Kozyr' S. M.
Intermixing “according to Ibragimov”. Estimate for rate of approach of family of integral functionals of solution of differential equation with periodic coefficients to family of the Wiener processes. Some applications. II
Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 303-318
In the first part of this work, we obtain estimates for the rate of approach of integrals of a family of "physical" white noises to a family of the Wiener processes. By using this result, we establish an estimate for the rate of approach of a family of solutions of ordinary differential equations, disturbed by some physical white noises, to a family of solutions of the corresponding Ito equations. We consider the case where the coefficient of random disturbance is separated from zero as well as the case where it is not separated from zero.
Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I
Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 733–753
We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space $C[0, T]$. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding Itô stochastic equation.
Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1025-1039
We consider the Merton problem of finding the strategies of investment and consumption in the case where the evolution of risk assets is described by the exponential model and the role of the main process is played by the integral of a certain stationary “physical” white noise generated by the centered Poisson process. It is shown that the optimal controls computed for the limiting case are ε-sufficient controls for the original system.