bluuebunny/crossref_metadata_2025 huggingface dataset:
Continuous-Time Skewed Multifractal Processes as a Model for Financial Returns
Authors: Bacry Emmanuel , Duvernet Laurent , Muzy Jean-François
Published: March 2016

We present the construction of a continuous-time stochastic process which has moments that satisfy an exact scaling relation, including odd-order moments. It is based on a natural extension of the multifractal random walk construction described in Bacry and Muzy (2003). This allows us to propose a continuous-time model for the price of a financial asset that reflects most major stylized facts observed on real data, including asymmetry and multifractal scaling.
Tightness for Maxima of Generalized Branching Random Walks
Authors: Fang Ming
Published: March 2016

We study generalized branching random walks on the real line R that allow time dependence and local dependence between siblings. Specifically, starting from one particle at time 0, the system evolves such that each particle lives for one unit amount of time, gives birth independently to a random number of offspring according to some branching law, and dies. The offspring from a single particle are assumed to move to new locations on R according to some joint displacement distribution; the branching laws and displacement distributions depend on time. At time n, F

n
(·) is used to denote the distribution function of the position of the rightmost particle in generation n. Under appropriate tail assumptions on the branching laws and offspring displacement distributions, we prove that F

n
(· - Med(F

n
)) is tight in n, where Med(F

n
) is the median of F

n
. The main part of the argument is to demonstrate the exponential decay of the right tail 1 - F

n
(· - Med(F

n
)).
An Application of the Backbone Decomposition to Supercritical Super-Brownian Motion with a Barrier
Authors: Kyprianou A. E. , Murillo-Salas A. , Pérez J. L.
Published: March 2016

We analyse the behaviour of supercritical super-Brownian motion with a barrier through the pathwise backbone embedding of Berestycki, Kyprianou and Murillo-Salas (2011). In particular, by considering existing results for branching Brownian motion due to Harris and Kyprianou (2006) and Maillard (2011), we obtain, with relative ease, conclusions regarding the growth in the right-most point in the support, analytical properties of the associated one-sided Fisher-Kolmogorov-Petrovskii-Piscounov wave equation, as well as the distribution of mass on the exit measure associated with the barrier.
Fractional Brownian Motion with H < 1/2 as a Limit of Scheduled Traffic
Authors: Araman Victor F. , Glynn Peter W.
Published: March 2016

In this paper we show that fractional Brownian motion with H < ½ can arise as a limit of a simple class of traffic processes that we call ‘scheduled traffic models’. To our knowledge, this paper provides the first simple traffic model leading to fractional Brownnian motion with H < ½. We also discuss some immediate implications of this result for queues fed by scheduled traffic, including a heavy-traffic limit theorem.
Uniqueness of Quasistationary Distributions and Discrete Spectra when ∞ is an Entrance Boundary and 0 is Singular
Authors: Littin C. Jorge
Published: March 2016

We study quasistationary distributions on a drifted Brownian motion killed at 0, when +∞ is an entrance boundary and 0 is an exit boundary. We prove the existence of a unique quasistationary distribution and of the Yaglom limit.
Joint Distributions of Counts of Strings in Finite Bernoulli Sequences
Authors: Huffer Fred W. , Sethuraman Jayaram
Published: March 2016

An infinite sequence (Y
1, Y
2,…) of independent Bernoulli random variables with P(Y

i
= 1) = a / (a + b + i - 1), i = 1, 2,…, where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z
1, Z
2, Z
3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z
1, Z
2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y
1, Y
2,…, Y

n
) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z
1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.
Predicting the Supremum: Optimality of ‘Stop at Once or Not at All’
Authors: Allaart Pieter C.
Published: March 2016

Let (X

t
)0 ≤ t ≤ T

be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (X

t
) ‘as close as possible’ to its eventual supremum M

T
:= sup0 ≤ t ≤ T

X

t
, when the reward for stopping at time τ ≤ T is a nonincreasing convex function of M

T
- X
τ. Under fairly general conditions on the process (X

t
), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T. For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.
The Noisy Secretary Problem and Some Results on Extreme Concomitant Variables
Authors: Krieger Abba M. , Samuel-Cahn Ester
Published: March 2016

