The large deviation principle for stochastic processes. I

We discuss the large deviation principle of stochastic processes as random elements of $l_{\infty}(T)$. We show that the large deviation principle in $l_{\infty}(T)$ is equivalent to the large deviation principle of the finite dimensional distributions plus an exponential asymptotic equicontinuity condition with respect to a pseudometric, which makes $T$ a totally bounded pseudometric space. This result allows us to obtain necessary and sufficient conditions for the large deviation principle of different types of stochastic processes. We discuss the large deviation principle of Gaussian and Poisson processes. As an application, we determine the integrability of the iterated fractional Brownian motion.