Limit behavior of the solution of Caputo-Hadamard fractional stochastic differential equations

In this letter, an averaging principle for Caputo-Hadamard fractional stochastic differential equations is established. It is showed the solution of the Caputo-Hadamard fractional stochastic differential equation converges to the solution of the averaged equation when the time scale parameter tends to zero. Compared to existing literatures, different estimates techniques are used to overcome the difficulties caused by the logarithmic term. It means that Khasminskii classical approach is extended to fractional stochastic differential equations of Caputo-Hadamard type.