bluuebunny/crossref_metadata_2025 huggingface dataset:
A Variable Step Size Riemannian Sum for an Itô Integral
Authors: Rapoo E.
Published: July 2016

We investigate the problem of using a Riemannian sum with random subintervals to approximate the iterated Itô integral ∫wdw - or, equivalently, solving the corresponding stochastic differential equation by Euler's method with variable step sizes. In the past this task has been used as a counterexample to illustrate that variable step sizes must be used with extreme caution in stochastic numerical analysis. This article establishes a class of variable step size schemes which do work.
On the Correlation Structure of a Lévy-Driven Queue
Authors: Es-Saghouani Abdelghafour , Mandjes Michel
Published: July 2016

In this paper we consider a single-server queue with Lévy input and, in particular, its workload process (Q

t
)
t≥0, with a focus on the correlation structure. With the correlation function defined as r(t) := cov(Q
0, Q

t
) / var(Q
0) (assuming that the workload process is in stationarity at time 0), we first determine its transform ∫0
∞
r(t)e-ϑt
dt. This expression allows us to prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show that r(·) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics of r(t), for large t, for the cases of light-tailed and heavy-tailed Lévy inputs.
Conditional Ordering of k-out-of-n Systems with Independent But Nonidentical Components
Authors: Zhao Peng , Li Xiaohu , Balakrishnan N.
Published: July 2016

By consideringk-out-of-nsystems with independent and nonidentically distributed components, we discuss stochastic monotone properties of the residual life and the inactivity time. We then present some stochastic comparisons of two systems based on the residual life and inactivity time.
Total Time on Test Transforms of Order n and Their Implications in Reliability Analysis
Authors: Nair N. Unnikrishnan , Sankaran P. G. , Kumar B. Vinesh
Published: July 2016

In this paper we study the properties of total time on test transforms of order n and examine their applications in reliability analysis. It is shown that the successive transforms produce either distributions with increasing or bathtub-shaped failure rates or distributions with decreasing or upside bathtub-shaped failure rates. The ageing properties of the baseline distribution is compared with those of transformed distributions, and a partial order based on nth-order transforms and their implications are discussed.
Double Optimal Stopping in the Fishing Problem
Authors: Karpowicz Anna
Published: July 2016

In this paper we consider the following problem. An angler buys a fishing ticket that allows him/her to fish for a fixed time. There are two locations to fish at the lake. The fish are caught according to a renewal process, which is different for each fishing location. The angler's success is defined as the difference between the utility function, which is dependent on the size of the fish caught, and the time-dependent cost function. These functions are different for each fishing location. The goal of the angler is to find two optimal stopping times that maximize his/her success: when to change fishing location and when to stop fishing. Dynamic programming methods are used to find these two optimal stopping times and to specify the expected success of the angler at these times.
On the Maximum Exceedance of a Sequence of Random Variables Over a Renewal Threshold
Authors: Ha Xuemiao , Tang Qihe , Wei Li
Published: July 2016

In this paper we study the tail behavior of the maximum exceedance of a sequence of independent and identically distributed random variables over a random walk. For both light-tailed and heavy-tailed cases, we derive a precise asymptotic formula, which extends and unifies some existing results in the recent literature of applied probability.
The Integral of the Supremum Process of Brownian Motion
Authors: Janson Svante , Petersson Niclas
Published: July 2016

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.
On the Height and Length of the Ancestral Recombination Graph
Authors: Pardoux Etienne , Salamat Majid
Published: July 2016

The goal of this paper is to provide formulae for the expectation and variance of the height and length of the ancestral recombination graph (ARG). While the formula for the expectation of the height is known (see, e.g. Krone and Neuhauser (1997)), the other formulae seem to be new. We obtain in particular (see Theorem 4.1) a very simple formula which expresses the expectation of the length of the ARG as a linear combination of the expectations of both the length of the coalescent tree and the height of the ARG. Finally, we study the speed at which the ARG comes down from infinity.
Local Limit Approximations for Markov Population Processes
Authors: Socoll Sanda N. , Barbour A.D.
Published: July 2016

In this paper we are concerned with the equilibrium distribution ∏
n
of the nth element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a (2+α)th moment condition on the jump distributions, we establish a bound of order O(n
-(α+1)/2√logn) on the difference between the point probabilities of ∏
n
and those of a translated Poisson distribution with the same variance. Except for the factor √logn, the result is as good as could be obtained in the simpler setting of sums of independent, integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.
Limit Theorems for Random Triangular URN Schemes
Authors: Aguech Rafik
Published: July 2016

