Posters

Uncovering Data Symmetries: Estimating Covariance Matrix in High-Dimensional Setting With ’gips’ R Package

Adam Przemysław Chojecki (Warsaw University of Technology, Faculty of Mathematics and Information Science)

Estimating high-dimensional covariance matrices is challenging when the sample has limited size. We present gips, an R package that automatically detects permutation symmetries among variables in a Gaussian model and uses them to constrain covariance estimation. By Bayesian inference over possible symmetry partitions, gips identifies groups of interchangeable dimensions and enforces these constraints to produce a reduced-variance estimator. In simulations, it accurately recovers true symmetries and matches popular shrinkage methods in accuracy while providing additional interpretation via symmetry structures.

This poster is based on Chojecki et al. \[2\], which presents the package in greater detail. gips is available on CRAN, complete with documentation \[3\], and examples, enabling straightforward application in data mining, computational biology, and other fields requiring structured covariance estimation.

Bibliography

\([1]\) Piotr Graczyk, Hideyuki Ishi, Bartosz Kołodziejek, Hélène Massam. Model selection in the space of Gaussian models invariant by symmetry. The Annals of Statistics, vol. 50, no. 3, pp. 1747–1774, June 2022. arXiv: 2004.03503; DOI: 10.1214/22-AOS2174.

\([2]\) Chojecki, A., Morgen, P., Kołodziejek, B. (2025). Learning Permutation Symmetry of a Gaussian Vector with gips in R. Journal of Statistical Software, vol. 112, no. 7, pp. 1–38. DOI: 10.18637/jss.v112.i07.

\([3]\) gips package documentation: przechoj.github.io/gips.

An asymptotically probabilistic method for a class of partial integrodifferential equations

Alioune Coulibaly (Université Amadou Mahtar Mbow de Diamniadio - Dakar - Sénégal)

In this paper, we consider a nonlocal boundary condition and examine the asymptotic behavior of the solution to a family of nonlocal partial differential equations in the half-space. Our approach is fully probabilistic and builds upon the works of Huang et al. (2022) and Diakhaby et al. (2016). Reflected stochastic differential equations, driven by multiplicative Lévy noise and with singular coefficients, play an important role in our method.
Let \(\left( b,c,\sigma\right) :\mathrm{\textbf{D}^{3}\times\mathbb{R}^{d}} \longrightarrow\mathbb{R}^{d}\), \(\left( \beta,\gamma,\varrho\right) :\mathrm{\partial\textbf{D}^{3}\times\mathbb{R}^{d-1}} \longrightarrow\mathbb{R}^{d}\), and \(\varepsilon\) be small positive. Our principal focus is the limit-solution, when \(\varepsilon\) goes to zero, of the following nonlocal partial differential equation (PDE) with rapidly oscillating coefficients : \[\label{eq1} \left\lbrace \begin{aligned} &\frac{\partial u^{\varepsilon}}{\partial t}(t,x)=\mathcal{L}^{\sigma,b,c}_{\varepsilon}u^{\varepsilon}(t,x)+\frac{\scriptstyle 1}{\scriptstyle \varepsilon}g\big(x_{\varepsilon},u^{\varepsilon}(t,x) \big).u^{\varepsilon}(t,x) , &x\in\mathrm{\textbf{D}},\\ & \mathcal{L}^{\varrho,\beta,\gamma}_{\varepsilon} u^{\varepsilon}(t,x)+\frac{\scriptstyle 1}{\scriptstyle \varepsilon}h\big( x_{\varepsilon},u^{\varepsilon}(t,x)\big).u^{\varepsilon}(t,x) =0, &x\in\partial\mathrm{\textbf{D}},\\ &u^{\varepsilon}(0,x)=u_{0}(x), &x\in\overline{\mathrm{\textbf{D}}}, \end{aligned} \right.\] where \(\mathrm{\textbf{D}}=\big\{\left(x^{1},\ldots,x^{d} \right) \in\mathbb{R}^{d} : x^{1}>0\big\}\). Definitively, the boundary \(\partial\mathrm{\textbf{D}}\) is supposed to be homeomorphic to \(\mathbb{R}^{d-1}\). Letting set \(x_{\varepsilon}:=\left(\frac{\scriptstyle x}{\scriptstyle \delta_{\varepsilon}}\right)_{\varepsilon>0},\ \delta_{\varepsilon}>0\), the family of linear integro-differential operators \(\mathcal{L}^{\varpi_{i},\tau_{i},\rho_{i}}_{\varepsilon}\) (\(i:=1,2\)) are given by \[\begin{split} \mathcal{L}^{\varpi_{i},\tau_{i},\lambda_{i}}_{\varepsilon}f(x) :=& \int_{\mathbb{R}^{d+1-i}_{*}}\bigg[f\left( x+\varepsilon\varpi_{i}\left(x_{\varepsilon},y\right)\right) - f(x) -\varepsilon\sum_{j=1}^{d}\varpi_{i}^{j}\left(x_{\varepsilon},y\right)\partial_{j}f(x)\bigg] \nu_{i}^{\varepsilon}(dy)\\%\boldsymbol{1}_{B} (y) &+\left(\frac{\scriptstyle \varepsilon}{\scriptstyle \delta_{\varepsilon}}\right)^{\alpha-1}\sum_{j=1}^{d}\tau_{i}^{j}\left(x_{\varepsilon}\right)\partial_{j}f(x)+\sum_{j=1}^{d}\lambda_{i}^{j}\left(x_{\varepsilon}\right)\partial_{j}f(x),\ x\in\mathbb{R}^{d+1-i}, \end{split}\] with \(\varpi_{i}\) in \(\bigl\{\sigma,\varrho\bigr\}\), \(\tau_{i}\) in \(\bigl\{b,\beta\bigr\}\) and \(\lambda_{i}\) in \(\bigl\{c,\gamma\bigr\}\) respectively.

Theorem. \([\)Coulibaly (2025)\(]\)
Suppose assumptions (A\(_1\))–(A\(_6\)) hold true. Then, for every \((t,x) \in\mathbb{R}^{*}_{+}\times\overline{\textrm{\textbf{D}}}\), \[\lim_{\varepsilon\downarrow 0}\varepsilon\log u^{\varepsilon}(t,x)=V^{*}(t,x)=\inf_{\tau\in\Theta}\sup_{\left\{\phi\in \mathcal{D}\left(\left[0,t\right],\overline{\textrm{\textbf{D}}}\right),\phi(0)=x,\phi(t)\in U_{0}\right\}}\Big\{\overline{\Pi}\tau-S_{0,\tau}\left(\phi\right)\Big\}.\]

Corollary. \(\quad\) \[\lim_{\varepsilon\downarrow 0}u^{\varepsilon}(t,x)=\left\{\begin{array}{l} 0 \textrm{\textit{ from any compact}} \textrm{ of the set }\Big\{(t,x)\in\mathbb{R}^{*}_{+}\times\overline{\textbf{D}};\ V^{*}(t,x)< 0\Big\},\\ 1 \textrm{\textit{ from any compact}}\textrm{ of the set } \Big\{(t,x)\in\mathbb{R}^{*}_{+}\times\overline{\textbf{D}};\ V^{*}(t,x)=0\Big\}. \end{array}\right.\]

Bibliography

\([1]\) Diakhaby A., Ouknine Y. (2016). Generalized BSDEs, weak convergence, and homogenization of semilinear PDEs with the Wentzell-type boundary condition. Stoch. Anal. Appl., 34, 496–509.
\([2]\) Coulibaly A., (2025). An asymptotically probabilistic method for a class of partial integrodifferential equations. AIMS Mathematics, 10(6): 13512–13523.
\([3]\) Huang Q., Duan J., Song R., (2022). Homogenization of nonlocal partial differential equations related to stochastic differential equations with Lévy noise, Bernoulli, 28, 1648–1674.

Risk-Sensitive First Exit Time Control with Varying and Constant Discount Factors on a General State Space with Approximation Algorithms

Amit Ghosh (Indian Institute of Technology Guwahati)

The risk-sensitive first exit time stochastic control problem for discrete-time on a general state space with state-dependent as well as constant discount factors has been analyzed. Under suitable assumptions, we prove the existence, Bellman characterization and uniqueness of optimal value function over randomized history-dependent policy space for bounded cost. We not only propose a Policy Improvement Algorithm (PIA) but also prove its convergence on general state space using stochastic representation of the optimal value function. To estimate the value function, a Value Iteration scheme is proposed. Lastly, we verified our convergence result through an example using MATLAB simulation.

Bibliography

\([1]\) Wei Q, Chen X. Continuous-time Markov decision processes under the risk-sensitive first passage discounted cost criterion. Journal of Optimization Theory and Applications. 2023 Apr;197(1):309-33.

\([2]\) Biswas A, Pradhan S. Ergodic risk-sensitive control of Markov processes on countable state space revisited. ESAIM: Control, Optimisation and Calculus of Variations. 2022;28:26.

