Searcharxiv⌕ Search

arXiv subjects

Vadim Shcherbakov

Publications and source records attributed to Vadim Shcherbakov.

At least 19 recordsLinked to original sources
arXiv2025

A diffusion limit for Markov chains with log-linear interaction on a graph

In this paper we establish a diffusion limit for a multivariate continuous time Markov chain whose components are indexed by vertices of a finite graph. The components take values in a common finite set of non-negative integers and evolve subject to a graph based log-linear interaction. We show that if the set of common values of the components expands to the set of all non-negative integers, then a time-scaled and normalised version of the Markov chain converges to a system of interacting Ornstein-Uhlenbeck processes reflected at the origin. This limit is akin to heavy traffic limits in queueing (and our model can be naturally interpreted as a queueing model). Our proof draws on developments in queueing theory and relies on martingale methods.

math.PR↗
arXiv2025

Explicit local volatility formula for Cheyette-type interest rate models

This paper addresses the approximation of the local volatility function in the Cheyette interest rate model. Its main contribution is an explicit analytical formula for approximating local volatility, derived by extending the classical Dupire framework to interest rate models. In particular, an implicit Dupire-like expression for local volatility is first derived for options written on the short rate. This expression is then approximated using a combination of perturbation methods and probabilistic techniques, resulting in a formula expressed in terms of time and strike derivatives of the Bachelier implied variance. The final formula naturally extends to multi-factor Cheyette models and provides a practical tool for model calibration.

q-fin.PR↗
arXiv2024

Superdiffusive planar random walks with polynomial space-time drifts

We quantify superdiffusive transience for a two-dimensional random walk in which the vertical coordinate is a martingale and the horizontal coordinate has a positive drift that is a polynomial function of the individual coordinates and of the present time. We describe how the model was motivated through an heuristic connection to a self-interacting, planar random walk which interacts with its own centre of mass via an excluded-volume mechanism, and is conjectured to be superdiffusive with a scale exponent $3/4$. The self-interacting process originated in discussions with Francis Comets.

math.PR↗
arXiv2024

Extreme ATM skew in a local volatility model with discontinuity: joint density approach

This paper concerns a local volatility model in which volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold value. The model is known, and a number of results have been obtained for it. In particular, option pricing formulas and a power law behaviour of the implied volatility skew have been established in the case when the threshold is taken at the money. In this paper we derive an alternative representation of option pricing formulas. In addition, we obtain an approximation of option prices by the corresponding Black-Scholes prices. Using this approximation streamlines obtaining the aforementioned behaviour of the skew. Our approach is based on the natural relationship of the model with Skew Brownian motion and consists of the systematic use of the joint distribution of this stochastic process and some of its functionals.

q-fin.MF↗
arXiv2023

Probabilistic models motivated by cooperative sequential adsorption

This survey concerns probabilistic models motivated by cooperative sequential adsorption (CSA) models. CSA models are widely used in physics and chemistry for modelling adsorption processes in which adsorption rates depend on the spatial configuration of already adsorbed particles. Corresponding probabilistic models describe random sequential allocation of particles either in a subset of Euclidean space, or at vertices of a graph. Depending on a technical setup these probabilistic models are stated in terms of spatial or integer-valued interacting birth-and-death processes. In this survey we consider several such models that have been studied in recent years.

math.PR↗
arXiv2022

Discrete SIR model on a homogeneous tree and its continuous limit

We study a discrete Susceptible-Infected-Recovered (SIR) model for the spread of infectious disease on a homogeneous tree and the limit behavior of the model in the case when the tree vertex degree tends to infinity. We obtain the distribution of the time it takes for a susceptible vertex to get infected in terms of a solution of a non-linear integral equation under broad assumptions on the model parameters. Namely, infection rates are assumed to be time-dependent, and recovery times are given by random variables with a fairly arbitrary distribution. We then study the behavior of the model in the limit when the tree vertex degree tends to infinity, and infection rates are appropriately scaled. We show that in this limit the integral equation of the discrete model implies an equation for the susceptible population compartment. This is a master equation in the sense that both the infectious and the recovered compartments can be explicitly expressed in terms of its solution. In other words, the master equation implies a continuous SIR model for the joint time evolution of all three population compartments.

math.PR↗
arXiv2022

Skew Brownian motion with dry friction: joint density approach

This note concerns distributions of Skew Brownian motion with dry friction and its occupation time. These distributions were obtained in [2] by using the Laplace transform and joint characteristic functions. We provide an alternative approach to deriving these distributions. Our approach is based on using the results for Skew Brownian motion obtained in [3].

