Stochastic differential system output control by the quadratic criterion. IV. Alternative numerical decision

In the study of the optimal control problem for the Ito diffusion process and the controlled linear output with a quadratic quality criterion, an intermediate result is resumed: for approximate calculation of the optimal solution, an alternative to classical numerical integration method based on computer simulation is proposed. The method allows applying statistical estimation to determine the coefficients $\beta_t(y)$ and $\gamma_t(y)$ of the previously obtained Bellman function $V_t(y,z)=\alpha_t z^2+\beta_t(y)z+\gamma_t(y)$, determining the optimal solution in the original problem of optimal stochastic control. The method is implemented on the basis of the properties of linear parabolic partial differential equations describing $\beta_t(y)$ and $\gamma_t(y)$ — their equivalent description in the form of stochastic differential equations and a theoretical-probability representation of the solution, known as A. N. Kolmogorov equation, or an equivalent integral form known as the Feynman–Katz formula. Stochastic equations, relations for optimal control and for auxiliary parameters are combined into one differential system, for which an algorithm for simulating a solution is stated. The algorithm provides the necessary samples for statistical estimation of the coefficients $\beta_t(y)$ and $\gamma_t(y)$. The previously performed numerical experiment is supplemented by calculations presented by an alternative method and a comparative analysis of the results.