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Publications (149)
In models with insufficient initial information, parameter estimation can be subject to statistical uncertainty, potentially resulting in suboptimal decision-making; however, delaying implementation to gather more information can also incur costs. This paper examines an extension of information-theoretic approaches designed to address this classica...
We propose a number of concepts and properties related to `weighted' statistical inference where the observed data are classified in accordance with a `value' of a sample string. The motivation comes from the concepts of weighted information and weighted entropy that proved useful in industrial/microeconomic and medical statistics. We focus on appl...
In this paper, we study the regularity of the value function associated with a stochastic control problem involving two controls that act simultaneously on a modulated multidimensional diffusion process. The first is a switching control modelling a random clock. Whenever the random clock rings, the generator matrix is replaced by another, leading t...
We study the problem of optimal risk policies and dividend strategies for an insurance company operating under the constraint that the timing of shareholder payouts is governed by the arrival times of a Poisson process. Concurrently, risk control is continuously managed through proportional reinsurance. Our analysis confirms the optimality of a per...
In this paper, we study the regularity of the value function associated with a stochastic control problem where two controls act simultaneously on a modulated multidimensional diffusion process. The first is a switching control modelling a random clock. Every time the random clock rings, the generator matrix is replaced by another, resulting in a d...
Theoretical‐information approach applied to the clinical trial designs appeared to bring several advantages when tackling a problem of finding a balance between power and expected number of successes (ENS). In particular, it was shown that the built‐in parameter of the weight function allows finding the desired trade‐off between the statistical pow...
We present a number of upper and lower bounds for the total variation distances between the most popular probability distributions. In particular, some estimates of the total variation distances in the cases of multivariate Gaussian distributions, Poisson distributions, binomial distributions, between a binomial and a Poisson distribution, and also...
This paper studies a mixed singular/switching stochastic control problem for a multidimensional diffusion with multiple regimes on a bounded domain. Using probabilistic partial differential equation and penalization techniques, we show that the value function associated with this problem agrees with the solution to a Hamilton–Jacobi–Bellman equatio...
This paper studies a mixed singular/switching stochastic control problem for a multidimensional diffusion with multiples regimes on a bounded domain. Using probabilistic, partial differential equation (PDE) and penalization techniques, we show that the value function associated with this problem agrees with the solution to a Hamilton-Jacobi-Bellman...
In this paper, we aim to determine an optimal insurance premium rate for health-care in deterministic and stochastic SEIR models. The studied models consider two standard SEIR centres characterised by migration fluxes and vaccination of population. The premium is calculated using the basic equivalence principle. Even in this simple set-up, there ar...
In many rare disease Phase II clinical trials, two objectives are of interest to an investigator: maximising the statistical power and maximising the number of patients responding to the treatment. These two objectives are competing, therefore, clinical trial designs offering a balance between them are needed. Recently, it was argued that response-...
This paper focuses on infinite-volume bosonic states for a quantum particle system (a quantum gas) in Rd. The kinetic energy part of the Hamiltonian is the standard Laplacian (with a boundary condition at the border of a ‘box’). The particles interact with each other through a two-body finite-range potential depending on the distance between them a...
This article studies the expected occupancy counts on an alphabet. Unlike the standard situation, where observations are assumed to be independent and identically distributed (iid), we assume that they follow a regime switching Markov chain. For this model, we 1) give finite sample bounds on the occupancy counts, and 2) provide detailed asymptotics...
Many human virus infections including those with the human immunodeficiency virus type 1 (HIV) are initiated by low numbers of founder viruses. Therefore, random effects have a strong influence on the initial infection dynamics, e.g., extinction versus spread. In this study, we considered the simplest (so-called, `consensus’) virus dynamics model a...
In this paper we aim to study an optimal insurance premium level for health-care in a deterministic and stochastic SIR models with migration fluxes and vaccination of population. The studied model considers two standard SIR centres connected via links and continuous migration fluxes. The premium is calculated using the basic equivalence principle....
The aim of this work is to investigate the optimal vaccine sharing between two susceptible, infected, removed (SIR) centres in the presence of migration fluxes of susceptibles and infected individuals during the mumps outbreak. Optimality of the vaccine allocation means the minimization of the total number of lost working days during the whole peri...
