Department of Mathematics

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Adaptive Stochastic Numerical Scheme in Parallel Random Walk Models for Transport Problems in Shallow Water
(Elsevier, 2009) Charles, Wilson M.; Van den Berg, E.; Lin, Hai X.; Heemink, Arnold W.
This paper deals with the simulation of transport of pollutants in shallow water using random walk models and develops several computation techniques to speed up the numerical integration of the stochastic differential equations (SDEs). This is achieved by using both random time stepping and parallel processing. We start by considering a basic stochastic Euler scheme for integration of the diffusion and drift terms of the SDEs, with a strong order 1 in the strong sense. The errors due to this scheme depend on the location of the pollutant; it is dominated by the diffusion term near boundaries, and by the deterministic drift further away from the boundaries. Using a pair of integration schemes, one of strong order 1.5 near the boundary and one of strong order 2.0 elsewhere, we can estimate the error and approximate an optimal step size for a given error tolerance. The resulting algorithm is developed such that it allows for complete flexibility of the step size, while guaranteeing the correct Brownian behaviour. Modelling pollutants by non-interacting particles enables the use of parallel processing in the simulation. We take advantage of this by implementing the algorithm using the MPI library. The inherent asynchronic nature of the particle simulation, in addition to the parallel processing, makes it difficult to get a coherent picture of the results at any given points. However, by inserting internal synchronisation points in the temporal discretisation, the code allows pollution snapshots and particle counts to be made at times specified by the user.
Application of Coloured Noise as a Driving Force in the Stochastic Differential Equations
(2010) Charles, Wilson M.
In this chapter we explore the application of coloured noise as a driving force to a set of stochastic differential equations(SDEs). These stochastic differential equations are sometimes called Random flight models as in A. W. Heemink (1990). They are used for prediction of the dispersion of pollutants in atmosphere or in shallow waters e.g Lake, Rivers etc. Usually the advection and diffusion of pollutants in shallow waters use the well known partial differential equations called Advection diffusion equations(ADEs)R.W.Barber et al. (2005). These are consistent with the stochastic differential equations which are driven by Wiener processes as in P.E. Kloeden et al. (2003). The stochastic differential equations which are driven by Wiener processes are called particle models. When the Kolmogorov’s forward partial differential equations(Fokker-Planck equation) is interpreted as an advection diffusion equation, the associated set of stochastic differential equations called particle model are derived and are exactly consistent with the advection-diffusion equation as in A. W. Heemink (1990); W. M. Charles et al. (2009). Still, neither the advection-diffusion equation nor the related traditional particle model accurately takes into account the short term spreading behaviour of particles. This is due to the fact that the driving forces are Wiener processes and these have independent increments as in A. W. Heemink (1990); H.B. Fischer et al. (1979). To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this chapter. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection-diffusion equation.
Coloured Noise for Dispersion of Contaminants in Shallow Waters
(Elsevier, 2009) Charles, Wilson M.; Heemink, Arnold W.; Van den Berg, E.
In this article, we explore the application of a set of stochastic differential equations called particle model in simulating the advection and diffusion of pollutants in shallow waters. The Fokker–Planck equation associated with this set of stochastic differential equations is interpreted as an advection–diffusion equation. This enables us to derive an underlying particle model that is exactly consistent with the advection–diffusion equation. Still, neither the advection–diffusion equation nor the related traditional particle model accurately takes into account the short-term spreading behaviour of particles. To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this article. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection–diffusion equation.
Computational Dynamics of Hydromagnetic Stagnation Flow towards a Stretching Sheet
(2010) Makinde, Oluwole D.; Charles, Wilson M.
This paper investigates the hydromagnetic stagnation flow of an incompressible viscous, electrically conducting fluid, towards a stretching sheet in the presence of axially increasing free stream velocity. The newton-Raphson shooting method, along with fourth-order Runge-Kutta integration algorithm, are employed to tackle the third order, nonlinear boundary layer equation governing the problem. The variational iteration method (VIM), coupled with Padé technique is also applied for a reliable treatment of the problem. The study shows that the series solution is obtained without restrictions on the nonlinearity behavio. The solutions are compared with the other available results in the literature, and a good agreement is found.
