Search

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 equation. In this way, we see that the regularity of the value function is $ \textrm{C}^{0,1}\cap \textrm{W}^{2,\infty}_{\textrm{loc}}$ .

This article studies the expected occupancy probabilities on an alphabet. Unlike the standard situation, where observations are assumed to be independent and identically distributed, we assume that they follow a regime-switching Markov chain. For this model, we (1) give finite sample bounds on the expected occupancy probabilities, and (2) provide detailed asymptotics in the case where the underlying distribution is regularly varying. We find that in the regularly varying case the finite sample bounds are rate optimal and have, up to a constant, the same rate of decay as the asymptotic result.

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&rsquo, ). The particles interact with each other through a two-body finite-range potential depending on the distance between them and featuring a hard core of diameter a&gt, 0. We introduce a class of so-called FK-DLR functionals containing all limiting Gibbs states of the system. As a justification of this concept, we prove that for d=2, any FK-DLR functional is shift-invariant, regardless of whether it is unique or not. This yields a quantum analog of results previously achieved by Richthammer.

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&rsquo, ) virus dynamics model and incorporated a delay between infection of a cell and virus progeny release from the infected cell. We then developed an equivalent stochastic virus dynamics model that accounts for this delay in the description of the random interactions between the model components. The new model is used to study the statistical characteristics of virus and target cell populations. It predicts the probability of infection spread as a function of the number of transmitted viruses. A hybrid algorithm is suggested to compute efficiently the system dynamics in state space domain characterized by the mix of small and large species densities.

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-adaptive designs such as families of multi-arm bandit (MAB) methods could provide the means for achieving this balance. Furthermore, response-adaptive designs based on a concept of context-dependent (weighted) information criteria were recently proposed with a focus on Shannon's differential entropy. The information-theoretic designs based on the weighted Renyi, Tsallis and Fisher informations are also proposed. Due to built-in parameters of these novel designs, the balance between the statistical power and the number of patients that respond to the treatment can be tuned explicitly. The asymptotic properties of these measures are studied in order to construct intuitive criteria for arm selection. A comprehensive simulation study shows that using the exact criteria over asymptotic ones or using information measures with more parameters, namely Renyi and Tsallis entropies, brings no sufficient gain in terms of the power or proportion of patients allocated to superior treatments. The proposed designs based on information-theoretical criteria are compared to several alternative approaches. For example, via tuning of the built-in parameter, one can find designs with power comparable to the fixed equal randomisation's but a greater number of patients responded in the trials.

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 betting. We focus on properties of the optimal portfolios and discuss a number of simple examples extending the well-known Kelly betting scheme. An important restriction is that the investment does not exceed the current capital value and allows the trader to cover the worst possible losses. The paper deals with a class of discrete-time models. A continuous-time extension is a topic of an ongoing study.

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 are non-intuitive results that illustrate how the premium depends on migration rates, the severity of a disease and the initial distribution of healthy and infected individuals through the centres. We investigate how the vaccination program affects the insurance costs by comparing the savings in benefits with the expenses for vaccination. We compare the results of deterministic and stochastic models.

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 operator corresponding to the generator of a non degenerate Brownian or stable driven Stochastic Differential Equation (SDE).

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. The required assumptions are written in terms of various expectations of the weight functions. Examples are weighted Ky Fan and weighted Hadamard inequalities involving determinants of positive-definite matrices, and weighted Cram\'{e}r-Rao inequalities involving the weighted Fisher information matrix., Comment: arXiv admin note: substantial text overlap with arXiv:1409.4102

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 period of epidemic outbreak $[0,t_f]$, which can be described by the functional $Q=\int _0^{t_f}I(t)\,{\textrm{d}}t$, where $I(t)$ stands for the number of infectives at time $t$. We explain the behaviour of the optimal allocation, which depends on the model parameters and the amount of vaccine available $V$.

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 process on the whole. In this study, we consider the simplest (so-called, 'consensus') virus dynamics model and develop a computational methodology for building an equivalent stochastic model based on Markov Chain accounting for random interactions between the components. The model is used to study the evolution of the probability densities for the virus and target cell populations. It predicts the probability of infection spread as a function of the number of the transmitted viruses. A hybrid algorithm is suggested to compute efficiently the dynamics in state space domain characterized by a mix of small and large species densities.