The classical secretary problem for selecting the best item is studied when the actual values of the items are observed with noise. One of the main appeals of the secretary problem is that the optimal strategy is able to find the best observation with a nontrivial probability of about 0.37, even when the number of observations is arbitrarily large. The results are strikingly different when the qualities of the secretaries are observed with noise. If there is no noise then the only information that is needed is whether an observation is the best among those already observed. Since the observations are assumed to be independent and identically distributed, the solution to this problem is distribution free. In the case of noisy data, the results are no longer distribution free. Furthermore, we need to know the rank of the noisy observation among those already observed. Finally, the probability of finding the best secretary often goes to 0 as the number of observations, n, goes to ∞. The results heavily depend on the behavior of p

n
, the probability that the observation that is best among the noisy observations is also best among the noiseless observations. Results involving optimal strategies if all that is available is noisy data are described and examples are given to elucidate the results.
A Note on ‘Improved Fréchet Bounds and Model-Free Pricing of Multi-Asset Options’ by Tankov (2011)
Authors: Bernard Carole , Jiang Xiao , Vanduffel Steven
Published: March 2016

Tankov (2011) improved the Fréchet bounds for a bivariate copula when its values on a compact subset of [0, 1]2 are given. He showed that the best possible bounds are quasi-copulas and gave a sufficient condition for these bounds to be copulas. In this note we give weaker sufficient conditions to ensure that the bounds are copulas. We also show how this can be useful in portfolio selection. It turns out that finding a copula as a lower bound plays a key role in determining optimal investment strategies explicitly for investors with some type of state-dependent constraints.
A Branching Process for Virus Survival
Authors: Cox J. Theodore , Schinazi Rinaldo B.
Published: March 2016

Quasispecies theory predicts that there is a critical mutation probability above which a viral population will go extinct. Above this threshold the virus loses the ability to replicate the best-adapted genotype, leading to a population composed of low replicating mutants that is eventually doomed. We propose a new branching model that shows that this is not necessarily so. That is, a population composed of ever changing mutants may survive.
Critical Path Statistics of Max-Plus Linear Systems with Gaussian Noise
Authors: Hook James
Published: March 2016

The critical paths of a max-plus linear system with noise are random variables. In this paper we introduce theedge criticalitieswhich measure how often the critical paths traverse each edge in the precedence graph. We also present theparallel path approximation, a novel method for approximating these new statistics as well as the previously studied max-plus exponent. We show that, for low amplitude noise, the critical paths spend most of their time traversing the deterministic maximally weighted cycle and that, as the noise amplitude is increased, the critical paths become more random and their distribution over the edges in the precedence graph approaches a highly uniform measure of maximal entropy.
On the Dynamics of Semimartingales with Two Reflecting Barriers
Authors: Pihlsgård Mats , Glynn Peter W.
Published: March 2016

We consider a semimartingaleXwhich is reflected at an upper barrierTand a lower barrierS, whereSandTare also semimartingales such thatTis bounded away fromS. First, we present an explicit construction of the reflected process. Then we derive a relationship in terms of stochastic integrals linking the reflected process and the local times at the respective barriers toX,S, andT. This result reveals the fundamental structural properties of the reflection mechanism. We also present a few results showing how the general relationship simplifies under additional assumptions onX,S, andT, e.g. if we takeX,S, andTto be independent martingales (which satisfy some extra technical conditions).
A Risk Model with Delayed Claims
Authors: Dassios Angelos , Zhao Hongbiao
Published: March 2016

In this paper we introduce a simple risk model with delayed claims, an extension of the classical Poisson model. The claims are assumed to arrive according to a Poisson process and claims follow a light-tailed distribution, and each loss payment of the claims will be settled with a random period of delay. We obtain asymptotic expressions for the ruin probability by exploiting a connection to Poisson models that are not time homogeneous. A finer asymptotic formula is obtained for the special case of exponentially delayed claims and an exact formula is obtained when the claims are also exponentially distributed.
Joint Distribution of Distances in Large Random Regular Networks
Authors: Salez Justin
Published: March 2016