Solution to Stochastic Loewner Equation with Several Complex Variables using Nevanlinna Theory

Andrej Srakar (University of Ljubljana)

Becker (1973) studied solutions to the Loewner differential equation in one complex variable using Carathéodory class of holomorphic functions. However, in several complex variables point singularities are removable and other approaches necessary. Pfaltzgraff generalized to higher dimensions the Loewner differential equation and developed existence and uniqueness theorems for its solutions. The existence and regularity theory has been considered by several authors, and applications given to the characterization of subclasses of biholomorphic mappings, univalence criteria, growth theorems and coefficient bounds for restricted classes of biholomorphic mappings. Duren et al. (2010) studied general form of solutions to the Loewner differential equation under common assumptions of holomorphicity and uniquely determined univalent subordination chains. To our knowledge, to date stochastic Loewner equation has not been studied in a several complex variable setting. We solve the equation in its ordinary and partial differential form by firstly appropriately defining Brownian motion in high dimensions, following Pitman and Yor (2018). We translate the problem in meromorphic form using generalizations of Nevanlinna theory for several complex variables (Noguchi and Winkelman, 2013) and previous work on invariance in the connections of Nevanlinna theory and stochastic processes (e.g. Atsuji, 1995). Finally, the equation is solved using general techniques from rough paths theory (see e.g. Hairer, 2013) and complex Feynman-Kac theorem (see e.g. Grothaus et al., 2010; Xu, 2015). Our solution allows to study stochastic phenomena such as Schramm-Loewner evolution in high dimensions (d>2) which answers one of important present open issues in probability theory and random geometry in particular. We shortly discuss convergence to a scaling limit for multidimensional lattice models as a power series problem on Hartogs domain. Connection between Nevanlinna theory in several complex variables and high-dimensional Brownian motions promises to provide a path to solving stochastic differential equations with several complex variables, which is at present an unaddressed issue in probability theory.

Convergence rate of Euler-Maruyama scheme for McKean-Vlasov SDEs with density-dependent drift

Anh-Dung Le (École nationale des ponts et chaussées)

In this paper, we study weak well-posedness of a McKean-Vlasov stochastic differential equations (SDEs) whose drift is density-dependent and whose diffusion is constant. The existence part is due to Hölder stability estimates of the associated Euler-Maruyama scheme. The uniqueness part is due to that of the associated Fokker-Planck equation. We also obtain convergence rate in weighted \(L^1\) norm for the Euler-Maruyama scheme.

Asymptotic Behaviour of Vertex-Shift Dynamics on Unimodular Networks

Bharath Roy Choudhury (Dioscuri Centre for Random Walks in Geometry and Topology, Jagiellonian University)

Vertex-shifts provide a framework for navigating on graphs by specifying, for each vertex, a unique “next” vertex. Such a shift induces dynamics on the space \(\mathcal{G}_*\) of rooted networks: starting from a probability measure \(\mathcal{P}\) on \(\mathcal{G}_*\), one obtains the set \(\mathcal{A}_{\mathcal{P}}\) of push-forward measures under the successive iterations of the vertex-shift. The accumulation points of \(\mathcal{A}_{\mathcal{P}}\), termed as \(f\)-probabilities, describe the long-term behaviour of these iterations. This paper establishes general sufficient conditions on \(f\)—namely periodicity and finiteness of its orbits—for the existence of \(f\)-probabilities. Furthermore, when \(f\)-probabilities exist, we give sufficient conditions under which they are either absolutely continuous or singular with respect to the initial measure \(\mathcal{P}\). In the special case where \(f\) is \(1\)-periodic and \(\mathcal{P}\) is unimodular, we derive the Radon-Nikodym derivative of the \(f\)-probability with respect to \(\mathcal{P}\).

Applications include the Parent vertex-shift on certain classes of unimodular Family Trees—where each vertex is mapped to its parent—yielding an explicit, unique \(f\)-probability. A second application considers the record vertex-shift on skip-free-to-the-left (also known as left-continuous) random walks with non-negative mean increments. In this setting, we derive the record probability, namely the \(f\)-probability associated to the record vertex-shift that maps each point on the trajectory of the random walk to its next record. The record probability describes the trajectory “seen from” the maximum value of the walk. These results unify dynamical and measure-theoretic perspectives on network navigation, extending tools from stochastic geometry and unimodular random graphs.

Some of the main results are as follows.

Theorem. Let \([\mathbf{G},\mathbf{o}]\) be a random network, \(\mathcal{P}\) be its distribution and \(f\) be an a.s. \(1\)-periodic vertex-shift on \([\mathbf{G},\mathbf{o}]\). Then, the \(f\)-probability \(\mathcal{P}^f\) of \(\mathcal{P}\) exists.

Moreover, if \([\mathbf{G},\mathbf{o}]\) is unimodular, then \(\mathcal{P}^f\) is absolutely continuous with respect to \(\mathcal{P}\). The limit function \(d^f_{\infty}(o)= \lim_{n \rightarrow \infty}d^f_n(o)\) exists and is almost surely finite under \(\mathcal{P}^f\). The Radon-Nikodym derivative of \(\mathcal{P}^f\) with respect to \(\mathcal{P}^f\) is given by \[\frac{d \mathcal{P}^f}{d \mathcal{P}}[G,o] = d_{\infty}(o) \mathbf{1}\{\text{o is a trap}\}.\]

Theorem. Let \([\mathbb{Z},0,X]\) be the marked rooted network whose vertices are \(\mathbb{Z}\), root is \(0\), and marks \(X=(X_n)_{n \in \mathbb{Z}}\) on edges \((n,n+1)_{n \in \mathbb{Z}}\) represent the increments of a skip-free-to-the-left random walk with \(\mathbb{E}[X_0] \geq 0\). Define the record vertex-shift \(R\) by \(R(i) = \inf\{j>i: \sum_{k=j}^{i-1}X_k \geq 0\}\) for all \(i \in \mathbb{Z}\). Then, the record probability \(\lim_{n \to \infty} [\mathbb{Z},R^n(0),X]\) exists. Moreover, this limit measure admits an explicit description via an encoding of a random Family Tree.

Bibliography

\([1]\) Bharath Roy Choudhury. Records of Stationary Processes and Unimodular Graphs. ArXiv, doi:10.48550/arXiv.2312.08121.

\([2]\) François Baccelli and Bharath Roy Choudhury. Genealogies of Records of Stochastic Processes with Stationary Increments as Unimodular Trees. ArXiv, doi:10.48550/arXiv.2403.05657.

\([3]\) François Baccelli, Mir-Omid Haji-Mirsadeghi, and Ali Khezeli. Eternal Family Trees and Dynamics on Unimodular Random Graphs. In Contemporary Mathematics, American Mathematical Society, Vol. 719, 2018, pp 85-127. doi:10.1090/conm/719/14471.

\([4]\) François Baccelli and Mir-Omid Haji-Mirsadeghi. Point-Map-Probabilities of a Point Process and Mecke’s Invariant Measure Equation. The Annals of Probability, Vol. 45, no. 3, 2017, pp 1723-1751.

Representation of a class of nonlinear SPDE driven by Lévy-space time noise

Boubaker Smii (King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia)

In this work, we discuss a class of nonlinear stochastic partial differential equations (SPDEs) driven by Lévy space-time noise. A perturbative strong solution is constructed using tree expansion, while the truncated moments of the solution are expressed as sums over a specific class of graphs. Additionally, we will discuss some applications.

Stable Thompson Sampling: Valid Inference via Variance Inflation

Budhaditya Halder (Rutgers University, New Brunswick)

We consider the problem of statistical inference when the data is collected via a Thompson Sampling-type algorithm. While Thompson Sampling (TS) is known to be both asymptotically optimal and empirically effective, its adaptive sampling scheme poses challenges for constructing confidence intervals for model parameters. We propose and analyze a variant of TS, called Stable Thompson Sampling, in which the posterior variance is inflated by a logarithmic factor. We show that this modification leads to asymptotically normal estimates of the arm means, despite the non-i.i.d. nature of the data. Importantly, this statistical benefit comes at a modest cost: the variance inflation increases regret by only a logarithmic factor compared to standard TS. Our results reveal a principled trade-off: by paying a small price in regret, one can enable valid statistical inference for adaptive decision-making algorithms.

Marcinkiewicz-Zygmund type strong law of large numbers for supOU processes

Danijel Grahovac (University of Osijek, Croatia)

Superpositions of Ornstein-Uhlenbeck type processes (supOU) form a versatile class of infinitely divisible stationary stochastic processes. These processes allow for independent modeling of the marginal distribution and the dependence structure. We present results on the almost sure growth of integrated supOU processes, establishing a Marcinkiewicz-Zygmund type strong law of large numbers. Unlike the classical version, the critical moment condition in our results depends not only on the finiteness of moments but also on the strength of dependence.

