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.
Stochastic differential equations play a prominent role in many application areas including finance, biology and epidemiology. By incorporating random elements to ordinary differential equation system, a system of stochastic differential equations (SDEs) arises. This leads to a more complex insight of the physical phenomena than their deterministic counterpart. However, most of the SDEs do not have an analytical solution where numerical method is the best way to resolve this problem. Recently, much work had been done in applying numerical methods for solving SDEs. A very general class of Stochastic Runge-Kutta, (SRK) had been studied and 2-stage SRK with order convergence of 1.0 and 4-stage SRK with order convergence of 1.5 were discussed. In this study, we compared the performance of Euler-Maruyama, 2-stage SRK and 4-stage SRK in approximating the strong solutions of stochastic logistic model which describe the cell growth of C. acetobutylicum P262. The MS-stability functions of these schemes were calculated and regions of MS-stability are given. We also perform the comparison for the performance of these methods based on their global errors.
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.