International Journal of Stochastic Analysis The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. A Comparative Numerical Study of the Spectral Theory Approach of Nishimura and the Roots Method Based on the Analysis of BDMMAP/G/1 Queue Tue, 17 Feb 2015 09:51:28 +0000 This paper considers an infinite-buffer queuing system with birth-death modulated Markovian arrival process (BDMMAP) with arbitrary service time distribution. BDMMAP is an excellent representation of the arrival process, where the fractal behavior such as burstiness, correlation, and self-similarity is observed, for example, in ethernet LAN traffic systems. This model was first apprised by Nishimura (2003), and to analyze it, he proposed a twofold spectral theory approach. It seems from the investigations that Nishimura’s approach is tedious and difficult to employ for practical purposes. The objective of this paper is to analyze the same model with an alternative methodology proposed by Chaudhry et al. (2013) (to be referred to as CGG method). The CGG method appears to be rather simple, mathematically tractable, and easy to implement as compared to Nishimura’s approach. The Achilles tendon of the CGG method is the roots of the characteristic equation associated with the probability generating function (pgf) of the queue length distribution, which absolves any eigenvalue algebra and iterative analysis. Both the methods are presented in stepwise manner for easy accessibility, followed by some illustrative examples in accordance with the context. Arunava Maity and U. C. Gupta Copyright © 2015 Arunava Maity and U. C. Gupta. All rights reserved. Asymptotic Stabilizability of a Class of Stochastic Nonlinear Hybrid Systems Wed, 11 Feb 2015 07:54:05 +0000 The problem of the asymptotic stabilizability in probability of a class of stochastic nonlinear control hybrid systems (with a linear dependence of the control) with state dependent, Markovian, and any switching rule is considered in the paper. To solve the issue, the Lyapunov technique, including a common, single, and multiple Lyapunov function, the hybrid control theory, and some results for stochastic nonhybrid systems are used. Sufficient conditions for the asymptotic stabilizability in probability for a considered class of hybrid systems are formulated. Also the stabilizing control in a feedback form is considered. Furthermore, in the case of hybrid systems with the state dependent switching rule, a method for a construction of stabilizing switching rules is proposed. Obtained results are illustrated by examples and numerical simulations. Ewelina Seroka Copyright © 2015 Ewelina Seroka. All rights reserved. A General Multidimensional Monte Carlo Approach for Dynamic Hedging under Stochastic Volatility Sun, 08 Feb 2015 14:05:28 +0000 We propose a feasible and constructive methodology which allows us to compute pure hedging strategies with respect to arbitrary square-integrable claims in incomplete markets. In contrast to previous works based on PDE and BSDE methods, the main merit of our approach is the flexibility of quadratic hedging in full generality without a priori smoothness assumptions on the payoff. In particular, the methodology can be applied to multidimensional quadratic hedging-type strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. In order to demonstrate that our methodology is indeed applicable, we provide a Monte Carlo study on generalized Föllmer-Schweizer decompositions, locally risk minimizing, and mean variance hedging strategies for vanilla and path-dependent options written on local volatility and stochastic volatility models. Daniel Bonetti, Dorival Leão, Alberto Ohashi, and Vinícius Siqueira Copyright © 2015 Daniel Bonetti et al. All rights reserved. A Stochastic Flows Approach for Asset Allocation with Hidden Economic Environment Tue, 27 Jan 2015 11:15:54 +0000 An optimal asset allocation problem for a quite general class of utility functions is discussed in a simple two-state Markovian regime-switching model, where the appreciation rate of a risky share changes over time according to the state of a hidden economy. As usual, standard filtering theory is used to transform a financial model with hidden information into one with complete information, where a martingale approach is applied to discuss the optimal asset allocation problem. Using a martingale representation coupled with stochastic flows of diffeomorphisms for the filtering equation, the integrand in the martingale representation is identified which gives rise to an optimal portfolio strategy under some differentiability conditions. Tak Kuen Siu Copyright © 2015 Tak Kuen Siu. All rights reserved. Yamada-Watanabe Results for Stochastic Differential Equations with Jumps Thu, 01 Jan 2015 09:34:03 +0000 Recently, Kurtz (2007, 2014) obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations covering also the case of stochastic differential equations with jumps. Following the original method of Yamada and Watanabe (1971), we give alternative proofs for the following two statements: pathwise uniqueness implies uniqueness in the sense