In this paper we study a generalized Pólya urn with balls of two colors and a random triangular replacement matrix. We extend some results of Janson (2004), (2005) to the case where the largest eigenvalue of the mean of the replacement matrix is not in the dominant class. Using some useful martingales and the embedding method introduced in Athreya and Karlin (1968), we describe the asymptotic composition of the urn after the nth draw, for large n.
The Early Stage Behaviour of a Stochastic SIR Epidemic with Term-Time Forcing
Authors: Britton Tom , Lindholm Mathias
Published: July 2016

The general stochastic SIR epidemic in a closed population under the influence of a term-time forced environment is considered. An ‘environment’ in this context is any external factor that influences the contact rate between individuals in the population, but is itself unaffected by the population. Here ‘term-time forcing’ refers to discontinuous but cyclic changes in the contact rate. The inclusion of such an environment into the model is done by replacing a single contact rate λ with a cyclically alternating renewal process with k different states denoted {Λ(t)}t≥0. Threshold conditions in terms of R
⋆ are obtained, such that R
⋆>1 implies that π, the probability of a large outbreak, is strictly positive. Examples are given where π is evaluated numerically from which the impact of the distribution of the time periods that Λ(t) spends in its different states is clearly seen.
An Inequality for Variances of the Discounted Rewards
Authors: Feinberg Eugene A. , Fei Jun
Published: July 2016

We consider the following two definitions of discounting: (i) multiplicative coefficient in front of the rewards, and (ii) probability that the process has not been stopped if the stopping time has an exponential distribution independent of the process. It is well known that the expected total discounted rewards corresponding to these definitions are the same. In this note we show that, the variance of the total discounted rewards is smaller for the first definition than for the second definition.
Large Deviations for Point Processes Based on Stationary Sequences with Heavy Tails
Authors: Hult Henrik , Samorodnitsky Gennady
Published: July 2016

In this paper we propose a framework that facilitates the study of large deviations for point processes based on stationary sequences with regularly varying tails. This framework allows us to keep track both of the magnitude of the extreme values of a process and the order in which these extreme values appear. Particular emphasis is put on (infinite) linear processes with random coefficients. The proposed framework provides a fairly complete description of the joint asymptotic behavior of the large values of the stationary sequence. We apply the general result on large deviations for point processes to derive the asymptotic decay of certain probabilities related to partial sum processes as well as ruin probabilities.
Duration Distribution of the Conjunction of Two Independent F Processes
Authors: Alodat M. T. , Al-Rawwash M. , Jebrini M. A.
Published: July 2016

In this paper we obtain an approximation for the duration distribution of the excursion set generated by the minimum of two independent F random processes above a high threshold u. Moreover, we obtain a closed-form approximation for the mean duration of the conjunction of these two F processes. As an illustration, we conduct a simulation study to compare the performances of the approximated distribution and the exact distribution.
Random effect bivariate survival models and stochastic comparisons
Authors: Gupta Ramesh C. , Gupta Rameshwar D.
Published: July 2016

In this paper we propose a general bivariate random effect model with special emphasis on frailty models and environmental effect models, and present some stochastic comparisons. The relationship between the conditional and the unconditional hazard gradients are derived and some examples are provided. We investigate how the well-known stochastic orderings between the distributions of two frailties translate into the orderings between the corresponding survival functions. These results are used to obtain the properties of the bivariate multiplicative model and the shared frailty model.
Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks
Authors: Iksanov Alexander , Meiners Matthias
Published: July 2016

We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. We include in particular the case of Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.
On estimation of the variances for critical branching processes with immigration
Authors: Ma Chunhua , Wang Longmin
Published: July 2016

The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 < α < 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).
Convergence Rate of Extremes for the General Error Distribution
Authors: Peng Zuoxiang , Nadarajah Saralees , Lin Fuming
Published: July 2016

Let {X

n
, n ≥ 1} be an independent, identically distributed random sequence with each X

n
having the general error distribution. In this paper we derive the exact uniform convergence rate of the distribution of the maximum to its extreme value limit.
Limit Theorems for the Inductive Mean on Metric Trees
Authors: Basrak Bojan
Published: July 2016

For random variables with values on binary metric trees, the definition of the expected value can be generalized to the notion of a barycenter. To estimate the barycenter from tree-valued data, the so-called inductive mean is constructed recursively using the weighted interpolation between the current mean and a new data point. We show the strong consistency of the inductive mean, but also that it, somewhat peculiarly, converges towards the true barycenter with different rates, and asymptotic distributions depending on the small variations of the underlying distribution.
A Central Limit Theorem for Reversible Processes with Nonlinear Growth of Variance
Authors: Zhao Ou , Woodroofe Michael , Volný Dalibor
Published: July 2016

