arXiv subjects

Olena Tymoshenko

Publications and source records attributed to Olena Tymoshenko.

9 recordsLinked to original sources

arXiv2026

A Psychometric and Practical Comparison of Standard Moodle-Based and STACK-Based Step-by-Step Tests in University Calculus

This paper compares two formats of online assessment in university Calculus: a standard Moodle-based step-by-step test and a STACK-based step-by-step test. Both tests assess integration by parts and divide the solution into consecutive response fields, but they differ in their validation mechanisms. The standard test relies on predefined scoring patterns, whereas the STACK-based test uses symbolic validation rules. The comparison is based on final manually verified scoring matrices from two student cohorts and follows a Classical Test Theory framework, including score distributions, reliability estimates, response-field-level indicators, and correlation-based measures. The results show high overall performance and ceiling effects in both formats. However, the STACK-based test required fewer manual corrections, showed higher internal consistency, and produced a more coherent relationship between solution steps and the total score.

math.HO↗

arXiv2026

Optimal Stopping for Systems Driven by the Brownian Sheet

We investigate optimal stopping problems for systems driven by the Brownian sheet. Our analysis is divided into two parts. In the first part we derive explicit solutions to two optimal stopping problems for the exponentially discounted Brownian sheet. The first problem consists in determining the optimal two-parameter first hitting point tau = (tau1,tau2) maximizing E[exp(-rho tau1 tau2) h(B(tau1,tau2))], where rho > 0 is a discount factor and h is a reward function. Restricting attention to first hitting points of levels, we obtain a closed-form characterization of the optimal stopping threshold. In particular, for linear rewards h(y)=y the optimal level is y_hat = (2 rho)^(-1/2). The second problem concerns optimal stopping of the integrated discounted Brownian sheet with payoff E[int_0^{tau1} int_0^{tau2} exp(-rho t x) B(t,x) dt dx]. We show that the optimal first hitting level is strictly positive and give an explicit representation of the value function in terms of the exponential integral function. The optimal threshold is characterized as the unique solution of a nonlinear equation derived from a Laplace transform identity for the product tau1 tau2. In the second and main part of the paper we develop a potential theoretic framework for two-parameter optimal stopping problems associated with stochastic partial differential equations driven by the Brownian sheet, proving that the value function is the least superharmonic majorant of the reward and establishing existence of optimal stopping points in the plane.

math.PR↗

arXiv2025

Extinction and Persistence in a Stochastic Mpox Model with Hawkes-type Self-Excitation

We develop a stochastic human-rodent compartment model for Mpox transmission that combines diffusion noise with Hawkes self-exciting jumps in the human infection dynamics. Including Hawkes processes allows, for instance, to model the short but significant spikes in transmission happening after crowded events. For the coupled human-rodent system, we prove global existence, uniqueness and positivity of solutions, derive a basic reproduction number R_0 that guarantees almost sure extinction when R_0 < 1, and obtain explicit persistence-in-the-mean conditions for both infected rodents and humans, which define persistence thresholds for the joint dynamics. Numerical experiments show how clustered human transmission events, environmental variability and control measures shift these thresholds and shape the frequency and size of Mpox outbreaks.

math.DS↗

arXiv2025

SPDE Games Driven by a Brownian Sheet with Applications to Pollution Minimization

This paper studies a nonzero-sum stochastic differential game in the context of shared spatial-domain pollution control. The pollution dynamics are governed by a stochastic partial differential equation (SPDE) driven by a Brownian sheet, capturing the stochastic nature of environmental fluctuations. Two players, representing different regions, aim to minimize their respective cost functionals, which balance pollution penalties with the cost of implementing control strategies. The nonzero-sum framework reflects the interdependent yet conflicting objectives of the players, where both cooperation and competition influence the outcomes. We derive necessary and sufficient conditions for Nash equilibrium strategies, using a maximum principle approach. This approach involves the introduction of a new pair of adjoint variables, (L_1, L_2), which do not appear in a corresponding formulation with the classical (1-parameter) Brownian motion. Finally, we apply our results to two case studies in pollution control, demonstrating how spatial and stochastic dynamics shape the equilibrium strategies.

math.OC↗

arXiv2024

Fokker-Planck equations for conditional McKean-Vlasov systems driven by Brownian sheets

We investigate conditional McKean-Vlasov equations driven by time-space white noise, motivated by the propagation of chaos in an N-particle system with space-time Ornstein-Uhlenbeck dynamics. The framework builds on the stochastic calculus of time-space white noise, utilizing tools such as the two-parameter Ito formula, Malliavin calculus, and orthogonal decompositions to analyze convergence and stochastic properties. Existence and uniqueness of solutions for the associated stochastic partial differential equations (SPDEs) are rigorously established. Additionally, an integral stochastic Fokker-Planck equation is derived for the conditional law, employing Fourier transform methods and stochastic analysis in the plane. The framework is further applied to a partial observation control problem, showcasing its potential for analyzing stochastic systems with conditional dynamics.

math.PR↗

arXiv2024

A Kalman filter for linear systems driven by time-space Brownian sheet

We study a linear filtering problem where the signal and observation processes are described as solutions of linear stochastic differential equations driven by time-space Brownian sheets. We derive a stochastic integral equation for the conditional value of the signal given the observation, which can be considered a time-space analogue of the classical Kalman filter. The result is illustrated with examples of the filtering problem involving noisy observations.

math.PR↗

arXiv2024

Insights on Stochastic Dynamics for Transmission of Monkeypox: Biological and Probabilistic Behaviour

The transmission of monkeypox is studied using a stochastic model taking into account the biological aspects, the contact mechanisms and the demographic factors together with the intrinsic uncertainties. Our results provide insight into the interaction between stochasticity and biological elements in the dynamics of monkeypox transmission. The rigorous mathematical analysis determines threshold parameters for disease persistence. For the proposed model, the existence of a unique global almost sure non-negative solution is proven. Conditions leading to disease extinction are established. Asymptotic properties of the model are investigated such as the speed of transmission.

math.DS↗

arXiv2024

Fokker-Planck equation for McKean-Vlasov SPDEs driven by time-space Brownian sheet

In this paper, we consider a McKean-Vlasov (mean-field) stochastic partial differential equations (SPDEs) driven by a Brownian sheet. We study the propagation of chaos for a space-time Ornstein-Uhlenbeck SPDE type. Subsequently, we prove the existence and uniqueness of a nonlinear McKean-Vlasov SPDE. Finally, we establish a Fokker-Planck equation for the law of the solution of the McKean-Vlasov type SPDE driven by a time-space Brownian sheet, and we provide some examples to illustrate the results obtained.

math.PR↗

arXiv2024

Optimal control of SPDEs driven by time-space Brownian motion

In this paper we study a Pontryagin type stochastic maximum principle for the optimal control of a system, where the state dynamics satisfy a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion (also called Brownian sheet). We first discuss some properties of a Brownian sheet driven linear SPDE which models the growth of an ecosystem. Further, applying time-space white noise calculus we derive sufficient conditions and necessary conditions of optimality of the control. Finally, we illustrate our results by solving a linear quadratic control problem and an optimal harvesting problem in the plane. We also study possible applications to machine learning.

math.OC↗