We further explore connections to weak limit theorems and demonstrate that the derived growth rates are optimal. Integrated supOU processes may exhibit intermittency, characterized by higher-order moments growing faster than predicted by limit theorems. However, our results demonstrate that large peaks, despite the unusual growth of moments, do not occur infinitely often. We also discuss possible extensions to mixed moving average processes.

The results presented are part of joint work with Péter Kevei (University of Szeged, Hungary), Nikolai N. Leonenko (Cardiff University, UK) and Murad S. Taqqu (Boston University, USA).

Bibliography

\([1]\) D. Grahovac, N.N. Leonenko, A. Sikorskii, M. Taqqu, The unusual properties of aggregated superpositions of Ornstein-Uhlenbeck type processes, Bernoulli 25/3 (2019), 2029-2050

\([2]\) D. Grahovac, N.N. Leonenko, M. Taqqu, Limit theorems, scaling of moments and intermittency for integrated finite variance supOU processes, Stochastic Processes and their Applications 129/12 (2019), 5113-5150

\([3]\) D. Grahovac, N.N. Leonenko, M. Taqqu, The multifaceted behavior of integrated supOU processes: The infinite variance case, Journal of Theoretical Probability 33 (2020), 1801-1831

\([4]\) D. Grahovac, N.N. Leonenko, M. Taqqu, Intermittency and Multiscaling in Limit Theorems, Fractals 30/7 (2022), 1-18

\([5]\) D. Grahovac, P. Kevei, Almost sure growth of integrated supOU processes, Bernoulli (2025), accepted

Functional convergence of self-normalized partial sums of linear processes with random coefficients

Danijel Krizmanic (University of Rijeka)

We derive a self-normalized functional limit theorem for strictly stationary linear processes with i.i.d. heavy-tailed innovations and random coefficients under the condition that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series. The convergence takes part in the space of càdlàg functions on \([0,1]\) with the Skorokhod \(M_{2}\) topology.

Bibliography

\([1]\) Danijel Krizmanić. "A functional limit theorem for self-normalized linear processes with random coefficients and i.i.d. heavy-tailed innovations." Lithuanian Mathematical Journal, vol. 63, 2023, pp. 32.

Stochastic Dynamic Machine Scheduling with Interruptible Set-up Times

Dongnuan Tian (Lancaster University)

We consider a problem in which a machine is scheduled dynamically within a network to process jobs at different demand points. Costs are accumulated as jobs wait to be processed. The time needed for a machine to travel between two nodes in the network represents the “set-up time” needed to switch from processing one type of job to another. By using a network formulation, we can model complex relationships in switching times between different types of activity and also allow the switching times to be interrupted. The problem can be formulated as a Markov decision process in which arrival times, service times and switching times are uncertain and the objective is to minimize the expected long-run average cost. However exact solutions using dynamic programming are not possible due to the complexity of the state space. Heuristics for certain special cases of the problem have been proposed in the literature and in this talk we discuss how to adapt these heuristics to our problem. We also discuss how forward-thinking strategies can be used to develop improved heuristics and show the results obtained by applying these methods to networks of various different configurations.

Differential equations driven by exponential Besov-Orlicz signals

František Hendrych (Charles University)

The poster describes the extention of the rough path theory to cover paths from the exponential Besov-Orlicz space \[B^\alpha_{\Phi_\beta,q}\quad\mbox{ for }\quad \alpha\in (1/3,1/2],\,\quad \Phi_\beta(x) \sim \mathrm{e}^{x^\beta}-1\quad\mbox{with}\quad \beta\in (0,\infty), \quad\mbox{and}\quad q\in (0,\infty],\] which is then used to treat nonlinear differential equations driven by such paths. The exponential Besov-Orlicz-type spaces, rough paths, and controlled rough paths are defined, a sewing lemma for such paths is given, and the existence and uniqueness of the solution to differential equations driven by these paths is stated. The results cover equations driven by paths of continuous local martingales with Lipschitz continuous quadratic variation (e.g. the Wiener process) or by paths of fractionally filtered Hermite processes in the \(n\)^th Wiener chaos with Hurst parameter \(H\in (1/3,1/2]\) (e.g. the fractional Brownian motion).

Fractional Brownian motions with random Hurst exponent

Hubert Woszczek (Wrocław University of Science and Technology)

We present several results concerning two types of fractional Brownian motions with a random Hurst exponent, which are inspired by recent biological experiments in single particle tracking. In both cases, we introduce basic probabilistic properties such as the q-th moment of the absolute value of the process, the autocovariance function, and the expectation of the time-averaged mean squared displacement. Furthermore, we analyze the ergodic properties of both processes. Alongside the theoretical analysis, we also provide a numerical study of the results. The poster is based on the results of \([1]\).

Bibliography

\([1]\) Hubert Woszczek, Agnieszka Wyłomańska, Aleksei Chechkin. "Riemann–Liouville Fractional Brownian Motion with Random Hurst Exponent." Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 35, no. 2, 2025, pp. 0243975.

Bell shaped sequences and first passages locations for two-dimensional random walks

Jacek Wszoła (Wrocław University of Science and Technology)

Bell-shaped functions have been studied since the 1940s and are widely used in probability theory. Our goal is to define and describe their discrete counterparts – bell-shaped sequences.

We will also consider a two-dimensional random walk \((X_n, Y_n)\) on the lattice \(\mathbb{Z}^2\), with diagonal jumps \((\pm 1, \pm1)\) and jump probabilities depending only on the position of the second coordinate \(Y_n\). The first passage location is a random variable \(X_\tau\), where \(\tau = \min\{n \geq 0: Y_n=0\}\) denotes the first passage time to zero. We will discuss results concerning the distribution of the first passage locations and its connections to bell-shaped sequences.

Bibliography

\([1]\) M. Kwaśnicki, J. Wszoła. ‘Bell-shaped sequences.’ Studia Mathematica, 271 (2023), 151-185.

\([2]\) M. Kwaśnicki, J. Wszoła. ‘Two-sided bell-shaped sequences.’ Preprint, ArXiv:2404.11274

\([3]\) J. Wszoła. ‘First passage locations for two-dimensional lattice random walks and the bell-shape.’ Preprint, ArXiv:2501.14393

The distributions of the mean of random vectors with fixed marginal distribution

Jacques Labuschagne (UNIWERSYTET ŁÓDZKI)

This poster will be based upon our publication, in which we used the recent results concerning the non-uniqueness of the center of the mix for completely mixable probability distributions we obtain the following result: For each \(d\in\mathbb N\) and each non-empty bounded Borel set \(B\subset\mathbb R^d\) there exists a \(d\)-dimensional probability distribution \(\boldsymbol\mu\) satisfying what follows: for each \(n\geq3\) and each probability distribution \(\boldsymbol\nu\) on \(B\) there exist \(d\)-dimensional random vectors \(\mathbf X_{\boldsymbol\nu,1},\mathbf X_{\boldsymbol\nu,2},\dots,\mathbf X_{\boldsymbol\nu,n}\) such that \(\frac1n(\mathbf X_{\boldsymbol\nu,1}+\mathbf X_{\boldsymbol\nu,2}+\dots+\mathbf X_{\boldsymbol\nu,n})\sim\boldsymbol\nu\) and \(\mathbf X_{\boldsymbol\nu,i}\sim\boldsymbol\mu\) for \(i=1,2,\dots,n\). We also show that the assumption regarding the boundedness of set \(B\) cannot be completely omitted, but it can be substantially weakened.

Bibliography

\([1]\) A. Komisarski, J. Labuschagne, The Distributions of the Mean of Random Vectors with Fixed Marginal Distribution, J. Theor. Probab. (2023), https://doi.org/10.1007/s10959-023-01277-2

Evolution of a quantitative trait in a metapopulation setting: Propagation of chaos meets adaptive dynamics

Josué Tchouanti Fotso (LIFEWARE, Inria Center of Saclay)

This work is about studying the role of migration as a key driver of the evolution of spatially structured populations. To this end, we consider a metapopulation setting where evolutionary processes at the level of each patch is modelled by a Moran model, which describes the evolution of a quantitative trait in a population of fixed size by two main mechanisms : trait resampling and mutations. Migrations are added in order to take into account interactions between patches and the question we would like to answer is: how do these migrations influence the long term evolution of the population at the level of a single patch and at the level of the entire metapopulation ?

For this purpose, we study several scaling limits of the model. Assuming rare mutations and migrations, we adapt a technique from Champagnat & Lambert \([1]\) in order to get a mean-field network of Trait Substitution Sequence (TSS) describing long-term successive dominant traits in each patch. We derive the propagation of chaos as the metapopulation becomes large. Patches are therefore i.i.d copies of each other, with a TSS described by a McKean-Vlasov pure jump process. In the limit where mutations have small effects and migration is further slowed down accordingly, we obtain the convergence of the TSS, in the new migration timescale, to the solution of a stochastic differential equation which can be referred to as a new canonical equation of adaptive dynamics. This equation includes an advection term representing selection, a diffusive term due to the genetic drift, and a jump term – representing the effect of migration – to a state distributed according to the law of the solution.