Kipnis and Varadhan (1986) showed that, for an additive functional, S

n
say, of a reversible Markov chain, the condition E[S

n

2] / n → κ ∈ (0, ∞) implies the convergence of the conditional distribution of S

n
/ √E[S

n

2], given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, E[S

n

2] = n
l(n), where l is a slowly varying function. It is shown by example that the conditional distributions of S

n
/ √E[S

n

2] need not converge to the standard normal distribution in this case; and sufficient conditions for convergence to a (possibly nonstandard) normal distribution are developed.
Simulation-Based Computation of the Workload Correlation Function in a Lévy-Driven Queue
Authors: Glynn Peter W. , Mandjes Michel
Published: July 2016

In this paper we consider a single-server queue with Lévy input, and, in particular, its workload process (Q

t
)
t≥0, focusing on its correlation structure. With the correlation function defined as r(t):= cov(Q
0, Q

t
) / varQ
0
(assuming that the workload process is in stationarity at time 0), we first study its transform ∫0
∞
r(t)e-ϑt
dt, both for when the Lévy process has positive jumps and when it has negative jumps. These expressions allow us to prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. For the light-tailed case, we estimate the behavior of r(t) for large t. We then focus on techniques to estimate r(t) by simulation. Naive simulation techniques require roughly (r(t))-2 runs to obtain an estimate of a given precision, but we develop a coupling technique that leads to substantial variance reduction (the required number of runs being roughly (r(t))-1). If this is augmented with importance sampling, it even leads to a logarithmically efficient algorithm.
Sample Path Large Deviations for Order Statistics
Authors: Duffy Ken R. , Macci Claudio , Torrisi Giovanni Luca
Published: July 2016

We consider the sample paths of the order statistics of independent and identically distributed random variables with common distribution function F. If F is strictly increasing but possibly having discontinuities, we prove that the sample paths of the order statistics satisfy the large deviation principle in the Skorokhod M
1 topology. Sanov's theorem is deduced in the Skorokhod M'1 topology as a corollary to this result. A number of illustrative examples are presented, including applications to the sample paths of trimmed means and Hill plots.
Discrete Models for Scattering Populations
Authors: Fayard Patrick , Field Timothy R.
Published: July 2016

Jakeman's random walk model with step number fluctuations describes the coherent amplitude scattered from a rough medium in terms of the summation of individual scatterers' contributions. If the scattering population conforms to a birth-death immigration model, the resulting amplitude is K-distributed. In this context, we derive a class of diffusion processes as an extension of the ordinary birth-death immigration model. We show how this class encompasses four different cross-section models commonly studied in the literature. We conclude by discussing the advantages of this unified description.
Contact Process with Destruction of Cubes and Hyperplanes: Forest Fires Versus Tornadoes
Authors: Lanchier N.
Published: July 2016

Nonspatial stochastic models of populations subject to catastrophic events result in the common conclusion that the survival probability of the population is nondecreasing with respect to the random number of individuals removed at each catastrophe. The purpose of this paper is to prove that such a monotonic relationship is not true for simple spatial models based on Harris' contact processes, whose dynamics are described by hypergraph structures rather than traditional graph structures. More precisely, it is proved that, for a wide range of parameters, the destruction of (infinite) hyperplanes does not affect the existence of a nontrivial invariant measure, whereas the destruction of large (finite) cubes drives the population to extinction, a result that we depict by using the biological picture: forest fires are more devastating than tornadoes. This indicates that the geometry of the subsets struck by catastrophes is somewhat more important than their area, thus the need to consider spatial rather than nonspatial models in this context.
Russo's Formula, Uniqueness of the Infinite Cluster, and Continuous Differentiability of Free Energy for Continuum Percolation
Authors: Jiang Jianping , Zhang Sanguo , Guo Tiande
Published: July 2016

A new formula for continuum percolation on the Euclidean space R

d
(d ≥ 2), which is analogous to Russo's formula for bond or site percolation, is proved. Using this formula, we prove the equivalence between uniqueness of the infinite cluster and continuous differentiability of the mean number of clusters per Poisson point (or free energy). This yields a new proof for uniqueness of the infinite cluster since the continuous differentiability of free energy has been proved by Bezuidenhout, Grimmett and Löffler (1998); a consequence of this new proof gives the continuity of connectivity functions.
A Two-Dimensional Risk Model with Proportional Reinsurance
Authors: Badescu Andrei L. , Cheung Eric C. K. , Rabehasaina Landy
Published: July 2016

