Estimating the time-varying parameters of SDE models by maximum principle

We present a parameter estimation method of stochastic differential equations with time-varying coefficients, where data can be observed at discrete points of time. Our objective is to develop the uniform mathematical technique to solve the parameter estimation problem for stochastic differential equations with both ordinary and fractional Brownian motions. This estimation principle is based on the replacement of a stochastic differential equation by a system of ordinary differential equations, which present the moment functions, and on the application of the Pontryagin´s maximum principle to find the optimal estimates of the time-varying coefficients of the initial equation. The key point is the constraints structural selection, which leads to major modifications of algorithms of analytical and numerical solutions. This estimation method is applied to study the North Atlantic herring population dynamics.