http://www.hindawi.comThe latest articles from Hindawi Publishing Corporation© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/189675The expected number of real zeros of the polynomial of the form a0+a1x+a2x2+⋯+anxn, where a0,a1,a2,…,an is a sequence of standard Gaussian random variables, is known. For n large it is shown that this expected number in (−∞,∞) is asymptotic to (2/π)log⁡n. In this paper, we show that this asymptotic value increases significantly to n+1 when we consider a polynomial in the form a0(n0)1/2x/1+a1(n1)1/2x2/2+a2(n2)1/2x3/3+⋯+an(nn)1/2xn+1/n+1 instead. We give the motivation for our choice of polynomial and also obtain some other characteristics for the polynomial, such as the expected number of level crossings or maxima. We note, and present, a small modification to the definition of our polynomial which improves our result from the above asymptotic relation to the equality.K. Farahmand, A. Grigorash, and B. McGuinness© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/254897Various upper bounds for the L2-norm of the Wick product of two measurable functions of a random variable X, having finite moments of any order, together with a universal minimal condition, are proven. The inequalities involve the second quantization operator of a constant times the identity operator. Some conditions ensuring that the constants involved in the second quantization operators are optimal, and interesting examples satisfying these conditions are also included.Alberto Lanconelli and Aurel I. Stan© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/735436Let F^n be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n. A central limit theorem is established for the target statistic F^n(ξ^n), where the underlying random vector forms an asymptotically stationary absolutely regular stochastic process, and ξ^n is an estimator of a multivariate parameter ξ by using a vector of U-statistics. The results obtained extend or generalize previous results from the stationary univariate case to the asymptotically stationary multivariate case. An example of asymptotically stationary absolutely regular multivariate ARMA process and an example of a useful estimation of F(ξ) are given in the applications.Echarif Elharfaoui and Michel Harel© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/275747We consider the martingale problem related to the solution of an SDE on the line. It is shown that the solution of this martingale problem can be approximated by solutions of the corresponding time-discrete martingale problems under some conditions. This criterion is especially expedient for establishing the convergence of population processes to SDEs. We also show that the criterion yields a weak Euler scheme approximation of SDEs under fairly weak assumptions on the driving force of the approximating processes.Henryk Zähle© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/160303We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup-norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.Charles El-Nouty© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/760214In 1997, Haruki and Rassias introduced two generalizations of the Poisson kernel in two dimensions and discussed integral formulas for them. Furthermore, they presented an open problem. In 1999, Kim gave a solution to that problem. Here, we give a solution to this open problem by means of a different method. The purpose of this paper is to give integral averages of two generalizations of the Poisson kernel, that is, we generalize the open problem.Serap Bulut© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/589480The existence of unbounded nonnegative solutions of a boundary value problem for nth-order differential equations defined on an infinite interval is obtained by means of the Mönch fixed-point theorem. An example is then presented to demonstrate the application of our results.Zhenbin Liu, Lishan Liu, Yonghong Wu, and Jing Zhao© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/415692We consider a model for a single link in a circuit-switched network. The link has C circuits, and the input consists of offered calls of two types, that we call primary and secondary traffic. Of the C links, R are reserved for primary traffic. We assume that both traffic types arrive as Poisson arrival streams. Assuming that C is large and R=O(1), the arrival rate of primary traffic is O(C), while that of secondary traffic is smaller, of the order O(C). The holding times of the primary calls are assumed to be exponentially distributed with unit mean. Those of the secondary calls are exponentially distributed with a large mean, that is, O(C). Thus, the primary calls have fast arrivals and fast service, compared to the secondary calls. The loads for both traffic types are comparable (O(C)), and we assume that the system is “critically loaded”; that is, the system's capacity is approximately equal to the total load. We analyze asymptotically the steady state probability that n1 (resp., n2) circuits are occupied by primary (resp., secondary) calls. In particular, we obtain two-term asymptotic approximations to the blocking probabilities for both traffic types.John A. Morrison and Charles Knessl© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/39171A monotonicity property and a refined estimate of Harnack inequality are derived for positive solutions of the Weinstein equation.Yifei Pan and Mei Wang© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/43091This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN(θ)=∑j=0N−1{αN−jcos(j+1/2)θ+βN−jsin(j+1/2)θ}, where αj and βj, j=0,1,2,…, N−1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials with complex variables and complex random coefficients, with a self-reciprocal property. We establish the relation between this type of random algebraic polynomials and the above random trigonometric polynomials, and we show that the required level crossings have the functionality form of cos(N+θ/2). We also discuss the relationship which exists and can be explored further between our random polynomials and random matrix theory.K. Farahmand© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/37848We analyze and compare three (s,S) inventory