International Journal of Stochastic Analysis
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The latest articles from Hindawi Publishing Corporation
© 2015 , Hindawi Publishing Corporation . All rights reserved.

A Stochastic Flows Approach for Asset Allocation with Hidden Economic Environment
Tue, 27 Jan 2015 11:15:54 +0000
http://www.hindawi.com/journals/ijsa/2015/462524/
An optimal asset allocation problem for a quite general class of utility functions is discussed in a simple twostate Markovian regimeswitching 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.

YamadaWatanabe Results for Stochastic Differential Equations with Jumps
Thu, 01 Jan 2015 09:34:03 +0000
http://www.hindawi.com/journals/ijsa/2015/460472/
Recently, Kurtz (2007, 2014) obtained a general version of the YamadaWatanabe 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 TwoDimensional Autoregressive Fields from Noisy Observations
Thu, 25 Dec 2014 00:10:03 +0000
http://www.hindawi.com/journals/ijsa/2014/247274/
This paper deals with the problem of twodimensional autoregressive (AR) estimation from noisy observations. The YuleWalker 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
http://www.hindawi.com/journals/ijsa/2014/534864/
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
http://www.hindawi.com/journals/ijsa/2014/628321/
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
http://www.hindawi.com/journals/ijsa/2014/746815/
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
http://www.hindawi.com/journals/ijsa/2014/575270/
This paper presents a closedform 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 DiscreteTime Queue with Balking, Reneging, and Working Vacations
Wed, 29 Oct 2014 00:00:00 +0000
http://www.hindawi.com/journals/ijsa/2014/358529/
This paper presents an analysis of balking and reneging in finitebuffer discretetime 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 ContinuousTime Markov Branching Processes
Sun, 19 Oct 2014 00:00:00 +0000
http://www.hindawi.com/journals/ijsa/2014/409345/
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
http://www.hindawi.com/journals/ijsa/2014/268086/
This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develop a semigroup expansion scheme for the CauchyDirichlet problem in the secondorder 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
http://www.hindawi.com/journals/ijsa/2014/152389/
In the present work we give a selfcontained 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
http://www.hindawi.com/journals/ijsa/2014/852962/
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 Eulertype 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 TwoMode MeanField Optimal Switching Problem for the Full Balance Sheet
Sun, 25 May 2014 09:28:32 +0000
http://www.hindawi.com/journals/ijsa/2014/159519/
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 twomode optimal switching problem of meanfield 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 PredatorPrey Interaction under Stochastic Environment
Tue, 08 Apr 2014 00:00:00 +0000
http://www.hindawi.com/journals/ijsa/2014/501836/
Previous experimental and theoretical studies suggest that predator’s interference in predatorprey relationship provides better descriptions of predator’s feeding over a range of predatorprey 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 whitenoise induced stochastic predatorprey model with the BeddingtonDeAngelis 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
http://www.hindawi.com/journals/ijsa/2014/892867/
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 PseudoBrownian Motion: First Exit Time from a OneSided or a TwoSided Interval
Wed, 26 Mar 2014 07:28:39 +0000
http://www.hindawi.com/journals/ijsa/2014/520136/
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 pseudoBrownian motion driven by the highorder heattype equation . We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudoBrownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudoBrownian 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
http://www.hindawi.com/journals/ijsa/2014/603692/
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 unitsum conservation law constraints are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the FokkerPlanck 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 WrightFisher, 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
http://www.hindawi.com/journals/ijsa/2014/793275/
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 Hilbertspace 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 BurkholderDavisGundy 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
http://www.hindawi.com/journals/ijsa/2014/201491/
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.

MeasureDependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion
Tue, 31 Dec 2013 17:44:11 +0000
http://www.hindawi.com/journals/ijsa/2013/868301/
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 functionaltype and those dependent upon a family of probability measures are incorporated into the initialboundary 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 secondorder 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 fixedpoint 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
http://www.hindawi.com/journals/ijsa/2013/952628/
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
http://www.hindawi.com/journals/ijsa/2013/961571/
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 treeindexed 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
http://www.hindawi.com/journals/ijsa/2013/798549/
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
http://www.hindawi.com/journals/ijsa/2013/240295/
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 timevarying parameters, which can be calculated online in realtime 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 QueueLength Dependent Vacations and PLimited Service in BMAP/G/1/N Systems: Stationary Distributions and Optimal Control
Sun, 10 Nov 2013 09:08:15 +0000
http://www.hindawi.com/journals/ijsa/2013/196372/
We consider a finitebuffer single server queueing system with queuelength dependent vacations where arrivals occur according to a batch Markovian arrival process (BMAP). The service discipline is Plimited service, also called Elimited 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. Queuelength 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 Elimited service can be treated as special cases of the Plimited 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
http://www.hindawi.com/journals/ijsa/2013/652364/
The class of probability distributions possessing the almostlackofmemory
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 almostlackofmemory property is
less known, but it is not less interesting. In this paper we give the copula of
the bivariate almostlackofmemory (BALM) distributions and discuss some
of its properties and applications. An example shows how the MarshalOlkin
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
http://www.hindawi.com/journals/ijsa/2013/431257/
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.

The LMI Approach for Stabilizing of Linear Stochastic Systems
Thu, 29 Aug 2013 08:56:12 +0000
http://www.hindawi.com/journals/ijsa/2013/281473/
Stochastic linear systems subjected both to Markov jumps and to multiplicative white noise are considered. In order to stabilize such type of stochastic systems, the socalled set of generalized discretetime algebraic Riccati equations has to be solved. The LMI approach for computing the stabilizing symmetric solution (which is in fact the equilibrium point) of this system is studied. We construct a new modification of the standard LMI approach, and we show how to apply the new modification. Computer realizations of all modifications are compared. Numerical experiments are given where the LMI modifications are numerically compared. Based on the experiments the main conclusion is that the new LMI modification is faster than the standard LMI approach.
Ivan Ivanov
Copyright © 2013 Ivan Ivanov. All rights reserved.

A Stochastic Diffusion Process for the Dirichlet Distribution
Wed, 10 Apr 2013 17:23:06 +0000
http://www.hindawi.com/journals/ijsa/2013/842981/
The method of potential solutions of FokkerPlanck equations is used to develop a transport equation for the
joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unitsum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate WrightFisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution.
J. Bakosi and J. R. Ristorcelli
Copyright © 2013 J. Bakosi and J. R. Ristorcelli. All rights reserved.

The Itô Integral with respect to an Infinite Dimensional Lévy Process: A Series Approach
Thu, 04 Apr 2013 11:15:21 +0000
http://www.hindawi.com/journals/ijsa/2013/703769/
We present an alternative construction of the infinite dimensional Itô integral with respect to a Hilbert space valued Lévy process. This approach is based on the wellknown theory of
realvalued stochastic integration, and the respective Itô integral is given by a series of Itô integrals with respect to standard Lévy processes. We also prove that this stochastic integral coincides with the Itô integral that has been developed in the literature.
Stefan Tappe
Copyright © 2013 Stefan Tappe. All rights reserved.