In this paper we consider an extension of the two-dimensional risk model introduced in Avram, Palmowski and Pistorius (2008a). To this end, we assume that there are two insurers. The first insurer is subject to claims arising from two independent compound Poisson processes. The second insurer, which can be viewed as a different line of business of the same insurer or as a reinsurer, covers a proportion of the claims arising from one of these two compound Poisson processes. We derive the Laplace transform of the time until ruin of at least one insurer when the claim sizes follow a general distribution. The surplus level of the first insurer when the second insurer is ruined first is discussed at the end in connection with some open problems.
Sensitivity Analysis in Markov Decision Processes with Uncertain Reward Parameters
Authors: Tan Chin Hon , Hartman Joseph C.
Published: July 2016

Sequential decision problems can often be modeled as Markov decision processes. Classical solution approaches assume that the parameters of the model are known. However, model parameters are usually estimated and uncertain in practice. As a result, managers are often interested in how estimation errors affect the optimal solution. In this paper we illustrate how sensitivity analysis can be performed directly for a Markov decision process with uncertain reward parameters using the Bellman equations. In particular, we consider problems involving (i) a single stationary parameter, (ii) multiple stationary parameters, and (iii) multiple nonstationary parameters. We illustrate the applicability of this work through a capacitated stochastic lot-sizing problem.
Small-Time Asymptotics of Option Prices and First Absolute Moments
Authors: Muhle-Karbe Johannes , Nutz Marcel
Published: July 2016

We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price processSfollows a general martingale. This is equivalent to studying the first centered absolute moment ofS. We show that ifShas a continuous part, the leading term is of order √Tin timeTand depends only on the initial value of the volatility. Furthermore, the term is linear inTif and only ifSis of finite variation. The leading terms for pure-jump processes with infinite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation ofSso that calculations are necessary only for the class of Lévy processes.
Some Inequalities of Linear Combinations of Independent Random Variables. I.
Authors: Xu Maochao , Hu Taizhong
Published: July 2016

In this paper we provide some sufficient conditions to stochastically compare linear combinations of independent random variables. The main results extend those given in Proschan (1965), Ma (1998), Zhao et al. (2011), and Yu (2011). In particular, we propose a new sufficient condition to compare the peakedness of linear combinations of independent random variables which may have heavy-tailed properties.
On the Zagreb Index of Random Recursive Trees
Authors: Feng Qunqiang , Hu Zhishui
Published: July 2016

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments ofZn, the Zagreb index of a random recursive tree of sizen, are obtained. We also show that the random process {Zn− E[Zn],n≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.
Moments of Random Sums and Robbins' Problem of Optimal Stopping
Authors: Gnedin Alexander , Iksanov Alexander
Published: July 2016

Robbins' problem of optimal stopping is that of minimising the expectedrankof an observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding thevalueof the stopped variable under the rule that yields the minimal expected rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank.
A note on the asymptotic behaviour of the extinction probability in supercritical population-size-dependent branching processes with independent and identically distributed random environments
Authors: Zhunwei Lu , Jagers Peter
Published: July 2016

In supercritical population-size-dependent branching processes with independent and identically distributed random environments, it is shown that under certain regularity conditions there exist constants 0 < α
1 ≤α
0 < + ∞ and 0 < C
1, C
2 < + ∞ such that the extinction probability starting with k individuals is bounded below by C
1
k
-α
0
and above by C
2
k
-α
1
for sufficiently large k. Moreover, a similar conclusion, which follows from a result of Höpfner, is presented along with some remarks.
Convolution equivalence and infinite divisibility
Authors: Pakes Anthony G.
Published: July 2016

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.
Limit theorem for continuous-time random walks with two time scales
Authors: Becker-Kern Peter , Meerschaert Mark M. , Scheffler Hans-Peter
Published: July 2016

Continuous-time random walks incorporate a random waiting time between random jumps. They are used in physics to model particle motion. A physically realistic rescaling uses two different time scales for the mean waiting time and the deviation from the mean. This paper derives the scaling limits for such processes. These limit processes are governed by fractional partial differential equations that may be useful in physics. A transfer theorem for weak convergence of finite-dimensional distributions of stochastic processes is also obtained.
Controlled Markov set-chains with discounting
Authors: Kurano Masami , Song Jinjie , Hosaka Masanori , Huang Youqiang
Published: July 2016

In the framework of discounted Markov decision processes, we consider the case that the transition probability varies in some given domain at each time and its variation is unknown or unobservable.

To this end we introduce a new model, named controlled Markov set-chains, based on Markov set-chains, and discuss its optimization under some partial order.