In this paper, we guarantee the existence and uniqueness (in the almost everywhere sense) of the solution to a Hamilton-Jacobi-Bellman (HJB) equation with gradient constraint and a partial integro-differential operator whose Lévy measure has bounded variation. This type of equation arises in a singular control problem, where the state process is a...
This article studies the expected occupancy probabilities on an alphabet. Unlike the standard situation, where observations are assumed to be independent and identically distributed (iid), we assume that they follow a regime switching Markov chain. For this model, we 1) give finite sample bounds on the occupancy probabilities, and 2) provide detail...
The aim of this work is to investigate the optimal vaccine sharing between two SIR centers in the presence of migration fluxes of susceptibles and infected individuals during the mumps outbreak. Optimality of the vaccine allocation means the minimization of the total number of lost working days during the whole period of epidemic outbreak $[0,t_f]$...
Following a series of works on capital growth investment, we analyse log-optimal portfolios where the return evaluation includes `weights' of different outcomes. The results are twofold: (A) under certain conditions, the logarithmic growth rate leads to a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional bettin...
This paper represents an extended version of an earlier note [10]. The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy context-dependent, through the weight function. We analyse analogs of the Fisher information inequality and entropy power inequality for the weighted entropy and discuss connections...
Many human infections with viruses such as human immunodeficiency virus type 1 (HIV - 1) are characterized by low numbers of founder viruses for which the random effects and discrete nature of populations have a strong effect on the dynamics, e.g., extinction versus spread. It remains to be established whether HIV transmission is a stochastic proce...
This paper represents an extended version of an earlier note [10]. The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy context-dependent, through the weight function. We analyse analogs of the Fisher information inequality and entropy power inequality for the weighted entropy and discuss connections...
Following a series of works on capital growth investment, we analyse log-optimal portfolios where the return evaluation includes `weights' of different outcomes. The results are twofold: (A) under certain conditions, the logarithmic growth rate leads to a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional bettin...
A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a...
A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a...
The paper considers a family of probability distributions depending on a parameter. The goal is to derive the generalized versions of Cramér-Rao and Bhattacharyya inequalities for the weighted covariance matrix and of the Kullback inequality for the weighted Kullback distance, which are important objects themselves [9, 23, 28]. The asymptotic forms...
The paper considers a family of probability distributions depending on a parameter. The goal is to derive the generalized versions of Cram´er-Rao and Bhattacharyya inequalities for the weighted covariance matrix and of the Kullback inequality for the weighted Kullback distance, which are important objects themselves. The asymptotic forms of these i...
In this paper, we guarantee the existence and uniqueness (in the almost everywhere sense) of the solution to a Hamilton-Jacobi-Bellman (HJB) equation with gradient constraint and a partial integro-differential operator whose L\'evy measure has bounded variation. This type of equation arises in a singular control problem, where the state process is...
The concept of weighted entropy takes into account values of different
outcomes, i.e., makes entropy context-dependent, through the weight function.
In this paper, we establish a number of simple inequalities for the weighted
entropies (general as well as specific), mirroring similar bounds on standard
(Shannon) entropies and related quantities. Th...
We consider the epidemic dynamics in stochastic interacting population centers coupled by a random migration. Both the epidemic and the migration processes are modelled by Markov chains. We derive explicit formulae for the probability distribution of the migration process, and explore the dependence of outbreak patterns on initial parameters, popul...
In this paper, we review Fisher information matrices properties in weighted version and discuss inequalities/bounds on it by using reduced weight functions. In particular, an extended form of the Fisher information inequality previously established in [6] is given. Further, along with generalized De-Bruijn's identity, we provide new interpretation...
We consider a Bayesian problem of estimating of probability of success in a series of conditionally independent trials with binary outcomes. We study the asymptotic behaviour of the differential entropy for a posterior probability density function conditional on x successes after n conditionally independent trials, when n --> infinity. Three partic...
We provide sharp error bounds for the difference between the transition
densities of some multidimensional Continuous Time Markov Chains (CTMC) and the
fundamental solutions of some fractional in time Partial (Integro) Differential
Equations (P(I)DEs). Namely, we consider equations involving a time fractional
derivative of Caputo type and a spatial...
