International Journal of Stochastic Analysis
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Malliavin Differentiability of Solutions of SPDEs with Lévy White Noise
Sun, 12 Mar 2017 00:00:00 +0000
http://www.hindawi.com/journals/ijsa/2017/9693153/
We consider a stochastic partial differential equation (SPDE) driven by a Lévy white noise, with Lipschitz multiplicative term . We prove that, under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the Lévy white noise. If is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.
Raluca M. Balan and Cheikh B. Ndongo
Copyright © 2017 Raluca M. Balan and Cheikh B. Ndongo. All rights reserved.

Semigroup Solution of PathDependent SecondOrder Parabolic Partial Differential Equations
Mon, 27 Feb 2017 00:00:00 +0000
http://www.hindawi.com/journals/ijsa/2017/2876961/
We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the secondorder pathdependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the pathdependent heat equation is equal to onehalf times the secondorder Malliavin derivative evaluated along the frozen path.
Sixian Jin and Henry Schellhorn
Copyright © 2017 Sixian Jin and Henry Schellhorn. All rights reserved.

Global Stability of Nonlinear Stochastic SEI Epidemic Model with Fluctuations in Transmission Rate of Disease
Mon, 23 Jan 2017 00:00:00 +0000
http://www.hindawi.com/journals/ijsa/2017/6313620/
We derive and analyze the dynamic of a stochastic SEI epidemic model for disease spread. Fluctuations in the transmission rate of the disease bring about stochasticity in model. We discuss the asymptotic stability of the infectionfree equilibrium by first deriving the closed form deterministic () and stochastic () basic reproductive number. Contrary to some author’s remark that different diffusion rates have no effect on the stability of the diseasefree equilibrium, we showed that even if no epidemic invasion occurs with respect to the deterministic version of the SEI model (i.e., ), epidemic can still grow initially (if ) because of the presence of noise in the stochastic version of the model. That is, diffusion rates can have effect on the stability by causing a transient epidemic advance. A threshold criterion for epidemic invasion was derived in the presence of external noise.
Olusegun Michael Otunuga
Copyright © 2017 Olusegun Michael Otunuga. All rights reserved.

Generalisation of Hajek’s Stochastic Comparison Results to Stochastic Sums
Mon, 05 Sep 2016 16:18:57 +0000
http://www.hindawi.com/journals/ijsa/2016/1018509/
Hajek’s univariate stochastic comparison result is generalised to multivariate stochastic sum processes with univariate convex data functions and for univariate monotonic nondecreasing convex data functions for processes with and without drift, respectively. As a consequence strategies for a class of multivariate optimal control problems can be determined by maximizing variance. An example is passport options written on multivariate traded accounts. The argument describes a narrow path between impossibilities of generalisations to jump processes or impossibilities of more general data functions.
Jörg Kampen
Copyright © 2016 Jörg Kampen. All rights reserved.

Asymptotic Time Averages and Frequency Distributions
Mon, 05 Sep 2016 06:27:26 +0000
http://www.hindawi.com/journals/ijsa/2016/2741214/
Consider an arbitrary nonnegative deterministic process (in a stochastic setting is a fixed realization, i.e., samplepath of the underlying stochastic process) with state space . Using a samplepath approach, we give necessary and sufficient conditions for the longrun time average of a measurable function of process to be equal to the expectation taken with respect to the same measurable function of its longrun frequency distribution. The results are further extended to allow unrestricted parameter (time) space. Examples are provided to show that our condition is not superfluous and that it is weaker than uniform integrability. The case of discretetime processes is also considered. The relationship to previously known sufficient conditions, usually given in stochastic settings, will also be discussed. Our approach is applied to regenerative processes and an extension of a wellknown result is given. For researchers interested in samplepath analysis, our results will give them the choice to work with the time average of a process or its frequency distribution function and go back and forth between the two under a mild condition.
Muhammad ElTaha
Copyright © 2016 Muhammad ElTaha. All rights reserved.