Also, a numerical example is given to explain the theoretical results and the computation.
The growth of general population-size-dependent branching processes year by year
Authors: Jagers Peter , Sagitov Serik
Published: July 2016

We study discrete-time population models where the nearest future of an individual may depend on the individual's life-stage (age and reproduction history) and the current population size. A criterion is given for whether there is a positive probability that the population survives forever. We identify the cases when population size grows exponentially and linearly and show that in the latter population size scaled by time is asymptotically Γ-distributed.
Use of an hourglass model in neuronal coding
Authors: Cottrell M. , Turova T. S.
Published: July 2016

We study a system of interacting renewal processes which is a model for neuronal activity. We show that the system possesses an exponentially large number (with respect to the number of neurons in the network) of limiting configurations of the ‘firing neurons’. These we call patterns. Furthermore, under certain conditions of symmetry we find an algorithm to control limiting patterns by means of the connection parameters.
Moments based approximation for the stationary distribution of a random walk in Z
+ with an application to the M/GI/1/n queueing system
Authors: Simonot F.
Published: July 2016

In this paper we consider an irreducible random walk in Z
+ defined by X(m+1) = max(0, X(m) + A(m+1)) with E{A} < 0 and for an s ≥ 0 where a
+ = max(0,a). Let π be the stationary distribution of X. We show that one can find probability distributions π
n
supported by {0,n} such that ||π
n
- π||1 ≤ Cn
-s
, where the constant C is computable in terms of the moments of A, and also that ||π
n
- π||1 = o(n
-s
). Moreover, this upper bound reveals exact for s ≥ 1, in the sense that, for any positive ε, we can find a random walk fulfilling the above assumptions and for which the relation ||π
n
- π||1 = o(n
-s-ε) does not hold. This result is used to derive the exact convergence rate of the time stationary distribution of an M/GI/1/n queueing system to the time stationary distribution of the corresponding M/GI/1 queueing system when n tends to infinity.
Large deviations in estimation of an Ornstein-Uhlenbeck model
Authors: Florens-Landais Danielle , Pham Huyên
Published: July 2016

A large deviation principle (LDP) with an explicit rate function is proved for the estimation of drift parameter of the Ornstein-Uhlenbeck process. We establish an LDP for two estimating functions, one of them being the score function. The first one is derived by applying the Gärtner–Ellis theorem. But this theorem is not suitable for the LDP on the score function and we circumvent this key point by using a parameter-dependent change of measure. We then state large deviation principles for the maximum likelihood estimator and another consistent drift estimator.
Super-replication in stochastic volatility models under portfolio constraints
Authors: Cvitanić Jakša , Pham Huyên , Touzi Nizar
Published: July 2016

We study a financial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints. The latter are given in terms of bounds imposed on the borrowing and short-selling of a ‘hedger’ in this market, and can be described by a closed convex set K. We find explicit characterizations of the minimal price needed to super-replicate European-type contingent claims in this framework. The results depend on whether the volatility is bounded away from zero and/or infinity, and also, on if we have linear dynamics for the stock price process, and whether volatility process depends on the stock price. We use a previously known representation of the minimal price as a supremum of the prices in the corresponding shadow markets, and we derive a PDE characterization of that representation.
Asymptotic analysis of the general stochastic epidemic with variable infectious periods
Authors: Startsev A. N.
Published: July 2016

A generalisation of the classical general stochastic epidemic within a closed, homogeneously mixing population is considered, in which the infectious periods of infectives follow i.i.d. random variables having an arbitrary but specified distribution. The asymptotic behaviour of the total size distribution for the epidemic as the initial numbers of susceptibles and infectives tend to infinity is investigated by generalising the construction of Sellke and reducing the problem to a boundary crossing problem for sums of independent random variables.
Some new results involving the NBU(2) class of life distributions
Authors: Li Xiaohu , Kochar Subhash C.
Published: July 2016

Some new results about the NBU(2) class of life distributions are obtained. Firstly, it is proved that the decreasing with time of the increasing concave ordering of the excess lifetime in a renewal process leads to the NBU(2) property of the interarrival times. Secondly, the NBU(2) class of life distributions is proved to be closed under the formation of series systems. Finally, it is also shown that the NBU(2) class is closed under convolution operation.
The size order of the state vector of a continuous-time homogeneous Markov system with fixed size
Authors: Kipouridis I. , Tsaklidis G.
Published: July 2016

The variation of the state vectors
p
(t) = (p

i
(t)) of a continuous-time homogeneous Markov system with fixed size is examined. A specific time t
0 after which the size order of the elements p

i
(t) becomes stable provides a criterion of the system's convergence rate. A method is developed to find t
0 and a quickly evaluated lower bound for t
0. This method is based on the geometric characteristics and the volumes of the attainable structures. Moreover, a condition concerning the selection of starting vectors
p
(0) is given so that the vector functions
p
(t) retain the same size order for every time greater than a given time t.
Some optimal stopping problems with nontrivial boundaries for pricing exotic options
Authors: Guo Xin , Shepp Larry
Published: July 2016