See the details in Lambert et al. \([2]\).

Bibliography

\([1]\) N. Champagnat, A. Lambert. Evolution of discrete populations and the canonical diffusion of adaptative dynamics. The Annals of Applied Probability, 17(1):102–155, 2007.

\([2]\) A. Lambert, H. Leman, H. Morlon, J. Tchouanti. Evolution of a trait distributed over a large fragmented population : Propagation of chaos meets adaptive dynamics. Preprint arxiv:2503.13154, 2025.

Pricing options on the cryptocurrency futures contracts

Julia Kończal (Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, 50-370 Wrocław, Poland)

The cryptocurrency options market is notable for its high volatility and lower liquidity compared to traditional markets. These characteristics introduce significant challenges to traditional option pricing methodologies. Addressing these complexities requires advanced models that can effectively capture the dynamics of the market. We explore which option pricing models are most effective in valuing cryptocurrency options. Specifically, we calibrate and evaluate the performance of the Black—Scholes, Merton Jump Diffusion, Variance Gamma, Kou, Heston, and Bates models. Our analysis focuses on pricing vanilla options on futures contracts for Bitcoin (BTC) and Ether (ETH). We find that the Black–Scholes model exhibits the highest pricing errors. In contrast, the Kou and Bates models achieve the lowest errors, with the Kou model performing the best for the BTC options and the Bates model for ETH options. The results highlight the importance of incorporating jumps and stochastic volatility into pricing models to better reflect the behavior of these assets.

Asymptotically Distribution-free Goodness-of-Fit Testing for Point Processes

Justin Baars (Universiteit van Amsterdam)

Consider an observation of a multivariate temporal point process \(N\) with law \(\mathcal P\) on the time interval \([0,T]\). To test the null hypothesis that \(\mathcal P\) belongs to a given parametric family, we construct a convergent compensated counting process to which we apply an innovation martingale transformation. We prove that the resulting process converges weakly to a standard Wiener process. Consequently, taking a suitable functional of this process yields an asymptotically distribution-free goodness-of-fit test for point processes. For several standard tests based on the increments of this transformed process, we establish consistency under alternative hypotheses. Finally, we assess the performance of the proposed testing procedure through a Monte Carlo simulation study and illustrate its practical utility with two real-data examples.

Doubly stochastic resetting

Kacper Taźbierski (WUST)

Stochastic resetting is a very potent branch of stochastic processes. It can model sudden (or not!) returns of the process or some of its properties to their initial state.

One of the forms of stochastic resetting is the so-called partial resetting, where at the moment of the reset the value of the process changes according to the rule \(X(\tau)= c X(\tau^-)\), \(c\in [0,1]\), and the process loses its memory. We will discuss the problems of complete and memory resetting, which are the special cases where \(c=0\) and \(c=1\). The main focus is put on the case where the resetting mechanism is guided by a Cox process. There, some interesting anomalous diffusion properties emerge.

Theorem 1. \([1]\) Let the underlying stochastic process \(X(t)\) display anomalous diffusion with exponent \(\alpha\) for large times and a memory resetting mechanism be guided by a mixed Poisson process with random intensity \(R\) and PDF displaying power law \(p_R(r)\stackrel{r\downarrow 0}{\sim}L(r)r^{\varrho-1}\). Here \(L\) is a slowly varying function and \(\varrho>0\). Then the process under resetting displays anomalous behavior with exponent \(\alpha-\varrho\) if \(\varrho<\alpha-1\) and normal diffusion otherwise.

Theorem 2. \([2]\) Let the underlying stochastic process \(X(t)\) display anomalous diffusion with exponent \(\alpha\) for large times and a complete resetting mechanism be guided by a Cox process with random intensity function \(r(t)\) scaling like \(t^{\varrho-1}\) for \(\varrho>0\). Then the process under resetting displays anomalous behavior with exponent \(\alpha(1-\varrho)\).

Bibliography

\([1]\) Taźbierski, K., Magdziarz, M., Metzler, R. Stochastic memory resetting. In preparation

\([2]\) Taźbierski, K., Magdziarz, M., Doubly stochastic resetting. In preparation

Identification of the heavy-tailed behaviour using modified Greenwood statistic - univariate and multivariate case.

Katarzyna Skowronek (Politechnika Wrocławska)

In this work, we present a methodology based on a modified Greenwood statistic for statistical testing in various scenarios for univariate and multivariate data. Classical Greenwood statistic is defined for positive random variables, while it’s modified version is defined for all real-valued random variables. One of the most important properties of the classical and modified Greenwood statistics is their stochastic monotonicity within the star-shaped ordered distribution. We utilize this property to propose statistical tests for the normal distribution for the classes of \(\alpha\)-stable and Student’s t distributions, and for infinite variance in the classes of Pareto and Student’s t distributions.
Moreover, a relatively simple form of the modified Greenwood statistic allows a straightforward extension to the multivariate case. We prove that the stochastic monotonicity of the modified Greenwood statistic holds for multivariate star-shaped ordered distributions. Therefore, we expand the proposed tests from the univariate to the multivariate case. In this work, we limit our analysis to the two- and three-dimensional distributions.
To validate the efficiency of the proposed methods, we demonstrate the results of the proposed methods via Monte Carlo simulations. In addition, we compare introduced tests with the methods known in the literature.

Bibliography

\([1]\) K. Skowronek, M. Arendarczyk, R. Zimroz, A. Wyłomańska: Modified Greenwood statistic and its application for statistical testing, Journal of Computational and Applied Mathematics 452, 116122, 2024

\([2]\) K. Skowronek, M. Arendarczyk, A. K. Panorska, T. J. Kozubowski, A. Wyłomańska: Testing and estimation of the index of stability of univariate and bivariate symmetric alpha-stable distributions via modified Greenwood statistic, Journal of Computational and Applied Mathematics 467, 116587, 2025

Neural network correction for numerical solutions of stochastic differential equations

Marcin Miśkiewicz (Wrocław University of Science and Technology)

Neural networks, known for their universal approximation capabilities, are widely used to solve ordinary and partial differential equations. In addition to directly approximating solutions, deep learning models can be incorporated into conventional numerical schemes to reduce the associated local truncation error. In this work, we explore the idea of a similar enhancement in the context of non-deterministic systems by introducing a neural network as a correction term in the Euler–Maruyama and Milstein methods for SDEs. Without assuming any prior knowledge of the exact solution, a single training run for a given equation enables the subsequent generation of more accurate trajectories. As demonstrated numerically on several test SDEs, augmenting the baseline scheme reduces the global approximation error in both the weak and strong sense.

Learning optimal search strategies

Maximilian Philipp Thiel (Friedrich Schiller University Jena)

We investigate a repeated search problem in a stochastic environment. In particular, we deal with the so-called parking lot problem in continuous time. We model the arrival of free parking lots by an (unknown) inhomogeneous Poisson process. The search for a free parking lot near a desired location is repeated and optimal stopping rules, i.e. the time at which the first free parking space is taken, are to be determined based on the available observations. The analysis of the problem therefore requires results and approaches from stochastic control theory, reinforcement learning and, in this context, bandit problems.
We develop an algorithm to determine optimal stopping rules under uncertainty. The algorithm achieves an asymptotically logarithmic regret rate under mild conditions on the Poisson process. Using a minimax criterion from statistical decision theory, we can show that a logarithmic rate is a lower bound for the regret in this problem, which in turn means that our algorithm is asymptotically optimal.