Controlling Ultimate Ruin Probability by Quota-Share Reinsurance Arrangements
(2013) Kasozi, Juma; Charles, Wilson M.; Mayambala, Fred
A basic insurance model is perturbated by a diffusion. We take this model to represent the wealth dynamics of an insurance company. The model is compounded by another return on investments process of the Black-Scholes type. Both models form the risk process used in this work. Further, to manage her risk levels, the company enters into quota-share reinsurance arrangements with a reinsurer. We derive a second-order Volterra integro-differential equation which we transforminto a linear Volterra integral equation of the second kind. We have solved the equations numerically using the block-by-block method for different retention levels for the chosen parameters. Results show that quota-share reinsurance improves the survival of the insurer
Dividend Maximization in the Cramer-Lundberg Model using Homotopy Analysis Method
(2011) Kasozi, Juma; Mayambala, Fred; Charles, Wilson M.
Problem statement: We used the Homotopy Analysis Method (HAM) to numerically compute the value function of the dividend payment in the basic insurance process. Approach: The process is a constant income stream from premiums which is subtracted a claim process of the Poisson type. Further, an allowance for payment of dividends to share holders was incorporated. Results: The case when the claims are exponential has an analytical solution. The HAM was then applied to the resulting Hamilton-Jacobi-Bellman equation and the numerical results obtained were compared to the theoretical results in order to check the validity of the method. Conclusion: The HAM was then applied to the model to check for other claim size distributions. The results obtained are very encouraging.
Dividend Payouts in a Perturbed Risk Process Compounded By Investments of the Black-Scholes Type
(2013) Kasozi, Juma; Charles, Wilson M.
This work addresses the issue of dividend payouts of an insurer whose portfolio is exposed to insurance risk. The insurance risk arises from the perturbed classical surplus process commonly known as the Cramér-Lundberg model in the insurance literature. To enhance her financial base, the insurer invests into assets whose price dynamics are governed by a Black-Scholes model. We derive a linear Volterra integral equation of the second kind and solve the equations for each chosen barrier, thus generating corresponding dividend value functions. We have obtained the optimal barrier that maximises the expected discounted dividend payouts prior to ruin.
A Mathematical Approach to a Stocks Portfolio Selection: The Case of Uganda Securities Exchange (USE)
(Scientific Research, 2013) Mayanja, Fredrick; Mataramvura, Sure; Charles, Wilson M.
In this paper, we present the problem of portfolio optimization under investment. This area of investment is traced with works of Professor Markowitz way back in 1952. First, we determine the probability distribution of the Uganda Securities Exchange (USE) stocks returns. Secondly, we develop unrestricted portfolio optimization model based on the classical Modern Portfolio Optimization (MPT) model, and then we incorporate certain restrictions typical of the USE trading or investment environment and hence, develop the modified restricted model. Thirdly, we explore the possibility of diversification under a portfolio of averagely correlated assets. Determination of the model parameters and model development is all done using Excel spreadsheets. We explicitly go through the mathematics of the solution methods for both models. Validation of the models is done using the USE stocks daily trading data, in which case we use a random sample of 6 stocks out of the 13 stocks listed at the USE. To start with, we prove that USE stocks log returns are normally distributed. Data analysis results and the frontier curves show that our modified (restricted) model is valid as the solutions are all consistent with the theoretical foundations of the classical MPT-model but inferior to the unrestricted model. To make the model more useful, accurate and easy to apply and robust, we programme the model using Visual Basic for Applications (VBA). We therefore recommend that before applying investment models such as the MPT, model modifications must be made so as to adapt them to particular investment environments. Moreover, to make them useful so as to serve the intended purpose, the models should be programmed so as to make implementation less cumbersome.
Modelling Stock Returns Volatility on Uganda Securities Exchange
(2014) Weke, Patrick G. O.; Namugaya, Jalira; Charles, Wilson M.
Stock returns volatility of daily closing prices of the Uganda Securities Exchange(USE) all share index over a period of 04/01/2005 to 18/12/2013 is Modelled. We employ different univariate Generalised Autoregressive Conditional Heteroscedastic(GARCH) models; both symmetric and asymmetric. The models include; GARCH(1,1), GARCH-M, EGARCH(1,1) and TGARCH(1,1). Quasi Maximum Likelihood(QML) method was used to estimate the models and then the best performing model obtained using two model selection criteria; Akaike Information criterion(AIC) and Bayesian Information criterion(BIC). Overall, the GARCH(1; 1) model outperformed the other competing models. This result is analogous with other studies, that GARCH(1; 1) is best.