Following the paper by Algoet--Cover (1988), we analyse log-optimal
portfolios where return evaluation includes `weights' of different outcomes.
The results are twofold: (A) under certain conditions, logarithmic growth rate
is a supermartingale, and (B) the optimal (martingale) investment strategy is a
proportional betting; it does not depend on th...
Epidemic dynamics in a stochastic network of interacting epidemic centers is
considered. The epidemic and migration processes are modelled by Markov's
chains. Explicit formulas for probability distribution of the migration process
are derived. Dependence of outbreak parameters on initial parameters,
population, coupling parameters is examined analy...
Consider a Bayesian problem of estimating of probability of success in a
series of trials with binary outcomes. We study the asymptotic behaviour of
weighted differential entropies for posterior probability density function
(PDF) conditional on $x$ successes after $n$ trials, when $n \to \infty$. In
the first part of work Shannon's differential ent...
We analyse an analog of the entropy-power inequality for the weighted
entropy.
We consider the Bayesian problem of estimating the success probability in a series of conditionally independent trials with binary outcomes. We study the asymptotic behaviour of the weighted differential entropy for posterior probability density function conditional on x successes after n conditionally independent trials when n → ∞. Suppose that on...
Probability and statistics are as much about intuition and problem solving as they are about theorem proving. Consequently, students can find it very difficult to make a successful transition from lectures to examinations to practice because the problems involved can vary so much in nature. Since the subject is critical in so many applications from...
We study an epidemic propagation between $M$ population centra. The novelty
of the model is in analyzing the migration of host (remaining in the same
centre) and guest (migrated to another centre) populations separately. Even in
the simplest case $M=2$, this modification is justified because it gives a more
realistic description of migration proces...
We study the Bartlett spectrum of the randomized Hawkes process and demonstrate hat it behaves very differently from the case of a classical Hawkes process. In particular, the Bartlett spectrum could have a singularity near the origin which indicates a long-range depende ce property.
We present an axiomatic approach to earthquake forecasting in terms of
multi-component random fields on a lattice. This approach provides a method for
constructing point estimates and confidence intervals for conditional
probabilities of strong earthquakes under conditions on the levels of
precursors. Also, it provides an approach for setting multi...
We establish a new upper bound for the Kullback-Leibler divergence of two discrete probability distributions which are close in a sense that typically the ratio of probabilities is nearly one and the number of outliers is small.
This paper continues the work Y. Suhov, M. Kelbert. FK-DLR states of a
quantum bose-gas, arXiv:1304.0782 [math-ph], and focuses on infinite-volume
bosonic states for a quantum system (a quantum gas) in a plane. We work under
similar assumptions upon the form of local Hamiltonians and the type of the
(pair) interaction potential as in the reference...
The paper focuses on infinite-volume bosonic states for a quantum particle
system (a quantum gas) in a Euclidean space. The kinetic energy part of the
Hamiltonian is the standard Laplacian (with a Dirichlet's boundary condition at
the border of a `box'). The particles interact with each other through a
two-body finite-range potential depending on t...
This is the first of a series of papers considering symmetry properties of quantum systems over 2D graphs or manifolds, with continuous spins, in the spirit of the Mermin--Wagner theorem. In the model considered here (quantum rotators) the phase space of a single spin is a $d-$dimensional torus, and spins (or particles) are attached to sites of a g...
This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin is where is a -dimensional unit torus with a flat metric. The phase space of spins is , the subspac...
This fundamental monograph introduces both the probabilistic and algebraic aspects of information theory and coding. It has evolved from the authors' years of experience teaching at the undergraduate level, including several Cambridge Maths Tripos courses. The book provides relevant background material, a wide range of worked examples and clear sol...
We consider infinite random casual Lorentzian triangulations emerging in
quantum gravity for critical values of parameters. With each vertex of the
triangulation we associate a Hilbert space representing a bosonic particle
moving in accordance with standard laws of Quantum Mechanics. The particles
interact via two-body potentials decaying with the...
We establish a Mermin--Wagner type theorem for Gibbs states on infinite
random Lorentzian triangulations (LT) arising in models of quantum gravity.
Such a triangulation is naturally related to the distribution $\sf P$ of a
critical Galton--Watson tree, conditional upon non-extinction. At the vertices
of the triangles we place classical spins taking...