of probability law, and weak existence together with pathwise uniqueness implies strong existence for stochastic differential equations with jumps. Mátyás Barczy, Zenghu Li, and Gyula Pap Copyright © 2015 Mátyás Barczy et al. All rights reserved. Adaptive Algorithm for Estimation of Two-Dimensional Autoregressive Fields from Noisy Observations Thu, 25 Dec 2014 00:10:03 +0000 This paper deals with the problem of two-dimensional autoregressive (AR) estimation from noisy observations. The Yule-Walker equations are solved using adaptive steepest descent (SD) algorithm. Performance comparisons are made with other existing methods to demonstrate merits of the proposed method. Alimorad Mahmoudi Copyright © 2014 Alimorad Mahmoudi. All rights reserved. On Henstock Method to Stratonovich Integral with respect to Continuous Semimartingale Thu, 18 Dec 2014 00:10:15 +0000 We will use the Henstock (or generalized Riemann) approach to define the Stratonovich integral with respect to continuous semimartingale in space. Our definition of Stratonovich integral encompasses the classical definition of Stratonovich integral. Haifeng Yang and Tin Lam Toh Copyright © 2014 Haifeng Yang and Tin Lam Toh. All rights reserved. Strong Law of Large Numbers for Hidden Markov Chains Indexed by an Infinite Tree with Uniformly Bounded Degrees Tue, 09 Dec 2014 06:59:15 +0000 We study strong limit theorems for hidden Markov chains fields indexed by an infinite tree with uniformly bounded degrees. We mainly establish the strong law of large numbers for hidden Markov chains fields indexed by an infinite tree with uniformly bounded degrees and give the strong limit law of the conditional sample entropy rate. Huilin Huang Copyright © 2014 Huilin Huang. All rights reserved. Optimal Foreign Exchange Rate Intervention in Lévy Markets Wed, 26 Nov 2014 00:10:03 +0000 This paper considers an exchange rate problem in Lévy markets, where the Central Bank has to intervene. We assume that, in the absence of control, the exchange rate evolves according to Brownian motion with a jump component. The Central Bank is allowed to intervene in order to keep the exchange rate as close as possible to a prespecified target value. The interventions by the Central Bank are associated with costs. We present the situation as an impulse control problem, where the objective of the bank is to minimize the intervention costs. In particular, the paper extends the model by Huang, 2009, to incorporate a jump component. We formulate and prove an optimal verification theorem for the impulse control. We then propose an impulse control and construct a value function and then verify that they solve the quasivariational inequalities. Our results suggest that if the expected number of jumps is high the Central Bank will intervene more frequently and with large intervention amounts hence the intervention costs will be high. Masimba Aspinas Mutakaya, Eriyoti Chikodza, and Edward T. Chiyaka Copyright © 2014 Masimba Aspinas Mutakaya et al. All rights reserved. A Note on the Distribution of Multivariate Brownian Extrema Sun, 16 Nov 2014 11:47:53 +0000 This paper presents a closed-form solution for the joint probability of the endpoints and minimums of a multidimensional Wiener process for some correlation matrices. This is the only explicit expressions found in the literature for this joint probability. The analysis can only be carried out for special correlation structures as it is related to the fundamentals regions of irreducible spherical simplexes generated by reflections and the link to the method of images. This joint distribution can be used in financial mathematics to obtain prices of credit or market related products in high dimension. The solution could be generalized to account for stochastic volatility and other stylized features of the financial markets. Marcos Escobar and Julio Hernandez Copyright © 2014 Marcos Escobar and Julio Hernandez. All rights reserved. A Discrete-Time Queue with Balking, Reneging, and Working Vacations Wed, 29 Oct 2014 00:00:00 +0000 This paper presents an analysis of balking and reneging in finite-buffer discrete-time single server queue with single and multiple working vacations. An arriving customer may balk with a probability or renege after joining according to a geometric distribution. The server works with different service rates rather than completely stopping the service during a vacation period. The service times during a busy period, vacation period, and vacation times are assumed to be geometrically distributed. We find the explicit expressions for the stationary state probabilities. Various system performance measures and a cost model to determine the optimal service rates are presented. Moreover, some queueing models presented in the literature are derived as special cases of our model. Finally, the influence of various parameters on the performance characteristics is shown numerically. Veena Goswami Copyright © 2014 Veena Goswami. All rights reserved. Limit Properties of Transition Functions of Continuous-Time Markov Branching Processes Sun, 19 Oct 2014 00:00:00 +0000 Consider the Markov Branching Process with continuous time. Our focus is on the limit properties of transition functions of this process. Using differential analogue of the Basic Lemma we prove local limit theorems for all cases and observe invariant properties of considering process. Azam A. Imomov Copyright © 2014 Azam A. Imomov. All rights reserved. A Semigroup Expansion for Pricing Barrier Options Sun, 14 Sep 2014 13:08:59 +0000 This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develop a semigroup expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs) arising in barrier option pricing. As an application, we propose a concrete approximation formula under a stochastic volatility model and demonstrate its validity by some numerical experiments. Takashi Kato, Akihiko Takahashi, and Toshihiro Yamada Copyright © 2014 Takashi Kato et al. All rights reserved. Backward Stochastic Differential Equations Approach to Hedging, Option Pricing, and Insurance Problems Thu, 11 Sep 2014 06:36:48 +0000 In the present work we give a self-contained introduction to financial mathematical models characterized by noise of Lévy type in the framework of the backward stochastic differential equations theory. Such techniques will be then used to analyse an innovative model related to insurance and death processes setting. Francesco Cordoni and Luca Di Persio Copyright © 2014 Francesco Cordoni and Luca Di Persio. All rights reserved. Efficient Variable Step Size Approximations for Strong Solutions of Stochastic Differential Equations with Additive Noise and Time Singularity Wed, 02 Jul 2014 10:48:33 +0000 We consider stochastic differential equations with additive noise and conditions on the coefficients in those equations that allow a time singularity in the drift coefficient. Given a maximum step size, , we specify variable (adaptive) step sizes relative to which decrease as the time node points approach the singularity. We use an Euler-type numerical scheme to produce an approximate solution and estimate the error in the approximation. When the solution is restricted to a fixed closed time interval excluding the singularity, we obtain a global pointwise error of order . An order of error for any is obtained when the approximation is run up to a time within of the singularity for an appropriate choice of exponent . We apply this scheme to Brownian bridge, which is defined as the nonanticipating solution of a stochastic differential equation of the type under consideration. In this special case, we show that the global pointwise error is of order , independent of how close to the singularity the approximation is considered. Harry Randolph Hughes and Pathiranage Lochana Siriwardena Copyright © 2014 Harry Randolph Hughes and Pathiranage Lochana Siriwardena. All rights reserved. A Two-Mode Mean-Field Optimal Switching Problem for the Full Balance Sheet Sun, 25 May 2014 09:28:32 +0000 We consider the problem of switching a large number of production lines between two modes, high production and low production. The switching is based on the optimal expected profit and cost yields of the respective production lines and considers both sides of the balance sheet. Furthermore, the production lines are all assumed to be interconnected through a coupling term, which is the average of all optimal expected yields. Intuitively, this means that each individual production line is compared to the average of all its peers which acts as a benchmark. Due to the complexity of the problem, we consider the aggregated optimal expected yields, where the coupling term is approximated with the mean of the optimal expected yields. This turns the problem into a two-mode optimal switching problem of mean-field type, which can be described by a system of Snell envelopes where the obstacles are interconnected and nonlinear. The main result of the paper is a proof of a continuous minimal solution to the system of Snell envelopes, as well as the full characterization of the optimal switching strategy. Boualem Djehiche and Ali Hamdi Copyright © 2014 Boualem Djehiche and Ali Hamdi. All rights reserved. Influence of Gestation Delay and Predator’s Interference in Predator-Prey Interaction under Stochastic Environment Tue, 08 Apr 2014 00:00:00 +0000 Previous experimental and theoretical studies suggest that predator’s interference in predator-prey relationship provides better descriptions of predator’s feeding over a range of predator-prey abundances. Also biological delays and environmental stochasticity play an important role to describe the system and its values. In this present study, I consider a Gaussian white-noise induced stochastic predator-prey model with the Beddington-DeAngelis functional response and gestation delay. Stochastic stability is measured by second order moment terms by calculating the nonequilibrium fluctuation of the nondelayed system and Fourier transform technique depicts the fluctuation of stochastic stability by introducing time lag. Different dynamical behaviors for both situations have been illustrated numerically also. The biological implications