A BSDE with Delayed Generator Approach to Pricing under Counterparty Risk and Collateralization
Tue, 02 Aug 2016 06:36:12 +0000
http://www.hindawi.com/journals/ijsa/2016/1059303/
We consider a nonlinear pricing problem that takes into account credit risk and funding issues. The aforementioned problem is formulated as a stochastic forwardbackward system with delay, both in the forward and in the backward component, whose solution is characterized in terms of viscosity solution to a suitable type of pathdependent PDE.
Francesco Cordoni and Luca Di Persio
Copyright © 2016 Francesco Cordoni and Luca Di Persio. All rights reserved.

Stochastic Analysis of Gaussian Processes via Fredholm Representation
Sun, 31 Jul 2016 13:12:25 +0000
http://www.hindawi.com/journals/ijsa/2016/8694365/
We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations, and maximum likelihood estimations.
Tommi Sottinen and Lauri Viitasaari
Copyright © 2016 Tommi Sottinen and Lauri Viitasaari. All rights reserved.

Optimal Bounds for the Variance of SelfIntersection Local Times
Wed, 20 Jul 2016 08:55:32 +0000
http://www.hindawi.com/journals/ijsa/2016/5370627/
For a valued random walk , let be its local time at the site . For , define the fold selfintersection local time as . Also let be the corresponding quantities for the simple random walk in . Without imposing any moment conditions, we show that the variance of the selfintersection local time of any genuinely dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, . In particular, for any genuinely dimensional random walk, with , we have . On the other hand, in dimensions we show that if the behaviour resembles that of simple random walk, in the sense that , then the increments of the random walk must have zero mean and finite second moment.
George Deligiannidis and Sergey Utev
Copyright © 2016 George Deligiannidis and Sergey Utev. All rights reserved.

Analysis of a Priority Queue with PhaseType Service and Failures
Sun, 17 Jul 2016 12:25:23 +0000
http://www.hindawi.com/journals/ijsa/2016/9152701/
We consider a single server queue with two types of customers. We propose a discipline of flexible priority in access that combines the features of randomization and the threshold type control. We introduce a new class of distributions, phasetype with failures () distribution, that generalizes the wellknown phasetype () distribution to the case when failures can occur during service of a customer. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of type with the parameters depending on the type of a customer. Customers of both types can be impatient. Behavior of the system is described by the multidimensional Markov chain. Problem of existence and computation of the stationary distribution of this Markov chain is discussed in brief as well as the problem of computation of the key performance measures of the system. Numerical examples are presented that give some insight into behavior of the system performance measures under different values of the parameters defining the strategy of customers access to service.
Alexander Dudin and Sergei Dudin
Copyright © 2016 Alexander Dudin and Sergei Dudin. All rights reserved.

Multiserver Queue with Guard Channel for Priority and Retrial Customers
Thu, 03 Mar 2016 15:16:49 +0000
http://www.hindawi.com/journals/ijsa/2016/7168359/
This paper considers a retrial queueing model where a group of guard channels is reserved for priority and retrial customers. Priority and normal customers arrive at the system according to two distinct Poisson processes. Priority customers are accepted if there is an idle channel upon arrival while normal customers are accepted if and only if the number of idle channels is larger than the number of guard channels. Blocked customers (priority or normal) join a virtual orbit and repeat their attempts in a later time. Customers from the orbit (retrial customers) are accepted if there is an idle channel available upon arrival. We formulate the queueing system using a level dependent quasibirthanddeath (QBD) process. We obtain a Taylor series expansion for the nonzero elements of the rate matrices of the level dependent QBD process. Using the expansion results, we obtain an asymptotic upper bound for the joint stationary distribution of the number of busy channels and that of customers in the orbit. Furthermore, we develop an efficient numerical algorithm to calculate the joint stationary distribution.
Kazuki Kajiwara and Tuan PhungDuc
Copyright © 2016 Kazuki Kajiwara and Tuan PhungDuc. All rights reserved.