This is the first of a series of papers considering symmetry properties of quantum systems over 2D graphs or manifolds, with continuous spins, in the spirit of the Mermin--Wagner theorem. In the model considered here (quantum rotators) the phase space of a single spin is a $d-$dimensional torus, and spins (or particles) are attached to sites of a g...
We obtain, for spectrally negative Lévy processes X, uniform approximations for the finite time ruin probability ψ(t,u)=P u [T≤t],T=inf{t≥0:X(t)<0}, when u=X(0) and t tends to infinity such that v=u/t is constant, and the so-called Cramér light-tail condition is satisfied.
We consider the problem of disease surveillance, early detection, and the requirement for rapid response. If it is important to implement a certain intervention before a threshold number of individuals become infected, then we must determine the expected time (after detection) at which this will occur. We define this as the critical reaction time....
The evolution of an infectious disease outbreak in an isolated population is split into two stages: a stochastic Markov process describing the initial contamination and a linked deterministic dynamical system with random initial conditions for the continued development of the outbreak. The initial contamination stage is well approximated by the ran...
A 1D lattice of coupled susceptible/infected/removed (SIR) epidemic centres is considered numerically and analytically. We
describe a mechanism for the interaction between nodes in an SIR network, i.e. for the migration process of individuals between
epidemic centres with a finite-characteristic time. More specifically, we study a model for a weakl...
This article addresses the issue of the proof of the entropy power inequality (EPI), an important tool in the analysis of
Gaussian channels of information transmission, proposed by Shannon. We analyse continuity properties of the mutual entropy
of the input and output signals in an additive memoryless channel and discuss assumptions under which the...
We present a probabilistic representation for solutions of the Lauricella problem and exploit it to obtain the upper bounds on the growth of these solutions when the domain increases.
We present the higher-order Feynman-Kac formula for solving the equation (Δ+V) m u=0 in a domain D. Probabilistic representation implies some a priori bounds on the growth of solutions when the domain D extends to ℝ d . The estimations of moments of random time to reach a high level for Bessel processes are used to establish a weak dependence of po...
This article addresses the issue of the proof of the entropy power inequality (EPI), an important tool in the analysis of Gaussian channels of information transmission, proposed by Shannon. We analyse continuity properties of the mutual entropy of the input and output signals in an additive memoryless channel and discuss assumptions under which the...
In this paper, a class of spatiotemporal random field models defined as mean-square solutions of fractional versions of the
stochastic heat equation are considered. Different sampling schemes in space and time are introduced to solve the problem
of estimation of fully parameterized spatiotemporal random fields.
In this paper we consider a general hyperbolic branching diffusion on a Lobachevsky space Hd. The question is to evaluate the Hausdorff dimension of the limiting set on the boundary (i.e., absolute) ∂H d. In the case of a homogeneous branching diffusion, an elegant formula for the Hausdorff dimension was obtained by Lalley and Sellke [Probab. Theor...
Probability and Statistics are as much about intuition and problem solving as they are about theorem proving. Because of this, students can find it very difficult to make a successful transition from lectures to examinations to practice, since the problems involved can vary so much in nature. Since the subject is critical in many modern application...
A one-dimensional lattice of SIR (susceptible/infected/removed) epidemic centres is considered numerically and analytically. The limiting solutions describing the behaviour of the standard SIR model with a small number of initially infected individuals are derived, and expressions found for the duration of an outbreak. We study a model for a weakly...
This paper extends results of previous papers [LS] and [KPS], on the Hausdorff dimension of the limiting set of a homogeneous hyperbolic branching diffusion, to the case of a variable fission mechanism. More pre-cisely, we consider a non-homogeneous branching diffusion on a Lobachevsky space H d and assume that parameters of the process approach un...
A determination of the exact statistical limits in the accuracy of time measurements in scintillation counters is presented for the case where the total number of photons in the emission process is randomized in accordance with Poisson statistics. Simple asymptotic formulae for the first two statistical moments are derived for the transient emissio...
A determination of the exact statistical limits in the accuracy of time measurements in scintillation counters is presented. An expression for the variance of the timing distribution, where the number of photons in the signal is fixed, is first derived. This leads to an expression for the case where the total number of photons in the emission is ra...