Time Scale Transformation in Bivariate Pearson Diffusions: A Shift from Light to Heavy Tails

Nenad Šuvak (NA)

In applications that require addressing heavy-tailed behavior, heavy-tailed Pearson diffusions provide a natural alternative to well-known Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes. All three heavy-tailed Pearson diffusions, having inverse gamma, Fisher-Snedecor (\(F\)) and Student stationary distributions, can be constructed via an absolutely continuous time-change employed in a specific functional transformation of CIR or OU process. Moreover, time-change rates in stochastic clocks are continuous functionals of the CIR process. \(F\)-diffusion, heavy-tailed Pearson diffusion with a stationary \(F\) distribution, is derived from the bivariate CIR process \(\left( X_{1}(t), X_{2}(t) \right)_{t \geq 0}\) given by SDE \[\tag{1} \left\{ \begin{array}{l} dX_{1}(t) = -b \left( X_{1}(t) - \beta_{1} \right) \, dt + \sqrt{c X_{1}(t)} \, dW_{1}(t) \\ dX_{2}(t) = -b \left( X_{2}(t) - \beta_{2} \right) \, dt + \sqrt{c X_{2}(t)} \, dW_{2}(t), \end{array} \right.\] where \(\left( W_{1}(t) \right)_{t \geq 0}\) and \(\left( W_{2}(t) \right)_{t \geq 0}\) are independent standard Brownian motions. The parameters \(\beta_{1}\) and \(\beta_{2}\) are positive and not necessarily identical, while parameters \(b > 0\) and \(c > 0\), governing the speed of the mean reversion and volatility, are identical in both univariate CIRs. The motivation for defining the functional transformation of \(\left( X_{1}(t), X_{2}(t) \right)_{t \geq 0}\) comes from the relationship between two independent gamma distributions and the \(F\)-distribution. In particular, if \(X_{1} \sim \Gamma(\alpha \beta_{1}, \alpha)\) and \(X_{2} \sim \Gamma(\alpha \beta_{2}, \alpha)\) are independent, where \(\alpha = 2b/c\) and \(\beta_{2}, b, c > 0\), then the random variable \(Y = \beta_{2} X_{1}/\beta_{1} X_{2}\) has an \(F\) distribution with shape parameters \(2\alpha\beta_{1}\) and \(2\alpha\beta_{2}\), denoted by \(Y \sim F(2\alpha\beta_{1}, 2\alpha\beta_{2})\). The procedure for construction of time-changed \(F\)-diffusion is given in the following proposition.

Proposition. Let \(\left( X_{1}(t), X_{2}(t) \right)_{t \geq 0}\) be a two-dimensional CIR process with independent components given by SDE \((1)\), satisfying the condition \(\beta_{2} > (c/b) \max\{1, c/2\}\), and let \(\left( Y_{1}(t), Y_{2}(t) \right)_{t \geq 0}\) be its functional, given as follows: \[Y_{1}(t) = \displaystyle\frac{X_{1}(t)}{X_{2}(t)}, \quad Y_{2}(t) = X_{2}(t).\] If the time-change process \(\left( T(t) \right)_{t \geq 0}\) is defined as \[T(t) = \displaystyle\int_{0}^{t} \frac{1}{Y_{2}(s)} \, ds,\] then it has continuous inverse \((\tau_{t})_{t \geq 0}\) given by \[\tau_{t}(\omega) = \tau(t, \omega) = \inf{\{s \geq 0: T_{t} > t\}},\] and \(\left( Y_{1}(\tau_{t}) \right)_{t \geq 0}\) is \(F\)-diffusion driven by the Brownian motion \[\widetilde{W}^{*}_{1}(t) = \displaystyle\int_{0}^{\tau_{t}} \sqrt{\frac{1}{Y_{2}(s)}} \, d\widetilde{W}_{1}(s), \, \, \widetilde{W}_{1}(t) = \sqrt{\frac{4 Y_{2}(t)}{Y_{1}(t) \left( Y_{1}(t) + 4 \right)}} \left(\sqrt{\frac{Y_{1}(t)}{Y_{2}(t)}} W_{1}(t) - \sqrt{\frac{Y_{1}^{2}(t)}{4 Y_{2}(t)}} W_{2}(t) \right),\] where \(\left( W_{i}(t) \right)_{t \geq 0}\), \(i = 1, 2\), are independent standard Brownian motions driving process \((1)\).

Bibliography

\([1]\) Avram, F., Leonenko, N., Šuvak, N. "On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusions." Markov Process. Relat. Fields, vol. 19, no. 2, 2013, pp. 249-298.

\([2]\) Barndorff-Nielsen, O., Shiryaev, A. (2010). Change of Time and Change of Measure. World Scientific.

\([3]\) Gouriéroux, C., Jasiak, J. "Multivariate Jacobi process with application to smooth transitions." J. Econom., vol. 131, no. 1-2, 2006, pp 475-505.

\([4]\) Øksendal, B. "When is a stochastic integral a time change of a diffusion?" J. Theor. Probab. vol. 3, 1990, pp 207-226.

\([5]\) Šuvak N. "Time Scale Transformation in Bivariate Pearson Diffusions: A Shift from Light to Heavy Tails." Axioms., vol. 13, no. 11, 2024, pp. 765.

Risk Control in Federated Learning via Threshold Aggregation

Onrina Chandra (Rutgers University)

Introduction

In federated learning, multiple local servers collaboratively train models without sharing raw data. However, aggregating their predictions can result in uncontrolled risk due to data heterogeneity and model uncertainty. We propose a novel risk control framework that leverages knowledge transfer to calibrate and control risk across local servers. Our approach assumes that each server has a threshold parameter to bound the risk at a desired level of confidence, even when the local models are black-box. By aggregating these thresholds through a calibrated weighting scheme, our method guarantees that the overall risk remains below a target level with high probability.

Model Setup

Let \(K\) be the number of local servers. For each server \(k \in \{1, \dots, K\}\):

• \(\{(X^{(k)}_i, Y^{(k)}_i)\}_{i=1}^{n_k} \sim P_k\) are exchangeable observations.

• Data is split into training and calibration sets: \(I^{(k)}_{\text{train}}\) and \(I^{(k)}_{\text{cal}}\).

• A score function \(S(Y, X)\) is defined.

• Set-valued predictor \(\mathcal{T}^k_\lambda: \mathcal{X} \to \mathcal{Y}'\), e.g., \[\mathcal{T}^k_\lambda(X) = \{ Y : S(X, Y) \leq \lambda \}.\]

• Predictors are nested: \(\lambda_1 < \lambda_2 \Rightarrow \mathcal{T}^k_{\lambda_1}(x) \subset \mathcal{T}^k_{\lambda_2}(x)\).

• Loss function \(L_k(Y, \mathcal{T}^k_\lambda(X))\), e.g., \[L_k(Y, \mathcal{T}^k_\lambda(X)) = 1 - \frac{|Y \cap \mathcal{T}^k_\lambda(X)|}{|Y|}.\]

Local Risk

• Local risk for server \(k\) is \[R_k(\lambda) = \mathbb{E}_{(X,Y) \in I_{\text{cal}}^{(k)}} \left[ L_k(Y, \mathcal{T}^k_\lambda(X)) \right].\]

• For a fixed cutoff \(\alpha\), assume \(R_k(\lambda) \leq \alpha\) for all \(\lambda \in \Lambda\), for all \(k\).

• Fix \(M\) levels \(\delta_1, \delta_2, \ldots, \delta_M \in [0,1]\). Each server provides \(M\) threshold-risk pairs \(\{(\tilde{\lambda}^{(k)}_{\delta_m}, \delta_m)\}_{m=1}^M\). Given only the threshold-risk pairs \(\{(\tilde{\lambda}^{(k)}_{\delta_m}, \delta_m)\}_{m=1}^M\) and cutoff \(\alpha\), can we control the global risk at a level \(\gamma\)? We employ conformal prediction to aggregate the values \(\{ \{ \tilde{\lambda}^{(k)}_{\delta_m}, \delta_m \}_{m=1}^M \}_{k=1}^K\) and define a parameter \(\tilde{\lambda}_\gamma\) such that \[\mathbb{P}_{\{I^{(k)}_{\text{cal}}\}_{k=1}^K} \left( R(\tilde{\lambda}_\gamma) \leq \alpha \right) \geq 1 - \gamma\]

Bibliography

• Bates, Stephen, et al. "Distribution-free, risk-controlling prediction sets." Journal of the ACM (JACM), 68.6 (2021).

• Mohri, Christopher, and Tatsunori Hashimoto. "Language models with conformal factuality guarantees."

• Lu, Charles, et al. "Federated conformal predictors for distributed uncertainty quantification."

• Angelopoulos, Anastasios N., et al. "Conformal risk control." arXiv preprint arXiv:2208.02814 (2022).

Load Balancing in Heterogeneous Systems

Ozge Tekin (University of Edinburgh)

We analyze load balancing systems where servers are heterogeneous at an individual server level. For these systems, each server maintains its own queue and a join-the-shortest queue (JSQ) mechanism is employed, i.e., the jobs are routed to one of the shortest queues upon arrival. The JSQ policy is enhanced by a hybrid tie-breaking rule: when multiple servers share the shortest queue length, jobs may be routed to idle and busy servers according to potentially different policies, such as fastest-server-first or totally blind policies. Under Halfin–Whitt scaling, we prove that the diffusion-scaled system state converges to a two-dimensional reflected diffusion, with drift determined by the limiting measure-valued processes. To do so, we introduce measure-valued stochastic processes to describe cumulative idleness and queue occupancy among servers whose service rates lie in a given Borel subset of the support of the service rates. This result generalizes classical JSQ diffusion limits to heterogeneous-server settings. The limiting equations, together with explicit characterizations under specific policies, suggest how different tie-breaking rules may influence the system performance.