We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime where the degree is kept fixed while the number of vertices tends to ∞. The marginal distribution of an individual entry is now well understood, thanks to the work of Bhamidi, van der Hofstad and Hooghiemstra (2010). The purpose of this note is to show that the whole array, suitably recentered, converges in the weak sense to an explicit infinite random array. Our proof consists in analyzing the invasion of the network by several mutually exclusive flows emanating from different sources and propagating simultaneously along the edges.
Emergent Structures in Large Networks
Authors: Aristoff David , Radin Charles
Published: March 2016

We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to the emergent multipartite structure, separated by a phase transition from a region of disordered graphs. An essential feature is the formalism of graph limits as developed by Lovász et al. for dense random graphs.
The Correct Asymptotic Variance for the Sample Mean of a Homogeneous Poisson Marked Point Process
Authors: Garner William , Politis Dimitris N.
Published: March 2016

The asymptotic variance of the sample mean of a homogeneous Poisson marked point process has been studied in the literature, but confusion has arisen as to the correct expression due to some technical intricacies. This note sets the record straight with regards to the variance of the sample mean. In addition, a central limit theorem in the general d-dimensional case is also established.
An Application of the Coalescence Theory to Branching Random Walks
Authors: Athreya K. B. , Hong Jyy-I
Published: March 2016

In a discrete-time single-type Galton--Watson branching random walk {Z

n
, ζ
n
}
n≤ 0, where Z

n
is the population of the nth generation and ζ
n
is a collection of the positions on ℝ of the Z

n
individuals in the nth generation, let Y

n
be the position of a randomly chosen individual from the nth generation and Z

n
(x) be the number of points in ζ
n
that are less than or equal to x for x∈ℝ. In this paper we show in the explosive case (i.e. m=E(Z
1∣ Z
0=1)=∞) when the offspring distribution is in the domain of attraction of a stable law of order α,0 <α<1, that the sequence of random functions {Z

n
(x)/Z

n
:−∞<x<∞} converges in the finite-dimensional sense to {δ
x
:−∞<x<∞}, where δ
x
≡ 1
{N≤ x} and N is an N(0,1) random variable.
Tail Properties and Asymptotic Expansions for the Maximum of the Logarithmic Skew-Normal Distribution
Authors: Liao Xin , Peng Zuoxiang , Nadarajah Saralees
Published: March 2016

We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)1/2.
Stochastic Orderings and Ageing Properties of Residual Life Lengths of Live Components in (n-k+1)-Out-Of-n Systems
Authors: Balakrishnan Narayanaswamy , Barmalzan Ghobad , Haidari Abedin
Published: March 2016

Suppose that a system consists ofnindependent and identically distributed components and that the life lengths of thencomponents areXi,i= 1, …,n. Fork∈ {1, …,n- 1}, letX(k)1, …,X(k)n-kbe the residual life lengths of the live components following thekth failure in the system. In this paper we extend various stochastic ordering results presented in Bairamov and Arnold (2008) on the residual life lengths of the live components in an (n-k+ 1)-out-of-nsystem, and also present a new result concerning the multivariate stochastic ordering of live components in the two-sample situation. Finally, we also characterize exponential distributions under a weaker condition than those introduced in Bairamov and Arnold (2008) and show that some special ageing properties of the original residual life lengths get preserved by residual life lengths.
On the Deterioration of Nonrepairable Multistate Strongly Coherent Systems
Authors: Natvig Bent
Published: March 2016

In the present paper, results given in Natvig (1990) are generalized to a multistate, strongly coherent, nonrepairable system of independent components by considering the reduction in remaining system time above a certain state due to a jump downwards of a component. This reduction also equals the increase in remaining system time above a certain state due to a minimal repair of the component at its time of jump downwards. The expected value of the sum of such reductions/increases for the different possible jumps downwards of the component is the building block of the Natvig measure of the importance of the component in the multistate case. Hence, the whole distributions of these reductions/increases are arrived at, not only the expectations, throwing more light on the consequences for the system of the deterioration of the components.
On the Signatures of Ordered System Lifetimes
Authors: Balakrishnan N. , Volterman William
Published: March 2016

