In this work, a one-step method of Euler-Maruyama (EMM) type has been developed for the solution of general first order stochastic differential equations (SDEs) using It么 integral equation as basis tool. The effect of varying stepsizes on the numerical solution is also examined for the SDEs. Two problems of first order SDEs are solved. Absolute errors for the problems are obtained from which the mean absolute errors (MAEs) are calculated. Comparison of variation in stepsizes is achieved using the MAEs. The results show that the MAEs decrease as the stepsize decreases. The strong orders of convergence and the residuals for the problems are respectively obtained using Least Square Fit. This work produces numerical values for the solution to the problems at discritised points in a given interval which differ from the existing methods of EMM type where results are obtained by simulation.
This paper examines the effect of varying stepsizes in finding the approximate solution of stochastic differential equations (SDEs). One step Milstein method (MLSTM) for solution of general first order stochastic differential equations (SDEs) has been derived using It么 Lemma and Euler-Maruyama Method as supporting tools. Two problems in the form of first order SDEs have been considered. The method of solution used is one step Milstein method. The absolute errors were calculated using the exact solution and numerical solution. Comparison of varying the stepsizes was achieved using mean absolute error criterion. The results showed that the mean absolute error due to approximation decreases as the stepsizes decreases. The order of convergence is approximately 1, which indicates the accuracy of the method. Also, the effect of varying stepsizes can also be identified using graphical method constructed for various stepsizes.
This study employs the Fireman Method in the solutions of optimal control problems of the form, The Hamiltonian Principle was employed for the analytical solutions of the given optimal control problems. It has been observed that this method converges close to the analytical solution for some class of problems.
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