We solve the following three optimal stopping problems for different kinds of options, based on the Black-Scholes model of stock fluctuations. (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more difficult than the closely related one for the Russian option, and we show that for a class of utility functions the free boundary is governed by a nonlinear ordinary differential equation. (ii) A new type of stock option, for a company, where the company provides a guaranteed minimum as an added incentive in case the market appreciation of the stock is low, thereby making the option more attractive to the employee. We show that the value of this option is given by solving a nonalgebraic equation. (iii) A new call option for the option buyer who is risk-averse and gets to choose, a priori, a fixed constant l as a ‘hedge’ on a possible downturn of the stock price, where the buyer gets the maximum of l and the price at any exercise time. We show that the optimal policy depends on the ratio of x/l, where x is the current stock price.
A note on convergence of probability measures
Authors: Grey D. R.
Published: July 2016

We present, discuss and prove an apparently unfamiliar result in the convergence of probability measures.
An improved approximation for assessing the statistical significance of molecular sequence features
Authors: Mercier S. , Cellier D. , Charlot D.
Published: July 2016

Using random walk theory, we first establish explicitly the exact distribution of the maximal partial sum of a sequence of independent and identically distributed random variables. This result allows us to obtain a new approximation of the distribution of the local score of one sequence. This approximation improves the one given by Karlin et al., which can be deduced from this new formula. We obtain a more accurate asymptotic expression with additional terms. Examples of application are given.
Problèmes de ruine en théorie du risque à temps discret avec horizon fini
Authors: Picard Philippe , Lefèvre Claude , Coulibaly Ibrahim
Published: July 2016

We consider a discrete-time risk model which describes the evolution of the reserves of an insurance company at periodic dates fixed in advance. The amount of loss per unit of time corresponds to independent and identically distributed random variables with arithmetic distribution, and the process of the receipt of premiums is assumed to be deterministic, nonnegative but not uniform (instead of being constant and equal to 1 as in the standard, compound binomial model). For this model, we determine the probability of ruin (or of non-ruin), as well as the distribution of the severity of the eventual ruin, with some finite horizon. A compact and efficient exact expression is found by bringing up-to-date a generalised family of Appell polynomials. The method used is illustrated with some numerical examples.
The equilibrium distribution for a generalized Sankoff-Ferretti model accurately predicts chromosome size distributions in a wide variety of species
Authors: De Arkendra , Ferguson Michael , Sindi Suzanne , Durrett Richard
Published: July 2016

Sankoff and Ferretti (1996) introduced several models of the evolution of chromosome size by reciprocal translocations, where for simplicity they ignored the existence of centromeres. However, when they compared the models to data on six organisms they found that their short chromosomes were too short, and their long chromosomes were too long. Here, we consider a generalization of their proportional model with explicit chromosome centromeres and introduce fitness functions based on recombination probabilities and on the length of the longest chromosome arm. We find a simple formula for the stationary distribution for our model which fits the data on chromosome lengths in many, but not all, species.
Multivariate product-type lower bounds
Authors: Block Henry W. , Chen Tuhao
Published: July 2016

Univariate probability inequalities have received extensive attention. It has been shown that under certain conditions, product-type bounds are valid and sharper than summation-type bounds. Although results concerning multivariate inequalities have appeared in the literature, product-type bounds in a multivariate setting have not yet been studied. This note explores an approach using graph theory and linear programming techniques to construct product-type lower bounds for the probability of the intersection among unions of k sets of events.
The annihilating process on random trees and the square lattice
Authors: Sudbury Aidan
Published: July 2016

An annihilating process is an interacting particle system in which the only interaction is that a particle may kill a neighbouring particle. Since there is no birth and no movement, once a particle has no neighbours its site remains occupied for ever. The survival probability is calculated for a random tree and for the square lattice. A connection is made between annihilating processes and the adsorption of molecules onto surfaces. A one-dimensional adsorption problem is solved in the case in which the two neighbours do not act independently.
Properties of classes of life distribution based on the conditional variance
Authors: Stoyanov Jordan , Al-Sadi M. H. M.
Published: July 2016

We consider two classes of life distribution, V

D
and V

I
, the members of which are defined in terms of the conditional variance σ
2(t) of the remaining lifetime of a system: a life distribution F belongs to V

D
if is a decreasing function and to V

I
if is increasing. We study closure properties of these classes under relevant reliability operations such as mixing, convolution and formation of coherent systems. We show, for example, that the class V

D
is not closed under convolution or mixing, and that the class V

I
is not closed under formation of coherent systems.
Some fluctuation identities for Lévy processes with jumps of the same sign
Authors: Zhou Xiaowen
Published: July 2016