The idea of the system signature is extended here to the case of ordered system lifetimes arising from a test of coherent systems with a signature. An expression is given for the computation of the ordered system signatures in terms of the usual system signature for system lifetimes. Some properties of the ordered system signatures are then established. Closed-form expressions for the ordered system signatures are obtained in some special cases, and some illustrative examples are presented.
A New Look at Urban Water Storage in a Series of Connected Dams
Authors: Howlett Phil , Pearce Charles , Piantadosi Julia
Published: March 2016

We propose a discrete state-space model for storage of urban stormwater in two connected dams using an optimal pump-to-fill policy to transfer water from the capture dam to the holding dam. We assume stochastic supply to the capture dam and independent stochastic demand from the holding dam. We find new analytic formulae to calculate steady-state probabilities for the contents of each dam and thereby enable operators to better understand system behaviour. We illustrate our methods by considering some particular examples and discuss extension of our analysis to a series of three connected dams.
Coexistence and Noncoexistence of Markovian Viruses and their Hosts
Authors: Björnberg Jakob E. , Broman Erik I.
Published: March 2016

Examining possibilities for the coexistence of two competing populations is a classic problem which dates back to the earliest ‘predator-prey’ models. In this paper we study this problem in the context of a model introduced in Björnberg et al. (2012) for the spread of a virus infection in a population of healthy cells. The infected cells may be seen as a population of ‘predators’ and the healthy cells as a population of ‘prey’. We show that, depending on the parameters defining the model, there may or may not be coexistence of the two populations, and we give precise criteria for this.
Percolation of Hard Disks
Authors: Aristoff D.
Published: March 2016

Random arrangements of points in the plane, interacting only through a simple hard-core exclusion, are considered. An intensity parameter controls the average density of arrangements, in analogy with the Poisson point process. It is proved that, at high intensity, an infinite connected cluster of excluded volume appears almost surely.
Cover Times and Generic Chaining
Authors: Lehec Joseph
Published: March 2016

A recent result of Ding, Lee and Peres (2012) expressed the cover time of the random walk on a graph in terms of generic chaining for the commute distance. Their argument is based on Dynkin's isomorphism theorem. The purpose of this article is to present an alternative approach to this problem, based only on elementary hitting time estimates and chaining arguments.
Undiscounted Markov Chain BSDEs to Stopping Times
Authors: Cohen Samuel N.
Published: March 2016

We consider backward stochastic differential equations in a setting where noise is generated by a countable state, continuous time Markov chain, and the terminal value is prescribed at a stopping time. We show that, given sufficient integrability of the stopping time and a growth bound on the terminal value and BSDE driver, these equations admit unique solutions satisfying the same growth bound (up to multiplication by a constant). This holds without assuming that the driver is monotone in y, that is, our results do not require that the terminal value be discounted at some uniform rate. We show that the conditions are satisfied for hitting times of states of the chain, and hence present some novel applications of the theory of these BSDEs.
The Limiting Behaviour of Hanski's Incidence Function Metapopulation Model
Authors: McVinish R. , Pollett P. K.
Published: March 2016

Hanski's incidence function model is one of the most widely used metapopulation models in ecology. It models the presence/absence of a species at spatially distinct habitat patches as a discrete-time Markov chain whose transition probabilities are determined by the physical landscape. In this analysis, the limiting behaviour of the model is studied as the number of patches increases and the size of the patches decreases. Two different limiting cases are identified depending on whether or not the metapopulation is initially near extinction. Basic properties of the limiting models are derived.
Exact Simulation for Diffusion Bridges: An Adaptive Approach
Authors: Dai Hongsheng
Published: March 2016

Exact simulation approaches for a class of diffusion bridges have recently been proposed based on rejection sampling techniques. The existing rejection sampling methods may not be practical owing to small acceptance probabilities. In this paper we propose an adaptive approach that improves the existing methods significantly under certain scenarios. The idea of the new method is based on a layered process, which can be simulated from a layered Brownian motion with reweighted layer probabilities. We will show that the new exact simulation method is more efficient than existing methods theoretically and via simulation.
Markov Chain Monte Carlo for Computing Rare-Event Probabilities for a Heavy-Tailed Random Walk
Authors: Gudmundsson Thorbjörn , Hult Henrik
Published: March 2016