of my analytical and numerical findings are discussed critically. Debaldev Jana Copyright © 2014 Debaldev Jana. All rights reserved. with Setup Time, Bernoulli Vacation, Break Down, and Delayed Repair Mon, 31 Mar 2014 07:21:43 +0000 We present a single server in which customers arrive in batches and the server provides service one by one. The server provides two heterogeneous service stages such that service time of both stages is different and mandatory to all arriving customers in such a way that, after the completion of first stage, the second stage should also be provided to the customers. The server may subject to random breakdowns with brake down rate and, after break down, it should be repaired but it has to wait for being repaired and such waiting time is called delay time. Both the delay time and repair time follow exponential distribution. Upon the completion of the second stage service, the server will go for vacation with probability or stay back in the system probability , if any. The vacation time follows general (arbitrary) distribution. Before providing service to a new customer or a batch of customers that joins the system in the renewed busy period, the server enters into a random setup time process such that setup time follows exponential distribution. We discuss the transient behavior and the corresponding steady state results with the performance measures of the model. G. Ayyappan and S. Shyamala Copyright © 2014 G. Ayyappan and S. Shyamala. All rights reserved. From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval Wed, 26 Mar 2014 07:28:39 +0000 Let be a positive integer, a positive constant and be a sequence of independent identically distributed pseudorandom variables. We assume that the ’s take their values in the discrete set and that their common pseudodistribution is characterized by the (positive or negative) real numbers for any . Let us finally introduce the associated pseudorandom walk defined on by and for . In this paper, we exhibit some properties of . In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation . We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess. Aimé Lachal Copyright © 2014 Aimé Lachal. All rights reserved. Diffusion Processes Satisfying a Conservation Law Constraint Tue, 04 Mar 2014 12:08:07 +0000 We investigate coupled stochastic differential equations governing N nonnegative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires a set of fluctuating variables to be nonnegative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the nonnegativity and the unit-sum conservation law constraints are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes. J. Bakosi and J. R. Ristorcelli Copyright © 2014 J. Bakosi and J. R. Ristorcelli. All rights reserved. SPDEs with -Stable Lévy Noise: A Random Field Approach Tue, 04 Feb 2014 12:18:58 +0000 This paper is dedicated to the study of a nonlinear SPDE on a bounded domain in , with zero initial conditions and Dirichlet boundary, driven by an -stable Lévy noise with , , and possibly nonsymmetric tails. To give a meaning to the concept of solution, we develop a theory of stochastic integration with respect to this noise. The idea is to first solve the equation with “truncated” noise (obtained by removing from the jumps which exceed a fixed value ), yielding a solution , and then show that the solutions coincide on the event , for some stopping times converging to infinity. A similar idea was used in the setting of Hilbert-space valued processes. A major step is to show that the stochastic integral with respect to satisfies a th moment inequality. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales. Raluca M. Balan Copyright © 2014 Raluca M. Balan. All rights reserved. The Relationship between the Stochastic Maximum Principle and the Dynamic Programming in Singular Control of Jump Diffusions Thu, 09 Jan 2014 13:11:32 +0000 The main objective of this paper is to explore the relationship between the stochastic maximum principle (SMP in short) and dynamic programming principle (DPP in short), for singular control problems of jump diffusions. First, we establish necessary as well as sufficient conditions for optimality by using the stochastic calculus of jump diffusions and some properties of singular controls. Then, we give, under smoothness conditions, a useful verification theorem and we show that the solution of the adjoint equation coincides with the spatial gradient of the value function, evaluated along the optimal trajectory of the state equation. Finally, using these theoretical results, we solve explicitly an example, on optimal harvesting strategy, for a geometric Brownian motion with jumps. Farid Chighoub and Brahim Mezerdi Copyright © 2014 Farid Chighoub and Brahim Mezerdi. All rights reserved. Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion Tue, 31 Dec 2013 17:44:11 +0000 We study a class of nonlinear stochastic partial differential equations arising in the mathematical modeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model. Mark A. McKibben Copyright © 2013 Mark A. McKibben. All rights reserved. Sharp Large Deviation for the Energy of -Brownian Bridge Sun, 08 Dec 2013 13:12:22 +0000 We study the sharp large deviation for the energy of -Brownian bridge. The full expansion of the tail probability for energy is obtained by the change of measure. Shoujiang Zhao, Qiaojing Liu, Fuxiang Liu, and Hong Yin Copyright © 2013 Shoujiang Zhao et al. All rights reserved. Some Limit Properties of the Harmonic Mean of Transition Probabilities for Markov Chains in Markovian Environments Indexed by Cayley's Trees Thu, 05 Dec 2013 18:20:23 +0000 We prove some limit properties of the harmonic mean of a random transition probability for finite Markov chains indexed by a homogeneous tree in a nonhomogeneous Markovian environment with finite state space. In particular, we extend the method to study the tree-indexed processes in deterministic environments to the case of random enviroments. Huilin Huang Copyright © 2013 Huilin Huang. All rights reserved. Foundations of the Theory of Semilinear Stochastic Partial Differential Equations Wed, 27 Nov 2013 08:38:41 +0000 The goal of this review article is to provide a survey about the foundations of semilinear stochastic partial differential equations. In particular, we provide a detailed study of the concepts of strong, weak, and mild solutions, establish their connections, and review a standard existence and uniqueness result. The proof of the existence result is based on a slightly extended version of the Banach fixed point theorem. Stefan Tappe Copyright © 2013 Stefan Tappe. All rights reserved. Online Stochastic Convergence Analysis of the Kalman Filter Thu, 21 Nov 2013 16:09:40 +0000 This paper presents modifications to the stochastic stability lemma which is then used to estimate the convergence rate and persistent error of the linear Kalman filter online without using knowledge of the true state. Unlike previous uses of the stochastic stability lemma for stability proof, this new convergence analysis technique considers time-varying parameters, which can be calculated online in real-time to monitor the performance of the filter. Through simulation of an example problem, the new method was shown to be effective in determining a bound on the estimation error that closely follows the actual estimation error. Different cases of assumed process and measurement noise covariance matrices were considered in order to study their effects on the convergence and persistent error of the Kalman filter. Matthew B. Rhudy and Yu Gu Copyright © 2013 Matthew B. Rhudy and Yu Gu. All rights reserved. Analysis of Queue-Length Dependent Vacations and P-Limited Service in BMAP/G/1/N Systems: Stationary Distributions and Optimal Control Sun, 10 Nov 2013 09:08:15 +0000 We consider a finite-buffer single server queueing system with queue-length dependent vacations where arrivals occur according to a batch Markovian arrival process (BMAP). The service discipline is P-limited service, also called E-limited with limit variation (ELV) where the server serves until either the system is emptied or a randomly chosen limit of customers has been served. Depending on the number of customers present in the system, the server will monitor his vacation times. Queue-length distributions at various epochs such as before, arrival, arbitrary and after, departure have been obtained. Several other service disciplines like Bernoulli scheduling, nonexhaustive service, and E-limited service can be treated as special cases of the P-limited service. Finally, the total expected cost function per unit time is considered to determine locally optimal values of or a maximum limit of as the number of customers served during a service period at a minimum cost. A. D. Banik Copyright © 2013 A. D. Banik. All rights reserved. The BALM Copula Thu, 26 Sep 2013 16:03:18 +0000 The class of probability distributions possessing the almost-lack-of-memory property appeared about 20 years ago. It reasonably took place in research and modeling, due to its suitability to represent uncertainty in periodic random environment. Multivariate version of the almost-lack-of-memory property is less known, but it is not less interesting. In this paper we give the copula of the bivariate almost-lack-of-memory (BALM) distributions and discuss some of its properties and applications. An example shows how the Marshal-Olkin distribution can be turned into BALM and what is its copula. Boyan Dimitrov and Nikolai Kolev Copyright © 2013 Boyan Dimitrov and Nikolai Kolev. All rights reserved. Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate Sun, 22 Sep 2013 12:10:25 +0000 We investigate a stochastic SIR epidemic model with specific nonlinear incidence rate. The stochastic model is derived from the deterministic epidemic model by introducing random perturbations around the endemic equilibrium state. The effect of random perturbations on the stability behavior of endemic equilibrium is discussed. Finally, numerical simulations are presented to illustrate our theoretical results. Jihad Adnani, Khalid Hattaf, and Noura Yousfi Copyright © 2013 Jihad Adnani et al. All rights reserved.