Some numerical methods for solving stochastic impulse control in natural gas storage facilities

The valuation of gas storage facilities is characterized as a stochastic impulse control problem with finite horizon resulting in Hamilton-Jacobi-Bellman (HJB) equations for the value function. In this context the two catagories of solving schemes for optimal switching are discussed in a stochastic control framework. We reviewed some numerical methods which include approaches related to partial differential equations (PDEs), Markov chain approximation, nonparametric regression, quantization method and some practitioners’ methods. This paper considers optimal switching problem arising in valuation of gas storage contracts for leasing the storage facilities, and investigates the recent developments as well as their advantages and disadvantages of each scheme based on dynamic programming principle (DPP

Parameter estimation in stochastic differential equations

Financial processes as processes in nature, are subject to stochastic fluctuations. Stochastic differential equations turn out to be an advantageous representation of such noisy, real-world problems, and together with their identification, they play an important role in the sectors of finance, but also in physics and biotechnology. These equations, however, are often hard to represent and to resolve. Thus we express them in a simplified manner of approximation by discretization and additive models based on splines. This defines a trilevel problem consisting of an optimization and a representation problem (portfolio optimization), and a parameter estimation (Weber et al. Financial Regression and Organization. In: Special Issue on Optimization in Finance, DCDIS-B, 2010). Two types of parameters dependency, linear and nonlinear, are considered by constructing a penalized residual sum of squares and investigating the related Tikhonov regularization problem for the first one. In the nonlinear case Gauss–Newton’s method and Levenberg–Marquardt’s method are employed in determining the iteration steps. Both cases are treated using continuous optimization techniques by the elegant framework of conic quadratic programming. These convex problems are well-structured, hence, allowing the use of the efficient interior point methods. Furthermore, we present nonparametric and related methods, and introduce into research done at the moment in our research groups which ends with a conclusion

Early Detection of Depression using Screening Tools and Electroencephalogram (EEG) Measurements

Mental illness refers to mental disorder that causes mild to severe disturbances in thoughts and behavior, resulting in the inability to cope with ordinary demands and daily life routines.  Among the wide spectrum of mental health conditions, depression was found to have the highest prevalence globally.  The current practice of depression detection depends on screening tools which are either physician-administered or self-administered and behavioral observations. Both methods are widely used but relies on subjective interpretation.  Thus, the method may lack the reliability and accuracy of detecting depression which may lead to incorrect diagnosis of medications. Many mental illness patients still suffer from their mental conditions as well as the side effects of medications during the treatment process.  Physiological measurement methods, such as electroencephalogram (EEG) measurements were found to be able to evaluate the condition of patients suffering from mental illness through recorded brain waves.  The paper sought to investigate the relationship between brainwaves and depression and to propose an alternative method to further evaluate the condition of patients who suffer from depression in a more accurate and effective way. Subjects underwent an initial screening to evaluate and classify their mental health condition based on the score obtained from the screening tools - Patient Health Questionnaire (PHQ)-9.  Subjects also underwent an EEG experiment with a given video-watching stimulus to evaluate their brain activity.  The study proves that increased depression severity subjects have shown distinctly lower Alpha waves at almost all electrode channels and comparatively lower Beta and Theta in the frontal region. This shows that there is a relationship between mental illness and brainwaves activities.  Despite that, the findings from the study showed that there were no strong association found between mean EEG amplitudes and the score from PHQ-9 which suggest that the current practice which only depends on subjective methods may not be sufficient for depression diagnosis. Using a more objective method showed that there are strong associations found between mean EEG amplitudes and the proposed EEG scoring especially in Alpha waves.  There were also strong association between the EEG scoring and the EEG amplitudes at all electrode channels in Alpha waves.  The use of EEG measurement may be considered as an effective and more accurate method to support the current practice in detecting early signs of mental illness in patients, specifically depression

Multi-Population mortality model: A practical approach

The growing number of multi-population mortality models in the recent years signifies the mortality improvement in developed countries. In this case, there exists a narrowing gap of sex-differential in life expectancy between populations; hence multi-population mortality models are designed to assimilate the correlation between populations. The present study considers two extensions of the single-population Lee-Carter model, namely the independent model and augmented common factor model. The independent model incorporates the information between male and female separately whereas the augmented common factor model incorporates the information between male and female simultaneously. The methods are demonstrated in two perspectives: First is by applying them to Malaysian mortality data and second is by comparing the significance of the methods to the annuity pricing. The performances of the two methods are then compared in which has been found that the augmented common factor model is more superior in terms of historical fit, forecast performance, and annuity pricing