In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.
Stochastic Modeling for Environmental Stress Screening
Authors: Cha Ji Hwan , Finkelstein Maxim
Published: March 2016

Environmental stress screening (ESS) of manufactured items is used to reduce the occurrence of future failures that are caused by latent defects by eliminating the items with these defects. Some practical descriptions of the relevant ESS procedures can be found in the literature; however, the appropriate stochastic modeling and the corresponding thorough analysis have not been reported. In this paper we develop a stochastic model for the ESS, analyze the effect of this operation on the population characteristics of the screened items, and also consider the relevant optimization issues.
Optimal Dynamic Risk Control for Insurers with State-Dependent Income
Authors: Zhou Ming , Cai Jun
Published: March 2016

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.
On the Joint Distribution of Stopping Times and Stopped Sums in Multistate Exchangeable Trials
Authors: Boutsikas M. V. , Antzoulakos D. L. , Rakitzis A. C.
Published: March 2016

Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, …, m}, and let S

T,i
be the stopped sum denoting the number of appearances of outcome 'i' in X
1, …, X

T
, 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, S

T,0, S

T,1, …, S

T,m
). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.
Synchronization via Interacting Reinforcement
Authors: Dai Pra Paolo , Louis Pierre-Yves , Minelli Ida G.
Published: March 2016

We consider a system of urns of Pólya type, containing balls of two colors; the reinforcement of each urn depends on both the content of the urn and the average content of all urns. We show that the urns synchronize almost surely, in the sense that the fraction of balls of a given color converges almost surely as time tends to ∞ to the same limit for all urns. A normal approximation for a large number of urns is also obtained.
On the Asymmetric Telegraph Processes
Authors: López Oscar , Ratanov Nikita
Published: March 2016

We study the one-dimensional random motionX=X(t),t≥ 0, which takes two different velocities with two different alternating intensities. The closed-form formulae for the density functions ofXand for the moments of any order, as well as the distributions of the first passage times, are obtained. The limit behaviour of the moments is analysed under nonstandard Kac's scaling.
Coupling Limit Order Books and Branching Random Walks
Authors: Simatos Florian
Published: March 2016

We consider a model for a one-sided limit order book proposed by Lakner, Reed and Stoikov (2013). We show that it can be coupled with a branching random walk and use this coupling to answer a nontrivial question about the long-term behavior of the price. The coupling relies on a classical idea of enriching the state space by artificially creating a filiation, in this context between orders of the book, which we believe has the potential of being useful for a broader class of models.
Uniform Asymptotics for Discounted Aggregate Claims in Dependent Risk Models
Authors: Yang Yang , Wang Kaiyong , Konstantinides Dimitrios G.
Published: March 2016

In this paper we consider some nonstandard renewal risk models with some dependent claim sizes and stochastic return, where an insurance company is allowed to invest her/his wealth in financial assets, and the price process of the investment portfolio is described as a geometric Lévy process. When the claim size distribution belongs to some classes of heavy-tailed distributions and a constraint is imposed on the Lévy process in terms of its Laplace exponent, we obtain some asymptotic formulae for the tail probability of discounted aggregate claims and ruin probabilities holding uniformly for some finite or infinite time horizons.
Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps
Authors: Zhu Lingjiong
Published: March 2016

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.
Approximation of Passage Times of γ-Reflected Processes with FBM Input
Authors: Hashorva Enkelejd , Ji Lanpeng
Published: March 2016

Define a γ-reflected process W
γ(t) = Y

H
(t) - γinf
s∈[0,t]
Y

H
(s), t ≥ 0, with input process {Y

H
(t), t ≥ 0}, which is a fractional Brownian motion with Hurst index H ∈ (0, 1) and a negative linear trend. In risk theory R
γ(u) = u - W
γ(t), t ≥ 0, is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes, numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserve u goes to ∞. In this paper we show that, for the γ-reflected process, the conditional (standardized) first and last passage times are jointly asymptotically Gaussian and completely dependent. An important contribution of this paper is that it links ruin problems with extremes of nonhomogeneous Gaussian random fields defined by Y