Bibliography

\([1]\) Patrick Eschenfeldt and David Gamarnik. "Join the shortest queue with many servers. The heavy-traffic asymptotics." Mathematics of Operations Research, vol. 43, no. 3, 2018, 867-886.

\([2]\) Burak Büke and Wenyi Qin. "Many-server queues with random service rates: A unified framework based on measure-valued processes." Mathematics of Operations Research, vol. 48, no. 2, 2023, 748–783.

\([3]\) Sanidhay Bhambay, Burak Büke and Arpan Mukhopadhyay. "Asymptotic Optimality of the Speed-Aware Join-the-Shortest-Queue in the Halfin-Whitt Regime for Heterogeneous Systems.", Stochastic Systems, 2025.

Path-dependent option pricing with two-dimensional PDE using MPDATA

Paweł Magnuszewski (AGH University of Krakow)

In this poster, we discuss a simple yet robust PDE method for evaluating path-dependent Asian-style options using the non-oscillatory forward-in-time second-order MPDATA finite-difference scheme. The valuation methodology involves casting the Black-Merton-Scholes equation as a transport problem by first transforming it into a homogeneous advection-diffusion PDE via variable substitution, and then expressing the diffusion term as an advective flux using the pseudo-velocity technique. As a result, all terms of the Black-Merton-Sholes equation are consistently represented using a single high-order numerical scheme for the advection operator. We detail the additional steps required to solve the two-dimensional valuation problem compared to MPDATA valuations of vanilla instruments documented in a prior study. Using test cases employing fixed-strike instruments, we validate the solutions against Monte Carlo valuations, as well as against an approximate analytical solution in which geometric instead of arithmetic averaging is used. The analysis highlights the critical importance of the MPDATA corrective steps that improve the solution over the underlying first-order "upwind" step. The introduced valuation scheme is robust: conservative, non-oscillatory, and positive-definite; yet lucid: explicit in time, engendering intuitive stability-condition interpretation and inflow/outflow boundary-condition heuristics. MPDATA is particularly well suited for two-dimensional problems as it is not a dimensionally split scheme. The documented valuation workflow also constitutes a useful two-dimensional case for testing advection schemes featuring both Monte Carlo solutions and analytic bounds. An implementation of the introduced valuation workflow, based on the PyMPDATA package and the Numba Just-In-Time compiler for Python, is provided as free and open source software.

From Text to Trends: The Feasibility of LLMs in Quantitative Finance

Rahul Tak (Bucharest University of Economic Studiies)

Predicting stock movements remains a formidable challenge due to the dynamic nature of financial markets and the impact of external factors like investor sentiment,macroeconomic events, etc. Conventional time-series statistical models frequently fail to capture the intricate nonlinear dependencies and latent market signals present in unstructured data. Recent advancements in Large Language Models (LLMs) offer new opportunities to enhance predictive accuracy by integrating textual data from financial news, social media, and earnings reports. In this study, we used the Chronos and LLaMA families of LLMs to evaluate the stock price forecasts generated by combining historical price data with news sentiment. We compare the performance of LLaMA-3.3, LLaMA-3.1, Chronos, and a benchmark ARIMA model to assess their efficacy in capturing temporal patterns and textual signals. Our findings reveal that while LLaMA-3.3 effectively extracts sentiment cues, all models exhibit limitations in accurately predicting market direction. Furthermore, our analysis suggest that market sentiment influences stock returns, particularly in driving short-term returns changes. By incorporating news sentiment into LLM prompts, we achieve improved forecasting performance compared to models relying solely on numerical data. These results underscore the importance of integrating structured (time series) and unstructured (sentiment) data for robust financial modeling. Our study suggests that LLM-driven sentiment analysis holds considerable promise for traders, analysts, and financial institutions seeking enhanced market insights. Future research should explore fine-tuning LLaMA models for domain-specific financial applications and improving interpretability in investment decision-making processes.
Keywords: LLMs, LLaMA, Chronos, News sentiment, Time series forecasting, Stock price

Bibliography

\([1]\) Alamsyah, A., Ayu, S. P., & Rikumahu, B. (2019). "Exploring relationship between headline news sentiment and stock return" In 2019 7th International Conference on Information and Communication Technology (ICoICT). (pp. 1-6). IEEE.

\([2]\) Dubey, A., Jauhri, A., Pandey, A., Kadian, A., Al-Dahle, A., Letman, A., ... & Ganapathy, R. (2024). "The llama 3 herd of models." arXiv preprint, arXiv:2407.21783.

\([3]\) Hossain, A., & Nasser, M. (2011). "Comparison of the finite mixture of ARMA-GARCH, back propagation neural networks and support-vector machines in forecasting financial returns." Journal of Applied Statistics, 38(3), 533-551.

\([4]\) Ansari, A. F., Stella, L., Turkmen, C., Zhang, X., Mercado, P., Shen, H., ... & Wang, Y. (2024). "Chronos: Learning the language of time series." arXiv preprint, arXiv:2403.07815.

\([5]\) Xu, Y., Liang, C., Li, Y., & Huynh, T. L. (2022). "News sentiment and stock return: Evidence from managers’ news coverages." Finance Research Letters, 48, 102959.

Weak convergence of stochastic integrals with applications to SPDEs

Salim Boukfal (Universitat Autònoma de Barcelona (UAB))

In the present work, see \([1]\) and \([2]\), We provide sufficient conditions for sequences of stochastic processes defined by stochastic integrals of the form \(\int_{D} f(x,u)\theta_n(u)du\), \(u \in D\) (being \(D\) a rectangular region), to weakly converge towards the stochastic integral with respect to the Brownian motion \(\int_{D} f(x,u)W(du)\) in the multidimensional parameter set case. We then apply these results to stablish the weak convergence of solutions of the stochastic Poisson equation.

Bibliography

\([1]\) Xavier Bardina and Salim Boukfal. “Weak convergence of stochastic integrals." 2025. arXiv: 2504.00733 [math.PR].

\([2]\) Xavier Bardina and Salim Boukfal. “Weak convergence of stochastic integrals with applications to SPDEs." 2025. arXiv: 2504.08317 [math.PR].

Learning payoffs while routing in skill-based queues

Sanne van Kempen (Eindhoven University of Technology)

Motivated by applications in service systems, we consider queueing systems where each customer must be handled by a server with the right skill set. We focus on optimizing the routing of customers to servers in order to maximize the total payoff of customer–server matches. In addition, customer—server dependent payoff parameters are assumed to be unknown a priori. We construct a machine learning algorithm that adaptively learns the payoff parameters while maximizing the total payoff and prove that it achieves polylogarithmic regret. Moreover, we show that the algorithm is asymptotically optimal up to logarithmic terms by deriving a regret lower bound. The algorithm leverages the basic feasible solutions of a static linear program as the action space. The regret analysis overcomes the complex interplay between queueing and learning by analyzing the convergence of the queue length process to its stationary behavior. We also demonstrate the performance of the algorithm numerically, and have included an experiment with time-varying parameters highlighting the potential of the algorithm in non-static environments.

Asymptotic results for dynamic contagion processes with different exciting functions and application to risk models.

Shamiksha (Indian Institute of Technology Delhi, Hauz Khas, New Delhi - 110016, India)

Hawkes process is a particular class of point process (introduced by Hawkes \([2]\)) which is characterized by the self-exciting and/or multi-exciting property, i.e., the occurrence of future events is influenced by the past events. Initially, Hawkes and Oakes \([3]\) constructed the cluster form representation of the Hawkes process from a homogeneous Poisson process and used the same exciting function for the production of subsequent offspring. These processes are widely used in several fields including seismology, finance, DNA modeling, neuroscience, queuing systems and many others.

In particular, the self-exciting property, and multi-exciting property property of the Hawkes processes provide a consistent model for examining the clustering of defaults during financial crises. However, defaults are also subject to external influences, thus in this regards, a new point process has been introduced by Dassios and Zhao \([1]\) known as the dynamic contagion process, which combines the characteristics of both self-excited and externally excited jumps.

In Pandey et al. \([4]\), we introduced an extension of the cluster representation of dynamic contagion process in which we used different exciting functions for different generations of offspring and studied their asymptotic properties including Central Limit Theorem and Large Deviation Principle. This consideration is important in a number of fields, e.g., in seismology, where main shocks produce aftershocks with possibly different intensities. Additionally, in a financial setting, we investigated the asymptotic behavior of the ruin probability in both finite and infinite time horizons. Specifically, we considered the risk model under the assumption that the dynamics of the contagion claims arrivals have different exciting functions and examined how it affects the ruin probability.

Bibliography

\([1]\) A. Dassios and H. Zhao, A dynamic contagion process, Advances in Applied Probability 43 (3) (2011) 814–846.

\([2]\) A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika 58 (1) (1971) 83–90.

\([3]\) A. G. Hawkes and D. Oakes, A cluster process representation of a self-exciting process, Journal of Applied Probability 11 (3) (1974) 493–503.