systems with positive service time and retrial of customers. In all of these systems, arrivals of customers form a Poisson process and service times are exponentially distributed. When the inventory level depletes to s due to services, an order of replenishment is placed. The lead-time follows an exponential distribution. In model I, an arriving customer, finding the inventory dry or server busy, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the inventory dry or server busy, returns to the orbit with probability δ and is lost forever with probability (1−δ). In addition to the description in model I, we provide a buffer of varying (finite) capacity equal to the current inventory level for model II and another having capacity equal to the maximum inventory level S for model III. In models II and III, an arriving customer, finding the buffer full, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the buffer full, returns to the orbit with probability δ and is lost forever with probability (1−δ). In all these models, the interretrial times are exponentially distributed with linear rate. Using matrix-analytic method, we study these inventory models. Some measures of the system performance in the steady state are derived. A suitable cost function is defined for all three cases and analyzed using graphical illustrations.A. Krishnamoorthy and K. P. Jose© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/31819We study qualitative properties of minimizers for a class of integral functionals, defined in a weighted space. In particular we obtain Hölder regularity up to the boundary for the minimizers of an integral functional of high order by using an interior local regularity result and a modified Moser method with special test function.S. Bonafede, V. Cataldo, and S. D'Asero© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/80750Given any finite set of trajectories of a Lipschitzian quantum stochastic differential inclusion (QSDI), there exists a continuous selection from the complex-valued multifunction associated with the solution set of the inclusion, interpolating the matrix elements of the given trajectories. Furthermore, the difference of any two of such solutions is bounded in the seminorm of the locally convex space of solutions.E. O. Ayoola and John O. Adeyeye© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/74191We estimate a lower bound for the number of real roots of a random alegebraic equation whose random coeffcients are dependent normal random variables.Takashi Uno© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/59204The main purpose of this paper is to give some common fixed point theorems of mappings and set-valued mappings of a symmetric space with some applications to probabilistic spaces. In order to get these results, we define the concept of E-weak compatibility between set-valued and single-valued mappings of a symmetric space.M. Aamri, A. Bassou, S. Bennani, and D. El Moutawakil© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/72326The paper develops a new class of financial market models. These models are based on generalized telegraph processes with alternating velocities and jumps occurring at switching velocities. The model under consideration is arbitrage-free and complete if the directions of jumps in stock prices are in a certain correspondence with their velocity and with the behaviour of the interest rate. A risk-neutral measure and arbitrage-free formulae for a standard call option are constructed. This model has some features of models with memory, but it is more simple.Nikita Ratanov© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/78343We consider a new class of complementarity problems for η-pseudomonotone maps and obtain an existence result for their solutions in real Hausdorff topological vector spaces. Our results extend the same previous results in this literature.A. P. Farajzadeh© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/68958We consider a class of queueing processes represented by a Skorokhod problem coupled with a controlled point process. Posing a discounted control problem for such processes, we show that the optimal value functions converge, in the fluid limit, to the value of an analogous deterministic control problem for fluid processes.Guodong Pang and Martin V. Day© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/69747We investigate a class of abstract stochastic evolution equations driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space. We establish the existence and uniqueness of a mild solution, a continuous dependence estimate, and various convergence and approximation results. Finally, the analysis of three examples is provided to illustrate the applicability of the general theory.Eduardo Hernandez, David N. Keck, and Mark A. McKibben© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/92723We construct a family of martingales with Gaussian marginal distributions. We give a weak construction as Markov, inhomogeneous in time processes, and compute their infinitesimal generators. We give the predictable quadratic variation and show that the paths are not continuous. The construction uses distributions Gσ having a log-convolution semigroup property. Further, we categorize these processes as belonging to one of two classes, one of which is made up of piecewise deterministic pure jump processes. This class includes the case where Gσ is an inverse log-Poisson distribution. The processes in the second class include the case where Gσ is an inverse log-gamma distribution. The richness of the family has the potential to allow for the imposition of specifications other than the marginal distributions.Kais Hamza and Fima C. Klebaner© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/81934This paper investigates the lim inf behavior of the sojourn time process and the escape rate process associated with the Cauchy process on the line. The monotone functions associated with the lower asymptotic growth rate of the sojourn time are characterized and the asymptotic size of the large values of the escape rate process is developed.A. Chukwuemeka Okoroafor© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/42640We prove that a class of fully coupled