Large Deviation Analysis of a Droplet Model Having a Poisson Equilibrium Distribution
Wed, 07 Oct 2015 07:04:19 +0000
http://www.hindawi.com/journals/ijsa/2015/287450/
In this paper we use large deviation theory to determine the equilibrium distribution of a basic droplet model that underlies a number of important models in material science and statistical mechanics. Given and , distinguishable particles are placed, each with equal probability , onto the sites of a lattice, where equals . We focus on configurations for which each site is occupied by a minimum of particles. The main result is the large deviation principle (LDP), in the limit and with , for a sequence of random, numberdensity measures, which are the empirical measures of dependent random variables that count the droplet sizes. The rate function in the LDP is the relative entropy , where is a possible asymptotic configuration of the numberdensity measures and is a Poisson distribution with mean , restricted to the set of positive integers satisfying . This LDP implies that is the equilibrium distribution of the numberdensity measures, which in turn implies that is the equilibrium distribution of the random variables that count the droplet sizes.
Richard S. Ellis and Shlomo Ta’asan
Copyright © 2015 Richard S. Ellis and Shlomo Ta’asan. All rights reserved.

On Continuous Selection Sets of NonLipschitzian Quantum Stochastic Evolution Inclusions
Tue, 28 Jul 2015 08:43:57 +0000
http://www.hindawi.com/journals/ijsa/2015/834194/
We establish existence of a continuous selection of multifunctions associated with quantum stochastic evolution inclusions under a general Lipschitz condition. The coefficients here are multifunctions but not necessarily Lipschitz.
Sheila Bishop
Copyright © 2015 Sheila Bishop. All rights reserved.

Pricing FX Options in the Heston/CIR JumpDiffusion Model with LogNormal and LogUniform Jump Amplitudes
Sun, 26 Jul 2015 06:17:28 +0000
http://www.hindawi.com/journals/ijsa/2015/258217/
We examine foreign exchange options in the jumpdiffusion version of the Heston stochastic volatility
model for the exchange rate with lognormal jump amplitudes and the volatility model with loguniformly distributed jump amplitudes. We assume that the domestic and foreign stochastic interest rates are governed by the CIR dynamics. The instantaneous volatility is correlated with the dynamics of the exchange rate return, whereas the domestic and foreign shortterm rates are assumed to be independent of the dynamics of the exchange rate and its volatility. The main result furnishes a semianalytical formula for the price of the foreign exchange European call option.
Rehez Ahlip and Ante Prodan
Copyright © 2015 Rehez Ahlip and Ante Prodan. All rights reserved.

A Generic Decomposition Formula for Pricing Vanilla Options under Stochastic Volatility Models
Wed, 17 Jun 2015 08:39:25 +0000
http://www.hindawi.com/journals/ijsa/2015/103647/
We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model, extending a previous decomposition formula for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used for this purpose are nonanticipative. In particular, we also see that equivalent results can be obtained by using Functional Itô Calculus. Using the same generalizing ideas, we also extend to nonexponential models the alternative call option price decomposition formula written in terms of the Malliavin derivative of the volatility process. Finally, we give a general expression for the derivative of the implied volatility under both the anticipative and the nonanticipative cases.
Raúl Merino and Josep Vives
Copyright © 2015 Raúl Merino and Josep Vives. All rights reserved.

Stochastic Nonlinear Equations Describing the Mesoscopic VoltageGated Ion Channels
Sun, 05 Apr 2015 16:01:57 +0000
http://www.hindawi.com/journals/ijsa/2015/658342/
We propose a stochastic nonlinear system to model the gating activity coupled with the membrane potential for a typical neuron. It distinguishes two different levels: a macroscopic one, for the membrane potential, and a mesoscopic one, for the gating process through the movement of its voltage sensors. Such a nonlinear system can be handled to form a HodgkinHuxleylike model, which links those two levels unlike the original deterministic HodgkinHuxley model which is positioned at a macroscopic scale only. Also, we show that an interacting particle system can be used to approximate our model, which is an approximation technique similar to the jump Markov processes, used to approximate the original HodgkinHuxley model.
Mauricio Tejo
Copyright © 2015 Mauricio Tejo. 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
http://www.hindawi.com/journals/ijsa/2015/958730/
This paper considers an infinitebuffer queuing system with birthdeath 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 selfsimilarity 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
http://www.hindawi.com/journals/ijsa/2015/231214/
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
http://www.hindawi.com/journals/ijsa/2015/863165/
We propose a feasible and constructive methodology which allows us to compute pure hedging strategies with respect to arbitrary squareintegrable 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 hedgingtype strategies for fully pathdependent 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 FöllmerSchweizer decompositions, locally risk minimizing, and mean variance hedging strategies for vanilla and pathdependent 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
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.