We consider a two-sided exit problem for a Lévy process with no positive jumps. The Laplace transform of the time when the process first exits an interval from above is obtained. It is expressed in terms of another Laplace transform for the one-sided exit problem. Applications of this result are discussed. In particular, a new expression for the solution to the two-sided exit problem is obtained. The joint distribution of the minimum and the maximum values of such a Lévy process is also studied.
Some risk management problems for firms with internal competition and debt
Authors: Guo Xin
Published: July 2016

Consider an optimization problem for a company with the following parameters: a constant liability payment rate (δ), an average return (μ) and a risk (σ) proportional to the size of the business unit, and an internal competition factor (α) between different units. The goal is to maximize the expected present value of the total dividend distributions, via controls (U

t
, Z

t
), where U

t
is the size of the business unit and Z

t
is the total dividend payoff up to time t. We formulate this as a stochastic control problem for a diffusion process X

t
and derive an explicit solution by solving the corresponding Hamilton-Jacobi-Bellman equation. The resulting optimal control policy involves a mixture of a nonlinear control for U

t
and a singular control for Z

t
. The optimal strategies are different for the cases δ < 0 and δ = 0. When δ > 0, it is optimal to play bold: the initial optimal investment size should be proportional to the debt rate δ. Under this optimal rule, however, the probability of bankruptcy in finite time is 1. When δ = 0, i.e. when the company is free of debt, the probability of going broke in finite time reduces to 0. Moreover, when δ = 0, the value function is singular at X
0 = 0. Our analytical result shows considerable consistency with daily business practices. For instance, it shows that ‘too many people is counter-productive’. In fact, the maximal optimal size of the business unit should be inversely proportional to α. This eliminates the redundant and simplistic technical assumption of a known uniform upper bound on the size of the firm.
Multivariate semi-Markov analysis of burst properties of multiconductance single ion channels
Authors: Ball F. , Milne R. K. , Yeo G. F.
Published: July 2016

Patch clamp recordings from ion channels often show periods of repetitive activity, known as bursts, which are noticeably separated from each other by periods of inactivity. Depending on the type of channel, such recordings may exhibit (conductance) levels between the closed (zero) level and the fully open level. Properties of bursts are less subject to problems that arise from recording than are properties for individual sojourns at different levels, and study of bursting behaviour provides important information about the finer structure of the underlying channel gating process. For a general finite state space continuous-time Markov chain model allowing one or more nonzero conductance levels, the present paper establishes results about the semi-Markov structure of a single burst and of a sequence of bursts, and uses this in a unified approach to properties of both theoretical and empirical bursts. The distribution and moments of particular burst properties, including the total charge transfer, the total sojourn time and the total number of visits to specified conductance levels during a burst, are derived. Various extensions are also described.
Proofs of the closure property of NBUC and NBU(2) under convolution
Authors: Hu Taizhong , Xie Huiliang
Published: July 2016

In this note, we give new proofs of the closure property of ageing classes NBUC and NBU(2) under convolution to make up the gaps in the proofs of Cao and Wang (1991) and of Li and Kochar (2001).
A family of densities derived from the three-parameter Dirichlet process
Authors: Carlton Matthew A.
Published: July 2016

The traditional Dirichlet process is characterized by its distribution on a measurable partition of the state space - namely, the Dirichlet distribution. In this paper, we consider a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.
Optimal stopping on patterns in strings generated by independent random variables
Authors: Bruss F. Thomas , Louchard Guy
Published: July 2016

Strings are generated by sequences of independent draws from a given alphabet. For a given pattern H of length l and an integer n ≥ l, our goal is to maximize the probability of stopping on the last appearance of the pattern H in a string of length n (if any), given that, if we choose to stop on an occurrence of H, we must do so right away. This contrasts with the goals of several other investigations on patterns in strings such as computing the expected occurrence time and the probability of finding exactly r patterns in a string of fixed length. Several motivations are given for our problem ranging from relatively simple best choice problems to more difficult stopping problems allowing for a variety of interesting applications. We solve this problem completely in the homogeneous case for uncorrelated patterns. However, several of these results extend immediately to the inhomogeneous case. In the homogeneous case, optimal success probabilities are shown to vary, depending on characteristics of the pattern, essentially between the value 1/e and a new asymptotic constant 0.619…. These results demonstrate a considerable difference between the true optimal success probability compared with what a first approximation by heuristic arguments would yield. It is interesting to see that the so-called odds algorithm which gives the optimal rule for l = 1 yields an excellent approximation of the optimal rule for any l. This is important for applications because the odds algorithm is very convenient. We finally give a detailed asymptotic analysis for the homogeneous case.
Correlation models with long-range dependence
Authors: Ma Chunsheng
Published: July 2016