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 a multidimensional jump-diffusion with jumps of finite variation and infinite activity. We verify, by means of {\epsilon}-penalized controls, that the value function associated with this problem satisfies the aforementioned HJB equation.

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 values in a torus $M$ of dimension $d$, with a given group action of a torus ${\tt G}$ of dimension $d'\leq d$. In the main body of the paper we assume that the spins interact via a two-body nearest-neighbor potential $U(x,y)$ invariant under the action of ${\tt G}$. We analyze quenched Gibbs measures generated by $U$ and prove that, for $\sf P$-almost all Lorentzian triangulations, every such Gibbs measure is ${\tt G}$-invariant, which means the absence of spontaneous continuous symmetry-breaking., Comment: 10n pages, 1 figure

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 multilevel alarm system and hypothesis testing for binary alarms. We use a method of comparison for different earthquake forecasts in terms of the increase of Shannon information. 'Forecasting' and 'prediction' of earthquakes are equivalent in this approach., 21 pages

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. Suppose a disease outbreak within a closed population N may be detected when a certain number I0 individuals are infected. Now assume that a specific intervention should be applied before this number increases to a level I1 > I0. The first time at which I1 is reached is called the critical reaction time. We derive simple asymptotic formulae for the expectation, variance and full distribution of the reaction time T(I1, I0, N) in the framework of a stochastic SI (susceptible/infected) model. Asymptotically these results are in agreement with the mean-field model and Gaussian approximations. Further corrections to these approximations are obtained that give a better fit to the tails of the distribution of reaction times, which are important for the modelling of extreme events.

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 expres- sions found for the duration of an outbreak. We study a model for a weakly mixed population distributed between the interacting centres. The centres are modelled as SIR nodes with interac- tion between sites determined by a diffusion-type migration process. Under the assumption of fast migration, a one-dimensional lattice of SIR nodes is studied numerically with deterministic and random coupling, and travelling wave-like solutions are found in both cases. For weak coupling, the main part of the travelling wave is well approximated by the limiting SIR solution. Explicit formulae are found for the speed of the travelling waves and compared with results of numeri- cal simulation. Approximate formulae for the epidemic propagation speed are also derived when coupling coefficients are randomly distributed, they allow us to estimate how the average speed in random media is slowed down.

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 emission process for the case in which both the total and the emitted number of photons are asymptotically large and their ratio is finite. Comparison with formulae obtained in [R.E. Post, L.I. Schiff, Phys. Rev. 80 (1950) 1113] and later in [M. Kelbert, I. Sazonov, A.G. Wright, Nucl. Instr. and Meth. A 564–1 (2006) 185] shows the formulae derived in the present work describe the moments more accurately and over a wider range of parameters.

Consider a Bayesian problem of success probability estimation 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 $n \to \infty$. Suppose that one is interested to know whether the coin is approximately fair with a high precision and for large $n$ is interested in the true frequency. In other words, the statistical decision is particularly sensitive in small neighbourhood of the particular value $\gamma=1/2$. For this aim the concept of weighted differential entropy is used. It is shown that when $x$ is a proportion of $n$ after an appropriate normalization the limiting distribution is Gaussian and the standard differential entropy of standardized RV converges to differential entropy of standard Gaussian random variable. Also, we found that the weight in suggested form does not change the asymptotic form of the Shannon and Renyi differential entropies, but changes the constants.

We consider the epidemic dynamics in stochastic interacting population centers coupled by random migration. Both the epidemic and the migration processes are modeled 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, population sizes and coupling parameters, using analytical and numerical methods. We show the importance of considering the movement of resident and visitor individuals separately. The mean field approximation for a general migration process is derived and an approximate method that allows the computation of statistical moments for networks with highly populated centers is proposed and tested numerically.

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 differential entropy for posterior probability density function conditional on $x$ successes after $n$ conditionally independent trials, when $n to infty$. Three particular cases are studied: $x$ is a proportion of $n$; $x$ $sim n^beta$, where $0

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 randomized in accordance with Poisson statistics.

