arXiv subjects

Olga Izyumtseva

Publications and source records attributed to Olga Izyumtseva.

9 recordsLinked to original sources

arXiv2026

Coloured Epidemic Models: Functional Law of Large Numbers and Propagation of Chaos

In this paper, we study a stochastic Susceptible-Infected-Removed (SIR) model where the infection and the recovery rates depend on individual covariates for susceptibility and infectiousness of the infector and the infectee. Such models allow explicit nonlinearity in the incidence term. They are also important from a practical perspective, as they allow for the incorporation of individual heterogeneity into the epidemic process. Statistical estimates for crucial epidemiological parameters, such as the basic reproduction number, herd immunity threshold, could be vastly different, and even biased, when the population heterogeneity is ignored in the mathematical model. We describe our epidemic model as an Interacting Particle System (IPS) of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures. Our main mathematical contributions are a Functional Law of Large Numbers (FLLN), which approximates the empirical random measure of the IPS by means of a deterministic measure-valued function, and the propagation of chaos phenomenon, which establishes asymptotic independence of the particles as the population size goes to infinity with an explicit construction of McKean--Vlasov type Kac's ``nonlinear process''. We also briefly mention how the propagation of chaos phenomenon leads to a product-form likelihood function, which forms the basis of the so-called Dynamic Survival Analysis (DSA) method for parameter inference based on sparse data.

math.PR↗

arXiv2026

From Individual-Based Stochastic Epidemics to Heterogeneous SIR Equations

We develop a stochastic framework for a broad class of heterogeneous SIR epidemic models. In the finite-population construction, each initially susceptible individual is assigned a fixed nonnegative susceptibility, and infection occurs when the accumulated population-level infection pressure exceeds an individual random threshold. Infectious periods are independent and exponentially distributed with a common recovery rate. For any susceptibility distribution with finite mean, we prove a uniform-on-compact law of large numbers for the susceptible, infectious, removed, and cumulative infection-pressure processes. In the limit, an individual with susceptibility lambda remains susceptible under cumulative pressure x with probability exp(-x lambda). It follows that the susceptible fraction is given by the Laplace transform of the initial susceptibility distribution, while the incidence rate is governed by the mean susceptibility among those who remain susceptible. The resulting limits recover several familiar heterogeneous SIR systems, including the classical power-law model, and also yield other closed nonlinear incidence forms. The framework therefore provides a unified probabilistic foundation for deterministic epidemic models with persistent individual heterogeneity.

math.PR↗

arXiv2026

Self-intersection local times for Volterra Gaussian processes in stochastic flows with interaction

In this paper, we study self-intersection local times for a stochastic process $x(u(\cdot),t)$, where $u$ is a Gaussian process of the form $u(t)=\int^t_0k(t,s)\mathrm{d}{w(s)}$, $k$ is a deterministic kernel of the Volterra type, $w$ is a Wiener process, and $x$ is a solution to the \emph{equation with interaction}. Equations with interaction are a class of interacting particle system described by stochastic differential equations whose coefficients depend on a random measure (initial distribution of particles) transformed by the flow of solutions. Considering the occupation measure of $u$ as the initial condition for the equation with interaction allows us to define a stochastic flow with interaction driven by self-intersection local times of the process $u$. The study of such stochastic differential equations whose coefficients carry information about the geometric properties of curves is new. They previously appeared only for deterministic differential equations and smooth curves, where the geometric characteristics typically considered are length, curvature, and so on. In this paper, we prove the existence of multiple self-intersection local times for the process $x(u(\cdot),t)$ and establish a ``change of variable formula" that allows us to describe self-intersection local times for the process $x(u(\cdot),t)$ in terms of the weighted self-intersection local times for the process $u.$ We describe the corresponding asymptotics of the self-intersection local times for $x(u(\cdot),t)$ for large $t$. Moreover, the existence of weighted self-intersection local times is established for a large class of unbounded weights, which is of independent interest.

math.PR↗

arXiv2026

Stochastic Analysis of Entanglement-assisted Quantum Communication Channels

We present a queueing model for a quantum communication network consisting of a primary queue and a service queue in which Bell pairs are formed and stored. The Bell pairs are inherently extremely short-lived rendering the service queue (the quantum queue) much faster than the primary queue. We study the asymptotic behaviour of this multi-scale queueing system via a stochastic averaging principle. We prove a Functional Law of Large Numbers (FLLN) and a Functional Central Limit Theorem (FCLT) for the standard queue averaging the dynamics of the fast service queue.

math.PR↗

arXiv2024

Local times of self-intersection and sample path properties of Volterra Gaussian processes

We study a Volterra Gaussian process of the form $X(t)=\int^t_0K(t,s)d{W(s)},$ where $W$ is a Wiener process and $K$ is a continuous kernel. In dimension one, we prove a law of the iterated logarithm, discuss the existence of local times and verify a continuous dependence between the local time and the kernel that generates the process. Furthermore, we prove the existence of the Rosen renormalized self-intersection local times for a planar Gaussian Volterra process.

math.PR↗

arXiv2019

Self-intersection local times of random fields in stochastic flows

In this article we study transformations of Gaussian field by stochastic flow on the plane. A stochastic flow is a solution to the equation with interaction whose coefficients depend on the occupation measure of the field. We consider nonsmooth Gaussian field, which has self-intersection local times of any multiplicity. In the article we prove the existence of self-intersection local times for the transformed field and study its asymptotics.

math.PR↗

arXiv2016

On local time for the solution to a white noise driven heat equation

In this article we discuss the existence of local time for a class of Gaussian processes which appears as the solutions to some stochastic evolution equations. We show that on small intervals such processes are Gaussian integrators generated by a continuously invertible operators. This allows us to conclude that the considered processes have a local time on any finite interval with respect to spatial variable.

math.PR↗

arXiv2016

Clark formula for local time for one class of Gaussian processes

In the article we present chaotic decomposition and analog of the Clark formula for the local time of Gaussian integrators. Since the integral with respect to Gaussian integrator is understood in Skorokhod sense, then there exist more than one Clark representation for the local time. We present different representations and discuss the representation with the minimal L_2-norm.

math.PR↗

arXiv2016

Moments estimates for local times of one class of Gaussian processes

In present paper we prove an existence and give a moments estimate for the local time of Gaussian integrators. Every Gaussian integrator is associated with a continuous linear operator in the space of square integrable functions via white noise representation. Hence, all properties of such process are completely characterized by properties of the corresponding operator. We describe the sufficient conditions on continuous linear non-invertible operator which allow the local time of the integrator to exists at any real point. Moments estimate for local time is obtained. A continuous dependence of local time of Gaussian integrators on generating them operators is established. The received statement improves our result presented in [1].

math.PR↗