Let \(\xi(t)\) and \(\Lambda(t)\), \(t\geq 0\), be independent stochastic processes on the same probability space. Assume that, with probability 1, \(\Lambda(t)\) takes nonnegative real values and has continuous trajectories starting at zero. The stochastic process \(\xi(\Lambda(t))\), \(t\geq 0\), is called an iterated process.
In this paper, the author derives new results on the asymptotic behavior of logarithms of small deviation probabilities for some iterated processes. He first fines a relationships between the logarithmic asymptotics of the probabilities \(\operatorname{P}(M(\Lambda(t))\leq y_t)\) and \(\operatorname{P}(M(t)\leq y_t)\), where \(M(t)\) is a stochastic process with positive nondecreasing trajectories and \(y_t\) is a function such that \(y_t\to \infty\) as \(t\to \infty\). Then the author takes the functional \(M(t):=\sup_{0\leq u \leq t}|\xi(u)|\) of trajectories of the process \(\xi(t)\) which has a known logarithmic asymptotic of small deviation probabilities. Since \[ M(\Lambda(t))=\sup_{0\leq u \leq t}|\xi(\Lambda(u))|, \] this allows him to find the asymptotic of logarithms of small deviations probabilities \[ \operatorname{P}(\sup_{0\leq u \leq t}|\xi(\Lambda(u))|\leq y_t) \] for \(\xi(t)\) belonging to various classes of stochastic processes. The author applies this results to the iterated processes generated by the compound Cox processes and compound renewal processes.