This paper is concerned with the correlation structure of a stationary discrete time-series with long memory or long-range dependence. Given a sequence of bounded variation, we obtain necessary and sufficient conditions for a function generated from the sequence to be a proper correlation function. These conditions are applied to derive various slowly decaying correlation models. To obtain correlation models with short-range dependence from an absolutely summable sequence, a simple method is introduced.
A decomposition for multistate monotone systems
Authors: Block Henry W. , Savits Thomas H.
Published: July 2016

A decomposition theorem for multistate structure functions is proven. This result is applied to obtain bounds for the system performance function. Another application is made to interpret the multistate structures of Barlow and Wu. Various concepts of multistate importance and coherence are also discussed.
A note on the branching random walk
Authors: Kaplan Norman
Published: July 2016

A well-known theorem in the theory of branching random walks is shown to hold when only Σj log jpj
<∞. This result was asserted by Athreya and Kaplan (1978), but their proof was incorrect.
Lower tail analysis of the distribution of the strength of load-sharing systems
Authors: Harlow D. G. , Smith R. L. , Taylor H. M.
Published: July 2016

The Weibull distribution for the strength of a long fiber is reviewed, with emphasis on its derivation via extreme-value theory. It is shown that a corresponding Weibull distribution, but with different scale and shape parameters, is appropriate for the lower tail distribution of a bundle of parallel fibers under general load-sharing assumptions. This result is applied to series-parallel systems. The paper contains discussion of the application of these results to the strength of fibrous materials. It is concluded that the Weibull approximation, developed for series-parallel systems, is appropriate for long thin materials consisting of a small number of parallel fibers.
A property of the generalized inverse Gaussian distribution with some applications
Authors: Embrechts Paul
Published: July 2016

An asymptotic convolution property for the generalized inverse Gaussian distribution with λ < 0 is proved. This result is applied to calculate the probability of ruin in the general risk model when these distributions are used to model claim sizes. Some related applications are discussed.
Recurrent composite events
Authors: Leslie R.T.
Published: July 2016

On a sequence of Bernoulli trials, the definition of a recurrent event ε involves the occurrence of a unique pattern of successes (S) and failures (F), the final element of which is the result of the nth trial. Success runs are the best known of such recurrent events, but Feller (1959, §13.8) mentions more complicated patterns, among which two types may be distinguished. The simpler involves a single more complex pattern such as SSFFSS; the second type involves a set of alternative events defining ε, which is said to occur when any one of the alternatives occurs at trial number n. Thus if ε stands for “either a success run of length r or a failure run of length ρ”, there are two alternatives in the set; the problem is elementary because the component events are “non-overlapping”.
On the existence of the expectation of randomly stopped sums
Authors: Samuel Ester
Published: July 2016

Let a+
= a if a > 0, a+
= 0 otherwise, and a–
= (– a)+.
A birth, death and migration process
Authors: Adke S. R.
Published: July 2016

A model proposed by Bailey (1968) for migratory individuals which reproduce according to a simple birth-death process is generalized to include time dependent birth and death rates.
A note on the queueing system GI/Ek
/1
Authors: Finch P. D.
Published: July 2016

In this note we adopt the notation and terminology of Kingman (1966) without further comment. For the general single server queue one has

For the queueing system GI/Ek/1 it is possible to make use of the particular nature of the service time distribution to evaluate the right-hand side of Equation (1) in terms of the k roots of a certain equation. This evaluation is carried out in detail in Prabhu (1965) to which reference should be made for the technicalities involved. A similar evaluation applies to the limiting distribution

when it exists. However, the resulting expression again involves the k roots of a certain equation. In this note we draw attention to an alternative procedure which does not involve the calculation of roots. We remark that a similar, but slightly different, procedure can be used in the study of the queueing system Ek/GI/1. Details of this will be presented in a separate note.
A sample path analysis of the M/M/1 queue
Authors: Baccelli Francois , Massey William A.
Published: July 2016

The exact solution for the transient distribution of the queue length and busy period of the M/M/1 queue in terms of modified Bessel functions has been proved in a variety of ways. Methods of the past range from spectral analysis (Lederman and Reuter (1954)), combinatorial arguments (Champernowne (1956)), to generating functions coupled with Laplace transforms (Clarke (1956)). In this paper, we present a novel approach that ties the computation of these transient distributions directly to the random sample path behavior of the M/M/1 queue. The use of Laplace transforms is minimized, and the use of generating functions is eliminated completely. This is a method that could prove to be useful in developing a similar transient analysis for queueing networks.