\([4]\) S. Pandey, D. Selvamuthu and P. Tardell, Asymptotic results for dynamic contagion processes with different exciting functions and application to risk models. Journal of Mathematical Analysis and Applications 547 (2025) 129392.

PERIODIC SOLUTION OF A STOCHASTIC EPIDEMIC MODEL WITH TWO DIFFERENT EPIDEMICS AND DIFFERENT TRANSMISSION MECHANISM

Shivam Kumar Mishra (Indian Institute of Technology Mandi, India)

In epidemiology, the host population may be at risk for more than one infectious disease. Researchers have been recently interested in this area of study. In the article, we have explored an epidemic model that combines SIRS and SIR, two different transmission techniques. The considered deterministic model has been perturbed stochastically at transmission rates. For the resulting stochastic model, the analysis has been done. We study the periodic solution for the stochastic system. We use the Lyapunov function and Khasminskii theory to establish that the nonautonomous periodic form of the system with white noise has a positive periodic solution.

Bibliography

\([1]\) Cantab, M. D., Hamer, W. H. (2006). The Milroy lectures on epidemic disease in England—The evidence of variability and persistence of type. The Lancet 167(4305): 569–574.

\([2]\) Kermack, W. O., McKendrick, A. G. (1991). Contributions to the mathematical theory of epidemics—-I, Bulletin of Mathematical Biology 53(1-2): 33-55.

\([3]\) Jin, Z., Li, G. (2005). Global stability of an SEI epidemic model with general contact rate. Chaos, Solitons & Fractals 23(3): 997–1004.

\([4]\) Meng, X., Wu, Z., Zhang, T. (2013). The dynamics and therapeutic strategies of a SEIS epidemic model. International Journal of Biomathematics 6(5):, 1793-5245.

\([5]\) Bao, K., Rong, L., Zhang, Q. (2019). Analysis of a stochastic SIRS model with interval parameters. Discrete and Continuous Dynamical Systems-B 24(9), 4827- 4849.

\([6]\) Gray, A., Greenhalgh, D., Hu, L., Mao, X., Pan, J. (2011). A Stochastic Differential Equation SIS Epidemic Model. SIAM Journal on Applied Mathematics 71(3): 876-902.

\([7]\) Chang, Z., Lu, X., Meng, X. (2017). Analysis of a novel stochastic SIRS epidemic model with two different saturated incidence rates. Physica A: Statistical Mechanics and its Applications 472: 103-116.

\([8]\) Meng, X. (2010). Stability of a novel stochastic epidemic model with double epidemic hypothesis. Applied Mathematics and Computation 217(2): 506-515.

\([9]\) Feng, T., Meng, X., Zhang, T., Zhao, S. (2016) Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis. Journal of Mathematical Analysis and Applications 433(1): 227–242.

Infinite-dimensional stochastic differential equations for Coulomb random point fields

Shota Osada (Kagoshima University)

The Coulomb random point field \(\mu\) is an infinite particle system in \(\mathbb{R}^d\) , \(d \geq 2\), defined by a sub-sequential limit of \[\notag \mu^N(d\xi) = \dfrac{1}{Z^{N}}\exp\left(-\beta \left\{\sum_{i=1}^{N}\Phi^N(x^i) + \sum_{j \neq k,\\j,k=1}^{N}\Psi(x^j-x^i)\right\}\right)d\mathbf{x}^N ,\] where \(\xi = \sum_{i=1}^{N}\delta_{x^i}\), \(\beta \geq 0\) is the inverse temperature, \(\{\Phi^N\}_{N \in \mathbb{N}}\) is a sequence of confining potentials, \(\Psi\) is the \(d\)-dimensional Coulomb potential defined by \[\Psi(x) = \dfrac{1}{d-2}\dfrac{1}{|x|^{d-2}} \,\,(d\geq 3) , \quad \Psi(x) =-\log{|x|} \,\, ( d=2).\]

This poster presents the existence of \((\mathbb{R}^d)^{\mathbb{N}}\)-valued diffusion process \(\mathbf{X}=(X^i)_{i \in \mathbb{N}}\) associated with the Coulomb random point field for each \(d\geq 2\) and \(\beta \geq 0\), constructed as the pathwise unique strong solution to the following infinite-dimensional stochastic differential equation (ISDE): \[\begin{aligned} \tag{1} &X_t^i - X_0^i = \int_{0}^{t}\sigma(X_u^i)dB_u^i + \frac{\beta}{2}\int_{0}^{t}\nabla\mathfrak{a}(X_u^i)du\\ & \quad -\dfrac{\beta}{2}\int_{0}^{t}\mathfrak{a}(X_u^i)\left(\nabla\Phi(X_u^i) +\lim_{R \to \infty}\sum_{|X_u^i-X_u^j|\leq R , j \neq i} \nabla\Psi(X_u^i-X_u^j)\right)du &(i \in \mathbb{N}).\end{aligned}\] Here, \(\mathfrak{a} \in C_b^2(\mathbb{R}^d ; \mathbb{R}^{d\times d})\) is a uniformly elliptic, bounded, symmetric matrix-valued function, and \(\sigma \in C_b^{1}(\mathbb{R}^d; \mathbb{R}^{d\times d})\) is a symmetric matrix such that \(\sigma^{t}\sigma = \mathfrak{a}\).
Let \(\ell: \{\xi \in \mathrm{Conf}(\mathbb{R}^d)|\xi(\mathbb{R}^d)=\infty\} \rightarrow (\mathbb{R}^d)^{\mathbb{N}}\) is a labeling map such that \(\xi = \sum_{i \in \mathbb{N}}\delta_{\ell(\xi)^i}.\)

Theorem. Let \(\mu\) be a sub-sequential limit \(\mu = \lim_{n \to \infty} \mu^{N_n}\) such that

1. \(\sup_{n \in \mathbb{N}}\mathbf{E}^{\mu^{N_n}}[\xi(S_R)]<\infty\) for each \(R\in \mathbb{N}\).

2. \(\mu(\{\xi(\mathbb{R}^d)=\infty\})=1\)

3. \(\mathbf{E}^{\mu}[\xi(S_R)]=O(R^p)\) as \(R \to \infty\) for some \(p >0\) .

4. \(\mathbf{E}^{\mu} \left[\dfrac{\xi(S_R)!}{(\xi(S_R)-m)!}\right]<\infty\) for all \(m,R \in \mathbb{N}\).

Then, ISDE \((1)\) has a unique strong solution \(\mathbf{X}=(X^i)_{i \in \mathbb{N}}\) with starting point \(\mathbf{X}_0 = \ell(\xi)\) for \(\mu\)-a.s. \(\xi\).

Windings of planar Stochastic Processes

Stavros Vakeroudis (Athens University of Economics and Business)

Windings of Planar Stochastic Processes
Stavros Vakeroudis
Athens University of Economics and Business, Greece, E-mail: svak@aueb.gr
Two-dimensional (planar) processes attract the interest of several researchers. This happens both because of their richness from a theoretical point of view and because their study turns out to be very fruitful in terms of applications (e.g. in Finance \([9]\), in Biology \([8]\) etc.). Here, we focus on the fine study of trajectories of planar processes, and in particular on their windings.

We survey several results concerning windings of two-dimensional processes, including planar Brownian motion (BM), complex-valued Ornstein-Uhlenbeck (OU) processes and planar stable processes (see e.g. \([5,6]\). We also present Spitzer’s asymptotic Theorem for each case.

Our starting point is the skew-product representation. Then, we introduce Bougerol’s celebrated identity in law \([7]\) which is very useful for the study of the windings of planar BM and of complex-valued OU processes, stating that, for \(u>0\) fixed, \[\sinh(\beta_{u}) \stackrel{(law)}{=} \hat{\beta}_{A_{u}(\beta)=\int^{u}_{0}ds\exp(2\beta_{s})} \ ,\] where \((\beta_{t},t\geq0)\) and \((\hat{\beta_{t}},t\geq0)\) are two independent linear Brownian motions, and the second one is also independent from \(A_{u}(\beta)\). However, this method cannot be applied for the case of planar stable processes \([1]\). So, we tackle this problem firstly by using new methods invoking the continuity of the composition function \([3]\) and secondly by applying new techniques from the theory of self-similar Markov processes \([4]\) having as a starting point the so-called Riesz–Bogdan–Żak transform introduced in \([2]\) which gives the law of the stable process when passed through the spatial Kelvin transform and an additional time change. This approach allows to study similarly one-dimensional and (possibly) higher-dimensional windings.

Bibliography

\([1]\) Jean Bertoin and Wendelin Werner, Stable windings. Ann. Probab. 24(3), 1996, pp. 1269–1279.

\([2]\) Krzysztof Bogdan and Tomasz Żak, On Kelvin Transformation. J. Theor. Probab. 19, 2010, pp. 89–120.

