Approximation of supremum and infimum processes as a stochastic approach to the providing of homeostasis

We consider the calculation of bounded functional of the trajectories of a stationary diffusion process. Since an analytical solution to this problem does not exist, it is necessary to use numerical methods. One possible direction for obtaining the numerical method is applying the Monte Carlo (MC) method. This involves reproducing the trajectory of a random process with subsequent averaging over the trajectories. To simplify the reproduction of the trajectory, the Girsanov transform is used in this paper. The main goal is to approximate the supremum and infimum processes, which allows us to more accurately compute the mathematical expectation of a function depending on the values of the supremum and infimum processes at the end of the time interval compared to the classical method. The method is based on randomly dividing the interval of the time axis by stopping times passages of the Wiener process, approximating the density to replace the measure, and using the MC method to calculate the expectation. One of the applications of the method is the task of keeping a random process in a given area — the problem of homeostasis.