forward-backward systems in infinite dimensions has a local unique solution. After studying the regularity property of the solution, we prove that for a peculiar class of systems arising in the theory of stochastic optimal control, the solution exists in arbitrary large time interval. Finally, we investigate the connection between the solution to the systems and a stochastic optimal control problem.Giuseppina Guatteri© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/78196We are concerned with the solutions of a special class of backward stochastic differential equations which are driven by a Brownian motion, where the uniform Lipschitz continuity is replaced by a stochastic one. We prove the existence and uniqueness of the solution in Lp with p>1.Jiajie Wang, Qikang Ran, and Qihong Chen© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/82753This is the first of the two companion papers which treat an infinite time horizon hereditary portfolio optimization problem in a market that consists of one savings account and one stock account. Within the solvency region, the investor is allowed to consume from the savings account and can make transactions between the two assets subject to paying capital gain taxes as well as a fixed plus proportional transaction cost. The investor is to seek an optimal consumption-trading strategy in order to maximize the expected utility from the total discounted consumption. The portfolio optimization problem is formulated as an infinite dimensional stochastic classical-impulse control problem. The quasi-variational HJB inequality (QVHJBI) for the value function is derived in this paper. The second paper contains the verification theorem for the optimal strategy. It is also shown there that the value function is a viscosity solution of the QVHJBI.Mou-Hsiung Chang© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/56924We generalize the Cauchy distribution so that we can have asymmetrical tails. This allows us to obtain unusual laws of large numbers involving weighted sums of these random variables. Unusual in the sense that even though in every case E|X|=∞, we can still obtain a nonzero limit for these weighted sums.André Adler© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/JAMSA/2006/81508We consider an (s,S) inventory system with random lead time and repeated demands of unsatisfied demands from the orbit. Whenever the inventory level falls to the level s, an order is placed to bring the level to S. The quantity ordered is M=S−s. Demands to the system are served immediately if there is a positive inventory. Otherwise it will go to a pool of unsatisfied customers called orbit. After a random amount of time, that demand is retried for service. We assume a Markovian setup for the time between consecutive arrivals, replenishments, and retrials. We obtained the condition for ergodicity of the system, steady state system size probabilities, expected length of the busy period of the system, expected inventory level, expected number of customers waiting in the orbit, expected waiting times, and so forth. A control problem is studied and some numerical illusrtations are provided.P. V. Ushakumari© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/40149This paper is the continuation of the paper entitled “Hereditary portfolio optimization with taxes and fixed plus proportional transaction costs I” that treats an infinite-time horizon hereditary portfolio optimization problem in a market that consists of one savings account and one stock account. Within the solvency region, the investor is allowed to consume from the savings account and can make transactions between the two assets subject to paying capital-gain taxes as well as a fixed plus proportional transaction cost. The investor is to seek an optimal consumption-trading strategy in order to maximize the expected utility from the total discounted consumption. The portfolio optimization problem is formulated as an infinite dimensional stochastic classical impulse control problem due to the hereditary nature of the stock price dynamics and inventories. This paper contains the verification theorem for the optimal strategy. It also proves that the value function is a viscosity solution of the QVHJBI.Mou-Hsiung Chang© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/82517We construct random iterative processes with errors for three asymptotically nonexpansive random operators and study necessary conditions for the convergence of these processes. The results presented in this paper extend and improve the recent ones announced by I. Beg and M. Abbas (2006), and many others.Somyot Plubtieng, Poom Kumam, and Rabian Wangkeeree© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/JAMSA/2006/39093We consider a k-out-of-N reliability system with identical components having exponential lifetimes. There is a single repairman who attends to failed components on a first-come first-served basis. The repair times are assumed to be of phase type. The system has K spares that can be used according to a probabilistic rule to extend the lifetime of the system. The system is analyzed using Markov chain theory and some interesting results are obtained. A few illustrative numerical examples are discussed.Srinivas R. Chakravarthy© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/JAMSA/2006/18130Pricing in mathematical finance often involves taking expected values under different equivalent measures. Fundamentally, one needs to first ensure the existence of ELMM, which in turn requires that the stochastic exponential of the market price of risk process be a true martingale. In general, however, this condition can be hard to validate, especially in stochastic volatility models. This had led many researchers to “assume the condition away,” even though the condition is not innocuous, and nonsensical results can occur if it is in fact not satisfied. We provide an applicable theorem to check the conditions for a general class of Markovian stochastic volatility models. As an example we will also provide a detailed analysis of the Stein and Stein and Heston stochastic volatility models.Bernard Wong and C. C. Heyde© 2008, Hindawi Publishing Corporation. All rights reserved.