1''), (2) the method based on Markov-Dobrushin's ergodic coefficient, and the new spectral method developed in recent publications by the speaker, as well as modifications of they both by iterations (the ''other methods''). We answer the question whether or not the ''other methods'' may provide the optimal or close to optimal convergence rate in the case of homogeneous Markov chains. The answer turns out to be positive for appropriate modifications of these ''other methods''. Their analogues for the non-homogeneous Markov chains will be also presented.

Elena Yarovaya, Lomonosov MSU, Russia

The behavior of a branching random walk depending on the lattice dimension

We study a continuous-time symmetric branching random walk on a multidimensional lattice Z^{d}, d ∈ ℕ, with a single branching source, i.e. a source of birth- and death of particles. Conditions for the “inheritance” of properties of the underlying random walks in the transition from Z^{d} to Z^{d+1} are established. A limit theorem is proved on the representation of the positive eigenvalue of the evolution operator of the mean particle numbers in a supercritical branching random walk in terms of the process parameters and the lattice dimension d as the branching source intensity tends to infinity. 

The recording: YouTube

March 18, 16:45 msk

Mikhail Zhitlukhin, Steklov Mathematical Institute of RAS, Russia

Problems of Statistical Sequential Analysis for Fractional Brownian Motion

The talk considers two classical problems of statistical sequential analysis: sequential hypothesis testing and the disorder detection (changepoint detection) problem. These problems have been extensively studied in the literature for standard Brownian motion and other diffusion processes. In our work they are investigated for fractional Brownian motion — a Gaussian process with dependent increments that generalizes standard Brownian motion and is widely used in applications. A key difficulty is that fractional Brownian motion is neither a Markov process nor a semimartingale, which prevents the direct application of classical methods of stochastic analysis. To address these problems, it is necessary to combine tools from the theory of Gaussian processes, fractional calculus, optimal control, and machine learning. The talk is based on joint work with A. N. Shiryaev and A. A. Muravlev.

The recording: YouTube

March 4, 16:45 msk

Oleg Vinogradov, Lomonosov MSU, Russia

Examples of "bridges" between some sections of probability theory

The report will indicate the connections between the various, at first glance, tasks of the theory of branching processes, the theory of ruin, the theory of queuing, and balloting for random flows, allowing new results to be obtained.

The recording: YouTube

February 18, 16:45 msk

Valentin Konakov, Lomonosov MSU, Russia

Local limit theorems and strong approximations for Robbins-Monro procedures

The parametrix method is a powerful analytical approach for constructing and analyzing fundamental solutions to parabolic equations and transition probability densities of solutions to stochastic differential equations. The "continuous" version of the method has a long history and dates back to the work of the Italian mathematician Eugenio Ella Levi (1907). However, the continuous version, which made it possible to develop a discrete analogue of the method, belongs to H. McKean, I. Singer (1967). A discrete version of the method was proposed in the article by K. and S. Molchanov (TV and MS, 1984), and a more detailed and general version in the work by K. and E. Mammen (PTRF, 2000). The method is effective for non-smooth (Hölder) coefficients of drift and diffusion. Modern research adapts the method to Kolmogorov degenerate diffusions and Markov chains. The work in question is motivated by the desire to find a concrete problem in which these methods work. The object of the study was the well-known stochastic approximation procedure proposed by Robbins and Monroe in 1951 and named after them. Markov chains related to this procedure were found and, apparently, for the first time, local limit theorems on convergence to the Gaussian diffusion process were obtained, Based on these results, strong invariance principles were obtained. The talk is based on joint works with Enno Mammen (Heidelberg university, Germany).

The recording: YouTube