Parallel and Distributed Simulation of Sediment Dynamics in Shallow Water Using Particle Decomposition Approach
(Elsevier, 2008) Charles, Wilson M.; Van den Berg, E.; Lin, Hai X.; Heemink, Arnold W.; Verlaan, Martin
This paper describes the parallel simulation of sediment dynamics in shallow water. By using a Lagrangian model, the problem is transformed to one in which a large number of independent particles must be tracked. This results in a technique that can be parallelised with high efficiency. We have developed a sediment transport model using three different sediment suspension methods. The first method uses a modified mean for the Poisson distribution function to determine the expected number of the suspended particles in each particular grid cell of the domain over all available processors. The second method determines the number of particles to suspend with the aid of the Poisson distribution function only in those grid cells which are assigned to that processor. The third method is based on the technique of using a synchronised pseudo-random-number generator to generate identical numbers of suspended particles in all valid grid cells for each processor. Parallel simulation experiments are performed in order to investigate the efficiency of these three methods. Also the parallel performance of the implementations is analysed. We conclude that the second method is the best method on distributed computing systems (e.g., a Beowulf cluster), whereas the third maintains the best load distribution.
A Particle Model for Simulation of Sediment Transport in Coastal Waters
(2004) Charles, Wilson M.; Heemink, Arnold W.; Verlaan, Martin
Stochastic Particle Models for Transport Problems in Coastal Waters
(2005) Charles, Wilson M.; Heemink, Arnold W.; Van den Berg, E.
In this paper transport processes in coastal waters are described by stochastic differential equations (SDEs). These SDEs are also called particle models (PMs). By interpreting a Fokker-Planck equation associated with the SDE as an advection diffusion equation (ADE), it is possible to derive the underlying PM which is exactly consistent with the ADE. Both the ADE and the related classical PM do not take into account accurately the short term spreading behaviour of particles. In the PM this shortcoming is due to the driving noise in the SDE which is modelled as a Brownian motion and therefore has independent increments. To improve the behaviour of the model shortly after the release of pollution we develop an improved PM forced by a coloured noise process representing the short-term correlated turbulent velocity of the particles. This way a more accurate and detailed short-term initial spreading behaviour of particles is achieved. For long-term simulations both the improved and classical PMs are consistent with the ADE. However, the improved PM is relatively easier to handle numerically than a corresponding ADE. In this paper both models are applied to a real life pollution problem in the Dutch coastal waters.
Unsteady Convection with Chemical Reaction and Radiative Heat Transfer past a Flat Porous Plate Moving Through a Binary Mixture
(Springer, 2011) Makinde, Oluwole D.; Olanrewaju, Philip O.; Charles, Wilson M.
In this paper, the problem of unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture in an optically thin environment is presented. The dimensionless governing equations for this investigation are solved numerically by the fourth-order Runge–Kutta integration scheme along with shooting technique. Numerical data for the local skin-friction coefficient, the local Nusselt number and the local Sherwood number have been tabulated for various values of parametric conditions. Graphical results for velocity, temperature and concentration profiles based on the numerical solutions are presented and discussed.
Variable Time Stepping in Parallel Particle Models for Transport Problems in Shallow Waters
(2006) Charles, Wilson M.; Van den Berg, E.; Lin, Hai X.; Heemink, Arnold W.
Stochastic differential equations (SDEs) are stochastic in nature. The SDEs under consideration are often called particle models (PMs). PMs in this article model the simulation of transport of pollutants in shallow waters. The main focus is the derivation and efficient implementation of an adaptive scheme for numerical integration of the SDEs in this article. The error determination at each integration time step near the boundary where the diffusion is dominant is done by a pair of numerical schemes with strong order 1 of convergence and that of strong order 1.5. When the deterministic is dominantwe use the aforementioned order 1 scheme and another scheme of strong order 2. An optimal stepsize for a given error tolerance is estimated. Moreover, the algorithm is developed in such a way that it allows for a completely flexible change of the time stepsize while guaranteeing correct Brownian paths. The software implementation uses the MPI library and allows for parallel processing. By making use of internal synchronisation points it allows for snapshots and particle counts to be made at given times, despite the inherent asynchronicity of the particles with regard to time.

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