H
, which we also investigate.
Automated State-Dependent Importance Sampling for Markov Jump Processes via Sampling from the Zero-Variance Distribution
Authors: Grace Adam W. , Kroese Dirk P. , Sandmann Werner
Published: March 2016

Many complex systems can be modeled via Markov jump processes. Applications include chemical reactions, population dynamics, and telecommunication networks. Rare-event estimation for such models can be difficult and is often computationally expensive, because typically many (or very long) paths of the Markov jump process need to be simulated in order to observe the rare event. We present a state-dependent importance sampling approach to this problem that is adaptive and uses Markov chain Monte Carlo to sample from the zero-variance importance sampling distribution. The method is applicable to a wide range of Markov jump processes and achieves high accuracy, while requiring only a small sample to obtain the importance parameters. We demonstrate its efficiency through benchmark examples in queueing theory and stochastic chemical kinetics.
Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns
Authors: Gapeev Pavel V. , Rodosthenous Neofytos
Published: March 2016

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.
Numerical Approximation of Stationary Distributions for Stochastic Partial Differential Equations
Authors: Bao Jianhai , Yuan Chenggui
Published: March 2016

In this paper we discuss an exponential integrator scheme, based on spatial discretization and time discretization, for a class of stochastic partial differential equations. We show that the scheme has a unique stationary distribution whenever the step size is sufficiently small, and that the weak limit of the stationary distribution of the scheme as the step size tends to 0 is in fact the stationary distribution of the corresponding stochastic partial differential equations.
Optimal Server Selection in a Queueing Loss Model with Heterogeneous Exponential Servers, Discriminating Arrivals, and Arbitrary Arrival Times
Authors: Ross Sheldon M.
Published: March 2016

We consider a multiple server queueing loss system where the service times of server i are exponential with rate μ
i
, where μ
i
decreases in i. Arrivals have associated vectors (X
1, …, X

n
) of binary variables, with X

i
= 1 indicating that server i
is eligible to serve that arrival. Arrivals finding no idle eligible servers are lost. Letting I

j
be the indicator variable for the event that the jth arrival enters service, we show that, for any arrival process, the policy that assigns arrivals to the smallest numbered idle eligible server stochastically maximizes the vector (I
1, …, I

r
) for every r if the eligibility vector of arrivals is either (a) exchangeable, or (b) a vector of independent variables for which P(X

i
= 1) increases in i.
A Note on a Lower Bound for the Multiplicative Odds Theorem of Optimal Stopping
Authors: Matsui Tomomi , Ano Katsunori
Published: March 2016

In this note we present a bound of the optimal maximum probability for the multiplicative odds theorem of optimal stopping theory. We deal with an optimal stopping problem that maximizes the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length N, where m and N are predetermined integers satisfying 1 ≤ m < N. This problem is an extension of Bruss' (2000) odds problem. In a previous work, Tamaki (2010) derived an optimal stopping rule. We present a lower bound of the optimal probability. Interestingly, our lower bound is attained using a variation of the well-known secretary problem, which is a special case of the odds problem.
Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph
Authors: Vaze Rahul , Iyer Srikanth
Published: March 2016

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R
2 of intensities λ and λ
E
, respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λ
E
of eavesdropper nodes, there exists a critical intensity λ
c
< ∞ such that for all λ > λ
c
the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λ
c
a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.
Identifiability of a Coalescent-Based Population Tree Model
Authors: RoyChoudhury Arindam
Published: March 2016

Identifiability of evolutionary tree models has been a recent topic of discussion and some models have been shown to be nonidentifiable. A coalescent-based rooted population tree model, originally proposed by Nielsenet al.(1998), has been used by many authors in the last few years and is a simple tool to accurately model the changes in allele frequencies in the tree. However, the identifiability of this model has never been proven. Here we prove this model to be identifiable by showing that the model parameters can be expressed as functions of the probability distributions of subsamples, assuming that there are at least two (haploid) individuals sampled from each population. This a step toward proving the consistency of the maximum likelihood estimator of the population tree based on this model.
Finite-Dimensional Distributions of a Square-Root Diffusion
Authors: Gordy Michael B.
Published: March 2016