Comparison of stochastic mortality model for wider age range

The incorporation of non-linear pattern of early ages has opened new research directions on improving the existing stochastic mortality model structure. Several authors have outlined the importance of encompassing the full age range in dealing with longevity risk exposure by not to ignore the dependence between young and old age. In this study, we consider the two extensions of Cairns, Blake and Dowd model that incorporate the irregularity profile seen at the mortality of lower ages which are Plat and O’Hare and Li. The models’ performances in terms of in-sample fitting and out-sample forecasts were examined and compared. The results indicated that O’Hare and Li model performs better as compared to Plat model

Parameter estimation of Stochastic Logistic Model : Levenberg-Marquardt Method

In this paper, we estimate the drift and diffusion parameters of the stochas- tic logisticmodels for the growth of Clostridium Acetobutylicum P262 using Levenberg- Marquardt optimization method of non linear least squares. The parameters are esti- mated for five different substrates. The solution of the deterministic models has been approximated using Fourth Order Runge-Kutta and for the solution of the stochastic differential equations, Milstein numerical scheme has been used. Small values of Mean Square Errors (MSE) of stochastic models indicate good fits. Therefore the use of stochastic models are shown to be appropriate in modelling cell growth of Clostridium Acetobutylicum P26

Parameter estimation of stochastic differential equation

Non-parametric modeling is a method which relies heavily on data and motivated by the smoothness properties in estimating a function which involves spline and non-spline approaches. Spline approach consists of regression spline and smoothing spline. Regression spline with Bayesian approach is considered in the first step of a two-step method in estimating the structural parameters for stochastic differential equation (SDE). The selection of knot and order of spline can be done heuristically based on the scatter plot. To overcome the subjective and tedious process of selecting the optimal knot and order of spline, an algorithm was proposed. A single optimal knot is selected out of all the points with exception of the first and the last data which gives the least value of Generalized Cross Validation (GCV) for each order of spline. The use is illustrated using observed data of opening share prices of Petronas Gas Bhd. The results showed that the Mean Square Errors (MSE) for stochastic model with parameters estimated using optimal knot for 1,000, 5,000 and 10,000 runs of Brownian motions are smaller than the SDE models with estimated parameters using knot selected heuristically. This verified the viability of the two-step method in the estimation of the drift and diffusion parameters of SDE with an improvement of a single knot selection

Modelling the Cervical Cancer Growth Process by Stochastic Delay Differential Equations

In this paper, the uncontrolled environmental factors are perturbed into the growth rate deceleration factor of the Gompertzian deterministic model. The growth process under Gompertz’s law is considered, thus lead to stochastic differential equations of Gompertzian with time delay. The Gompertzian deterministic model has proven to fit well with the clinical data of cancerous growth, however the performance of stochastic model towards clinical data is yet to be confirmed. The prediction quality of stochastic model is evaluated by comparing the simulated results with the clinical data of cervical cancer growth. The parameter estimation of stochastic models is computed by using simulated maximum likelihood method. 4-stage stochastic Runge-Kutta is applied to simulate the solution of stochastic model. Low values of root mean-square error (RMSE) of Gompertzian model with random effect indicate good fits

Stabilisation in distribution of hybrid ordinary differential equations by periodic noise

Many systems in the real world are periodic due to periodic phenomena in nature. Periodic hybrid stochastic differential equations are often used to model them. In many situations, it is inappropriate to study whether the solutions of periodic hybrid stochastic differential equations will converge to an equilibrium state (say, 0 or the trivial solution) but more appropriate to discuss whether the probability distributions of the solutions will converge to a stationary distribution, known as stability in distribution. This paper aims to determine whether or not a periodic stochastic state feedback control can make a given nonlinear periodic hybrid differential equation, which is not stable in distribution, to become stable in distribution. We will refer to this problem as stabilisation in distribution by periodic noise. There is little known on this problem so far. This paper initiates the study in this direction

Modelling the effect of hydraulic conductivity on one dimensional contaminant transport in RBF system

Riverbank filtration (RBF) system is a surface water technology that is based on the natural treatment of filtration instead of the use of chemicals, to pretreat surface water and provides public water supplies. Hydraulic conductivity value is one of the significant factors affecting the water quality in RBF systems. In this article, an analytical modelling is developed to investigate the effect of this parameter on one dimensional contaminant transport in RBF system. The model is solved by using Green’s function approach. The model is applied for the first RBF system conducted in Malaysia. Generally, the results show that increasing the hydraulic conductivity value lead to an increase in contaminant concentration in pumping well area