math.PR↗
arXiv2022

Balls-in-bins models with asymmetric feedback and reflection

Balls-in-bins models describe a random sequential allocation of infinitely many balls into a finite number of bins. In these models a ball is placed into a bin with probability proportional to a given function (feedback function), which depends on the number of existing balls in the bin. Typically, the feedback function is the same for all bins (symmetric feedback), and there are no constraints on the number of balls in the bins. In this paper we study versions of BB models with two bins, in which the above assumptions are violated. In the first model of interest the feedback functions can depend on a bin (BB model with asymmetric feedback). In the case when both feedback functions are power law and superlinear, a single bin receives all but finitely many balls almost surely, and we study the probability that this happens for a given bin. In particular, under certain initial conditions we derive the normal approximation for this probability, which generalizes the result in [5] obtained in the case of the symmetric feedback. The main part of the paper concerns the BB model with asymmetric feedback evolving subject to certain constraints on the numbers of allocated balls. The model can be interpreted as a transient reflecting random walk in a curvilinear wedge, and we obtain a complete classification of its long term behavior.

math.PR↗
arXiv2021

Linear competition processes and generalized Polya urns with removals

A competition process is a continuous time Markov chain that can be interpreted as a system of interacting birth-and-death processes, the components of which evolve subject to a competitive interaction. This paper is devoted to the study of the long-term behaviour of such a competition process, where a component of the process increases with a linear birth rate and decreases with a rate given by a linear function of other components. A zero is an absorbing state for each component, that is, when a component becomes zero, it stays zero forever (and we say that this component becomes extinct). We show that, with probability one, eventually only a random subset of non-interacting components of the process survives. A similar result also holds for the relevant generalized Polya urn model with removals.

math.PR↗
arXiv2020

Localisation in a growth model with interaction. Arbitrary graphs

This paper concerns the long term behaviour of a growth model describing a random sequential deposition of particles on a finite graph. The probability of allocating a particle at a vertex is proportional to a log-linear function of numbers of existing particles in a neighbourhood of a vertex. When this function depends only on the number of particles in the vertex, the model becomes a special case of the generalised Polya urn model. In this special case all but finitely many particles are allocated at a single random vertex almost surely. In our model interaction leads to the fact that, with probability one, all but finitely many particles are allocated at vertices of a clique.

math.PR↗
arXiv2019

Boundary effect in competition processes

This paper studies the long-term behaviour of a continuous time Markov chain formed by two non-negative integer valued components that evolve subject to a competitive interaction. In the absence of interaction the Markov chain is just a pair of independent linear birth processes with immigration. Interactions of interest include, as a special case, the famous Lotka-Volterra interaction. The Markov chain with another special case of interaction can be interpreted as an urn model with ball removals and is reminiscent, in a sense, of Friedman's urn model. We show that, with probability one, eventually one of the components of the process tends to infinity, while the other component oscillates between values $0$ and $1$ (between values $0$ and $2$ in a special case).

math.PR↗
arXiv2018

Long term behaviour of a reversible system of interacting random walks

This paper concerns the long-term behaviour of a system of interacting random walks labeled by vertices of a finite graph. The model is reversible which allows to use the method of electric networks in the study. In addition, examples of alternative proofs not requiring reversibility are provided.

math.PR↗
arXiv2017

Localisation in a growth model with interaction

This paper concerns the long term behaviour of a growth model describing a random sequential allocation of particles on a finite cycle graph. The model can be regarded as a reinforced urn model with graph-based interactions. It is motivated by cooperative sequential adsorption, where adsorption rates at a site depend on the configuration of existing particles in the neighbourhood of that site. Our main result is that, with probability one, the growth process will eventually localise either at a single site, or at a pair of neighbouring sites.

math.PR↗
arXiv2017

Long term behaviour of two interacting birth-and-death processes

In this paper we study the long term evolution of a continuous time Markov chain formed by two interacting birth-and-death processes. The interaction between the processes is modelled by transition rates which are functions with suitable monotonicity properties. This is in line with the approach proposed by Gauss G.F. and Kolmogorov A.N. for modelling interaction between species in ecology. We obtain conditions for transience/recurrence of the Markov chain and describe in detail its asymptotic behaviour in special transient cases. In particular, we find that in some of these cases the Markov chain escapes to infinity in an unusual way, and the corresponding trajectories can be rather precisely described.

math.PR↗
arXiv2016

A note on scaling limits for truncated birth-and-death processes with interaction

In this note we consider a Markov chain formed by a finite system of interacting birth-and-death processes on a finite state space. We study an asymptotic behaviour of the Markov chain as its state space becomes large. In particular, we show that the appropriately scaled Markov chain converges to a diffusion process, and derive conditions for existence of the stationary distribution of the limit diffusion process in special cases.

math.PR↗
arXiv2014

Long term behaviour of locally interacting birth-and-death processes

In this paper we study long-term evolution of a finite system of locally interacting birth-and-death processes labelled by vertices of a finite connected graph. A detailed description of the asymptotic behaviour is obtained in the case of both constant vertex degree graphs and star graphs. The model is motivated by modelling interactions between populations and is related to interacting particle systems, Gibbs models with unbounded spins, as well as urn models with interaction.

math.PR↗