Mathematics – Probability
Scientific paper
2011-01-28
Mathematics
Probability
47 pages
Scientific paper
The paper deals with the large deviation principle for a sequence of stochastic integrals and stochastic differential equations in infinite-dimensional settings. Let $\H$ be a separable Banach space. We consider a sequence of stochastic integrals $\{X_{n-}\cdot Y_n\}$, where $\{Y_n\}$ is a sequence of infinite-dimensional semimartingales indexed by $\H\times [0,\infty)$ and the $X_n$ are $\H$-valued cadlag processes. Assuming that $\{(X_n,Y_n)\}$ satisfies a large deviation principle, a uniform exponential tightness condition is described under which a large deviation principle holds for $\{(X_n,Y_n,X_{n-}\cdot Y_n)\}$. An expression for the rate function of the sequence of stochastic integrals $\{X_{n-}\cdot Y_n\}$ is given in terms of the rate function of $\{(X_n,Y_n)\}$. A similar result for stochastic differential equations also holds. Since many Markov processes can be represented as solutions of stochastic differential equations, these results, in particular, provide a new approach to the study of large deviation principle for a sequence of Markov processes.
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