326233 A New Method for Parameter Estimation in Stochastic Differential Equations
Stochastic differential equations (SDEs) are widely used in disciplines like finance, engineering, environment, physics, population dynamics and medicine. The predictive power of SDEs lies in the choice of the parameter values which can describe the real data effectively. Thus, the problem of finding an accurate and computationally feasible method for parameter estimation has been a key research area. Its importance has increased in recent years due to the application in wide range of fields like pharmaceutical problems, ecosystem models, medical data like EKG, blood pressure, sugar levels, wind and sound wave fluctuations, etc. The classical methods are based on likelihood estimation procedures. The new method is fairly simple in terms of the computation time and feasibility.
The SDE consist of two parts; the drift term or the deterministic part and the diffusion term or the stochastic component. The system involves an additional parameter due to the randomness, usually addressed as the ‘standard deviation’ when compared to its deterministic counterpart. The standard deviation is estimated by analyzing the available data and using Ito form of the stochastic differential equations. Thus, the parameter in the diffusion term is evaluated without any optimization or probabilistic methods. The parameters in the drift term are estimated by using kinetic inversion which involves non-linear programming optimization methods. However, instead of using the least square error minimization as the objective, the objective function is modified to include the previously evaluated parameter due to randomness. The results of the SDE are evaluated by using the estimated parameters in the model and solving the system for the expected value. The expected value is then compared with the actual data and the results from the deterministic model. The method yields better estimates when compared to the deterministic model. The results will also be compared to the existing methods for SDE parameter estimation.
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