This paper extends results of previous papers [S. Lalley and T. Sellke, Probab. Theory Related Fields, 108 (1997), pp. 171–192] and [F. I. Karpelevich, E. A. Pechersky, and Yu. M. Suhov, Comm. Math. Phys., 195 (1998), pp. 627–642] on the Hausdorff dimension of the limiting set of a homogeneous hyperbolic branching diffusion to the case of a variable fission mechanism. More precisely, we consider a nonhomogeneous branching diffusion on a Lobachevsky space ${\bf H}^d$ and assume that parameters of the process uniformly approach their limiting values at the absolute $\partial{\bf H}^d$. Under these assumptions, a formula is established for the Hausdorff dimension $h(\Lambda)$ of the limiting (random) set $\Lambda\subseteq \partial{\bf H}^d$, which agrees with formulas obtained in the papers cited above for the homogeneous case. The method is based on properties of the minimal solution to a Sturm–Liouville equation, with a potential taking two values, and elements of the harmonic analysis on ${\bf H}^d$.

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 demand function.

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)

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 insurance to telecommunications to bioinformatics, the authors have collected more than 200 worked examples and examination questions with complete solutions to help students develop a deep understanding of the subject rather than a superficial knowledge of sophisticated theories. With amusing stories and historical asides sprinkled throughout, this enjoyable book will leave students better equipped to solve problems in practice and under exam conditions.

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 processes. This becomes evident in a purely migration model with vanishing epidemic parameters. It is important to account for a certain number of guest susceptible present in non-host cenrta because these susceptible may be infected and return to the host node as infectives. The flux of such infectives is not negligible and is comparable with the flux of host infectives migrated to other centra, because the return rate of a guest individual will, by nature, tend to be high. It is shown that taking account of both fluxes of infectives noticeably increases the speed of epidemic spread in a 1D lattice of identical SIR centra., Comment: 19 pages, 2 figures

We say that n independent trajectories ξ1(t),…,ξn(t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξi(ti) and ξj(tj) is at least ɛ, for some indices i, j and for all large enough t1,…,tn, with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct−ν/2 then n trajectories of ξ are asymptotically separated provided \(\). Moreover, if \(\) for some α∈(0, 2)then n trajectories of ξ(α) are asymptotically separated, where ξ(α) is the α-process generated by −(−Δ)α/2.

for any e > 0. In the view of Khinchin’s result, inequality (1) has long been considered as one of a rather historical value. However, the recent results on Brownian motion on Riemannian manifolds give a new insight into it. In this note, we show that an analogue of (1), for the Brownian motion on Riemannian manifolds of the polynomial volume growth, is sharp and, therefore, cannot be replaced by an analogue of (2). Let M be a smooth connected geodesically complete non-compact Riemannian manifold. Denote by ∆ the Laplace operator of the Riemannian metric of M and by Wx(t) the minimal diffusion on M starting at the point x ∈ M and generated by the operator 12∆. We say that a positive function R(t) on (0,∞) is an upper radius for Wx(t) if, with probability 1, dist (Wx(t), x) ≤ R(t) , for all t large enough, where dist denotes the geodesic distance. For example, if M = Rd, then the d-dimensional law of the iterated logarithm implies that the function √ (2 + e) t log log t (3) ∗Supported by the EPSRC Advanced Fellowship B/94/AF/1782 †Partially supported by the collaborative grant of London Mathematical Society and by the EPSRC Visiting Fellowship

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 Liapunov’s function approach form the basis of considerations.

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 solutions to problems from real exam papers. It is a valuable teaching aid for undergraduate and graduate students, or for researchers and engineers who want to grasp the basic principles.

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 isℋ1=L2(M),whereMis ad-dimensional unit torusM=ℝd/ℤdwith a flat metric. The phase space ofkspins isℋk=L2sym(Mk), the subspace ofL2(Mk)formed by functions symmetric under the permutations of the arguments. The Fock spaceH=⊕k=0,1,…ℋkyields the phase space of a system of a varying (but finite) number of particles. We associate a spaceH≃H(i)with each vertexi∈Γof a graph(Γ,ℰ)satisfying a special bidimensionality property. (Physically, vertexirepresents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i)-Δ/2, the minus a half of the Laplace operator onM, responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentialsU(1)(x),x∈M, describing a field generated by a heavy atom, (b) two-body potentialsU(2)(x,y),x,y∈M, showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentialsV(x,y),x,y∈M, scaled along the graph distanced(i,j)between verticesi,j∈Γ, which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie groupGacts onM, represented by a Euclidean space or torus of dimensiond'≤d, preserving the metric and the volume inM. Furthermore, we suppose that the potentialsU(1),U(2), andVareG-invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian isG-invariant, provided that the thermodynamic variables (the fugacityzand the inverse temperatureβ) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.