A risk model of a joint business (insurer/re-insurer) is studied in the Large Devi- ations (LD) regime. In the model considered, the premium paid by the insurer to the re-insurer changes if the insurer's capital falls below a certain level P.A n op- timal premium arrangement for the both business participants is investigated. By a proper choice of...
Because probability and statistics are as much about intuition and problem solving, as they are about theorem proving, students can find it very difficult to make a successful transition from lectures to examinations and practice. Since the subject is critical in many modern applications, Yuri Suhov and Michael Kelbert have rectified deficiencies i...
This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.
This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.
This paper studies the use of simulation techniques to model complex situations in insurance business, where the risk process for old and new business are modeled by compound Poisson processes. The simulation is used to benchmark the solutions of the Hamilton-Jacobi-Bellman equation and to obtain results in the case of distributions, where solution...
The problem is how to compare the quality of different hypothesis tests in a Bayesian framework without introducing a loss function. Three different linear orders on the set of all possible hypothesis tests are studied. The most natural order estimates the Fisher information between indicators of event and decision.
We study a risk model where the insurer’s profit at a finite time horizon τ1 can be controlled by making a change of premium at an optimally chosen time τ < τ1. In the fluid approximation limit, this probabilistic control problem converges in probability to a deterministic problem, which we solve for specific claim size distributions and a unimodal...
Consider a branching diffusion process on R1 starting at the origin. Take a high level u>0 and count the number R(u,n) of branches reaching u by generation n. Let Fk,n(u) be the probability P(R(u,n)<k),k=1,2,…. We study the limit limn→∞Fk,n(u)=Fk(u). More precisely, a natural equation for the probabilities Fk(u) is introduced and the structure of t...
Introduction In this paper we consider a queueing network model under an arrivalsynchronization constraint. The model leads to a higher-order Lindley equation 11 . The network has N input nodes I 1 ; : : : ; I N which are fed with independent Poisson flows 1 ; : : : ; N of intensity . All tasks arriving in the input flows j have i.i.d. service time...
We relate the recurrence and transience of a branching diffusion process on a Riemannian manifold M to some properties of a linear elliptic operator onM (including spectral properties). There is a trade-off between the tendency of the transient Brownian motion to escape and the birth process of the new particles. If the latter has a high enough int...
We say that n independent trajectories ξ1(t),…,ξ
n
(t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ
i
(t
i
) and ξ
j
(t
j
) is at least ɛ, for some indices i, j and for all large enough t
1,…,t
n
, with probability 1. We prove sufficient conitions for asymptotic separationi...
This review presents a modern approach to intersections of Brownian paths. It exploits the fundamental link between intersection properties and percolation processes on trees. More precisely, a Brownians path is intersect-equivalent to certain fractal percolation. It means that the intersection probabilities of Brownian paths can be estimated up to...
Markov processes 15 7 Examples 18 7.1 Diffusions on manifolds : : : : : : : : : : : : : : : : : : : : : : : : : : 19 7.2 The ff-process : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 7.3 Random walk : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 8 Proofs 24 8.1 Intersections of trajectories with covering balls :...
We investigate the escape rate of the Brownian motion $W_x (t)$ on a complete noncompact Riemannian manifold. Assuming that the manifold has at most polynomial volume growth and that its Ricci curvature is bounded below, we prove that $$\dist (W_x (t), x) \leq \sqrt{Ct \log t}$$ for all large $t$ with probability 1. On the other hand, if the Ricci...
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We consider a multiphase service system with a Poisson input flow. Its intensity depends on the number of customers under
service. The stationary distribution for this system can be found in an explicit form. We study the rate of convergence to
this stationary distribution as well as the bounds for some mixing coefficients. Coupling arguments and L...
Wave motions in fluids may appear as a result of circumstances other than compressibility (acoustic waves) or stratification (internal waves). Waves in shear flows attract particular interest in hydrodynamics. They appear in the framework of the model of an incompressible fluid with constant density. Here we mention but a few of the large number of...
In this chapter we study the distortion of pulses propagating in fluids due to their internal structure (chemical composition, micro-inhomogeneity, etc.). This effect is known as physical dispersion (to distinguish from geometrical dispersion due to the boundary effects and macro-inhomogeneity of media). Acoustic relaxation is the internal process...