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t
1,…,t

n
, we find the conditional joint distribution of (X(t
1),…,X(t

n
)) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.
Limiting Behavior of the Target-Dependent Stochastic Sequential Assignment Problem
Authors: Baharian Golshid , Jacobson Sheldon H.
Published: March 2016

The stochastic sequential assignment problem assigns distinct workers to sequentially arriving tasks with stochastic parameters. In this paper the assignments are performed so as to minimize the threshold probability, which is the probability of the long-run reward per task failing to achieve a target value (threshold). As the number of tasks approaches infinity, the problem is studied for independent and identically distributed (i.i.d.) tasks with a known distribution function and also for tasks that are derived from r distinct unobservable distributions (governed by a Markov chain). Stationary optimal policies are presented, which simultaneously minimize the threshold probability and achieve the optimal long-run expected reward per task.
On the Conditional Residual Life and Inactivity Time of Coherent Systems
Authors: Parvardeh A. , Balakrishnan N.
Published: March 2016

In this paper we derive mixture representations for the reliability functions of the conditional residual life and inactivity time of a coherent system with n independent and identically distributed components. Based on these mixture representations we carry out stochastic comparisons on the conditional residual life, and the inactivity time of two coherent systems with independent and identical components.
Dynamic Reliability Modeling of Three-State Networks
Authors: Ashrafi S. , Asadi M.
Published: March 2016

This paper is an investigation into the reliability and stochastic properties of three-state networks. We consider a single-step network consisting of n links and we assume that the links are subject to failure. We assume that the network can be in three states, up (K = 2), partial performance (K = 1), and down (K = 0). Using the concept of the two-dimensional signature, we study the residual lifetimes of the networks under different scenarios on the states and the number of failed links of the network. In the process of doing so, we define variants of the concept of the dynamic signature in a bivariate setting. Then, we obtain signature based mixture representations of the reliability of the residual lifetimes of the network states under the condition that the network is in state K = 2 (or K = 1) and exactly k links in the network have failed. We prove preservation theorems showing that stochastic orderings and dependence between the elements of the dynamic signatures (which relies on the network structure) are preserved by the residual lifetimes of the states of the network (which relies on the network ageing). Various illustrative examples are also provided.
Extreme Analysis of a Random Ordinary Differential Equation
Authors: Liu Jingchen , Zhou Xiang
Published: March 2016

In this paper we consider a one dimensional stochastic system described by an elliptic equation. A spatially varying random coefficient is introduced to account for uncertainty or imprecise measurements. We model the logarithm of this coefficient by a Gaussian process and provide asymptotic approximations of the tail probabilities of the derivative of the solution.
Random Fields with Pólya Correlation Structure
Authors: Finlay Richard , Seneta Eugene
Published: March 2016

We construct random fields with Pólya-type autocorrelation function and dampened Pólya cross-correlation function. The marginal distribution of the random fields may be taken as any infinitely divisible distribution with finite variance, and the random fields are fully characterized in terms of their joint characteristic function. This makes available a new class of non-Gaussian random fields with flexible correlation structure for use in modeling and estimation.
Lumpings of Markov Chains, Entropy Rate Preservation, and Higher-Order Lumpability
Authors: Geiger Bernhard C. , Temmel Christoph
Published: March 2016

A lumping of a Markov chain is a coordinatewise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from their lumped images. Both are purely combinatorial criteria, depending only on the transition graph of the Markov chain and the lumping function. A lumping is strongly k-lumpable, if and only if the lumped process is a kth-order Markov chain for each starting distribution of the original Markov chain. We characterise strong k-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k-lumpable.
On the Time Spent in the Red by a Refracted Lévy Risk Process
Authors: Renaud Jean-François
Published: March 2016

In this paper we introduce an insurance ruin model with an adaptive premium rate, henceforth referred to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model the premium rate is increased as soon as the wealth process falls into the red zone and is brought back to its regular level when the wealth process recovers. The analysis is focused mainly on the time a refracted Lévy risk process spends in the red zone (analogous to the duration of the negative surplus). Building on results from [11] and [16], we identify the distribution of various functionals related to occupation times of refracted spectrally negative Lévy processes. For example, these results are used to compute both the probability of bankruptcy and the probability of Parisian ruin in this model with restructuring.
The Containment Condition and Adapfail Algorithms
Authors: Łatuszyński Krzysztof , Rosenthal Jeffrey S.
Published: March 2016

