Large deviations of semimartingales: A maxingale problem approach i. limits as solutions to a maxingale problem

We establish conditions on the predictable characteristics of semimartingales under which the semimartingales obey a large deviation principle for the Skorohod topology. The associated rate function may depend on the whole past of a trajectory. We use a new technique, which is a counterpart of the martingale problem approach in weak convergence theory. In connection with this, we develop an analogue of stochastic calculus on a space with rate function

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