\([3]\) Ron A. Doney and Stavros Vakeroudis, Windings of planar stable processes. Sém. Prob., Vol. XLV, Lecture Notes in Mathematics 2078, 2013, pp. 277–300.

\([4]\) Andreas E. Kyprianou and Stavros Vakeroudis, Stable windings at the Origin, Stoch. Proc. Appl. 128, 2018, pp. 4309-4325.

\([5]\) Stavros Vakeroudis, On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol’s identity. SIAM Theory Probab. Appl. 56(3), 2011, pp. 485–507 (originally published in 2011 in Teor. Veroyatnost. i Primenen., 56(3), pp. 566–591).

\([6]\) Stavros Vakeroudis, Bougerol’s identity in law and extensions. Probability Surveys 9, 2012, pp. 411–437.

\([7]\) Stavros Vakeroudis and Marc Yor, Integrability properties and Limit Theorems for the exit time from a cone of planar Brownian motion. Bernoulli 19(5A), 2012, pp. 2000–2009.

\([8]\) Stavros Vakeroudis, Marc Yor and David Holcman, The Mean First Rotation Time of a planar polymer. J. Stat. Phys. 143(6), 2011, pp. 1074–1095.

\([9]\) Marc Yor, Exponential Functionals of Brownian Motion and Related Processes. Berlin, Springer, 2001.

Ruin Probability Approximation for Bidimensional Brownian Risk Model with Tax.

Timofei Shashkov (University of Lausanne)

Abstract: Consider a bidimensional Brownian motion with independent components \[\bigl(B_{1}(t),B_{2}(t)\bigr), t\ge 0.\] For constants \(c_{1},c_{2}\in\mathbb{R}\) and reflection (tax) parameters \(\gamma_{1},\gamma_{2}\in[0,1]\), define the \(\boldsymbol\gamma\)-reflected process \[\bigl(X_{1}(t),X_{2}(t)\bigr) =\begin{pmatrix} B_{1}(t)-c_{1}t-\gamma_{1}\displaystyle\inf_{0\le s\le t}\!\bigl(B_{1}(s)-c_{1}s\bigr)\\[6pt] B_{2}(t)-c_{2}t-\gamma_{2}\displaystyle\inf_{0\le s\le t}\!\bigl(B_{2}(s)-c_{2}s\bigr) \end{pmatrix}, \qquad t\ge 0.\] In actuarial terms, \(c_{i}\) and \(\gamma_i\) correspond to the premium and tax rates of a line \(i\) respectively.

For a fixed time horizon \(T>0\) and a level ratio \(a \in \mathbb{R}\) we investigate the finite–time simultaneous ruin probability \[\mathbb{P}\left\{\exists\,t\in[0,T]: X_{1}(t)>u,\;X_{2}(t)>a\,u\right\}, \qquad u\to\infty,\] and derive its exact asymptotics as \(u\to\infty\).

Limit Theorems for the Infinite Occupancy Scheme

Valeriia Kotelnykova (University of California, Irvine; Taras Shevchenko National University of Kyiv)

The infinite occupancy scheme is defined as follows. Balls are thrown independently into an infinite array of boxes numbered \(1\), \(2,\ldots\), with a probability \(p_k\) of hitting the box \(k\). Here, \((p_k)_{k\in\mathbb{N}}\) is a discrete probability distribution with infinitely many \(p_k>0\).

We are interested in the number of occupied boxes, denoted by \(\mathcal{K}_n\), after \(n\) balls have been distributed. First, I will discuss the common assumptions imposed on the distribution \((p_k)_{k\in\mathbb{N}}\).

Then, I will provide an overview of the recent limit theorems for the number of occupied boxes \(\mathcal{K}_n\): a functional limit theorem \[2\]; laws of the iterated and single logarithm \[1\].

Bibliography

\([1]\) D. Buraczewski, A. Iksanov and V. Kotelnikova, Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin’s occupancy scheme. Stochastic Process. Appl. 183 (2025), Paper No. 104597.

\([2]\) A. Iksanov, Z. Kabluchko and V. Kotelnikova, A functional limit theorem for nested Karlin’s occupancy scheme generated by discrete Weibull-like distributions. J. Math. Anal. Appl. 507 (2022), 125798.

Accounting for reporting delays in real-time phylodynamic analyses with preferential sampling

Volodymyr Minin (University of California, Irvine)

The COVID-19 pandemic demonstrated that fast and accurate analysis of continually collected infectious disease surveillance data is crucial for situational awareness and policy making. Coalescent-based phylodynamic analysis can use genetic sequences of a pathogen to estimate changes in its effective population size, a measure of genetic diversity. These changes in effective population size can be connected to the changes in the number of infections in the population of interest under certain conditions. Phylodynamics is an important set of tools because its methods are often resilient to the ascertainment biases present in traditional surveillance data (e.g., preferentially testing symptomatic individuals). Unfortunately, it takes weeks or months to sequence and deposit the sampled pathogen genetic sequences into a database, making them available for such analyses. These reporting delays severely decrease precision of phylodynamic methods closer to present time, and for some models can lead to extreme biases. Here we present a method that affords reliable estimation of the effective population size trajectory closer to the time of data collection, allowing for policy decisions to be based on more recent data. Our work uses readily available historic times between sampling and reporting of sequenced samples for a population of interest, and incorporates this information into the sampling model to mitigate the effects of reporting delay in real-time analyses. We illustrate our methodology on simulated data and on SARS-CoV-2 sequences collected in the state of Washington in 2021 \[1\].

Coalescent models are continuous-time Markov chains used to model a genealogy from a sample of sequences \[2\]. Rodrigo et al. (1999) extended coalescent theory for heterochronous sampling \[3\]: \[\phantomsection\label{eq-coalescent-density}{ \begin{split} P(\boldsymbol{g} | \boldsymbol{s}, \boldsymbol{n}, N_e(t)) &= \prod_{k = 2}^{n}P(t_{k-1} | t_k, \boldsymbol{s}, N_e(t)) \\ &= \prod_{k = 2}^{n} \frac{A_{0, k}}{N_e(t_{k-1})}\exp \biggl\{ -\int_{I_{0, k}} \frac{A_{0, k}}{N_e(t)}dt - \sum\limits_{i \geq 1} \int_{I_{i,k}} \frac{A_{i, k}}{N_e(t)} dt \biggr\}. \end{split} }\] In the preferential sampling model, we model sampling events as a Poisson Process with intensity \(\lambda(t)\): \[\begin{split} \log\lambda(t) =& \beta_0 + \beta_1 \log[N_e(t)] + \beta_2 f_2(t) + ... + \beta_p f_p(t). \end{split}\] Altogether, the posterior we are interested in is \[\phantomsection\label{eq-reduced-posterior}{ Pr(\boldsymbol{\gamma}, \kappa, \boldsymbol{\beta} | \boldsymbol{g}, \boldsymbol{s}) \propto Pr(\boldsymbol{g} | \boldsymbol{\gamma}, \boldsymbol{s}) Pr(\boldsymbol{s} | \boldsymbol{\gamma}, \boldsymbol{\beta}) Pr(\boldsymbol{\gamma} | \kappa) Pr(\kappa) Pr(\boldsymbol{\beta}). }\]

Bibliography

\([1]\) Medina, CM, Palacios JA, Minin VM. "Accounting for reporting delays in real-time phylodynamic analyses with preferential sampling." PLoS Computational Biology, vol. 21, no. 5, 2025, pp. e1012970.

\([2]\) Kingman JFC. "The coalescent." Stochastics Processes and Applications, vol. 13, no. 3, 1982, pp. 235–248.

\([3]\) Rodrigo AG, Felsenstein J. "Coalescent approaches." The Evolution of HIV. 1999, pp. 233–72.

Stochastic Simulation for Transient Dynamics of Schrödinger’s Cat States

Yi Shi (University College London)

Schrödinger’s Cat states offer a valuable noise bias, with the potential to substantially simplify quantum error correction. However, simulating their transient dynamics is computationally demanding due to the high-dimensional, non-linear nature of these systems. This challenge is particularly acute when multiple interacting subsystems are involved, as the matrix dimension of the Lindblad master equation then expands exponentially. We address this by employing stochastic methods that utilize only a few stochastic variables per mode to accurately capture key observables in the transient dynamics.

Convergence of the loop-erased percolation explorer on UIHPT

Yuyang Feng (University of Chicago)

We study critical site percolation on a uniform infinite half-planar triangulation with a white-black boundary condition. Previous studies have shown the convergence of percolation interface-decorated maps to an \(\mathrm{SLE}_6\)-decorated \(\sqrt{8/3}\)-LQG surface under the local GHPU topology. In this work, we prove that the loop-erasure of the percolation interface also has a scaling limit and conjecture it to be an \(\mathrm{SLE}_{8/3}\)-type curve.