This short note investigates convergence of adaptive Markov chain Monte Carlo algorithms, i.e. algorithms which modify the Markov chain update probabilities on the fly. We focus on the containment condition introduced Roberts and Rosenthal (2007). We show that if the containment condition is not
satisfied, then the algorithm will perform very poorly. Specifically, with positive probability, the adaptive algorithm will be asymptotically less efficient then any nonadaptive ergodic MCMC algorithm. We call such algorithms AdapFail, and conclude that they should not be used.
Convergence to Stable Laws in the Space D
Authors: Roueff François , Soulier Philippe
Published: March 2016

We study the convergence of centered and normalized sums of independent and identically distributed random elements of the spaceDof càdlàg functions endowed with Skorokhod'sJ1topology, to stable distributions inD. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications; in particular, to the empirical process of the renewal-reward process.
State-Dependent Fractional Point Processes
Authors: Garra R. , Orsingher E. , Polito F.
Published: March 2016

In this paper we analyse the fractional Poisson process where the state probabilitiespkνk(t),t≥ 0, are governed by time-fractional equations of order 0 < νk≤ 1 depending on the numberkof events that have occurred up to timet. We are able to obtain explicitly the Laplace transform ofpkνk(t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on νkdiffers from that constructed from the fractional state equations (in the case of νk= ν, for allk, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.
Applying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy Processes to Path Functionals
Authors: Ferreiro-Castilla Albert , van Schaik Kees
Published: March 2016

In this paper we apply the recently established Wiener-Hopf Monte Carlo simulation technique for Lévy processes from Kuznetsov et al. (2011) to path functionals; in particular, first passage times, overshoots, undershoots, and the last maximum before the passage time. Such functionals have many applications, for instance, in finance (the pricing of exotic options in a Lévy model) and insurance (ruin time, debt at ruin, and related quantities for a Lévy insurance risk process). The technique works for any Lévy process whose running infimum and supremum evaluated at an independent exponential time can be sampled from. This includes classic examples such as stable processes, subclasses of spectrally one-sided Lévy processes, and large new families such as meromorphic Lévy processes. Finally, we present some examples. A particular aspect that is illustrated is that the Wiener-Hopf Monte Carlo simulation technique (provided that it applies) performs much better at approximating first passage times than a ‘plain’ Monte Carlo simulation technique based on sampling increments of the Lévy process.
On the Disorder Problem for a Negative Binomial Process
Authors: Buonaguidi Bruno , Muliere Pietro
Published: March 2016

We study the Bayesian disorder problem for a negative binomial process. The aim is to determine a stopping time which is as close as possible to the random and unknown moment at which a sequentially observed negative binomial process changes the value of its characterizing parameter p ∈ (0, 1). The solution to this problem is explicitly derived through the reduction of the original optimal stopping problem to an integro-differential free-boundary problem. A careful analysis of the free-boundary equation and of the probabilistic nature of the boundary point allows us to specify when the smooth fit principle holds and when it breaks down in favour of the continuous fit principle.
Maximal Percentages in Pólya's Urn
Authors: Schulte-Geers Ernst , Stadje Wolfgang
Published: March 2016

We show that the supremum of the successive percentages of red balls in Pólya's urn model is almost surely rational, give the set of values that are taken with positive probability, and derive several exact distributional results for the all-time maximal percentage.
Eigentime Identity for One-Dimensional Diffusion Processes
Authors: Cheng Li-Juan , Mao Yong-Hua
Published: March 2016

The eigentime identity for one-dimensional diffusion processes on the halfline with an entrance boundary at ∞ is obtained by using the trace of the deviation kernel. For the case of an exit boundary at ∞, a similar eigentime identity is presented with the aid of the Green function. Explicit equivalent statements are also listed in terms of the strong ergodicity or the uniform decay for diffusion processes.