Large Deviations Principle for Diffusions
Large Deviations Principle for Diffusions
Large deviations theory formulated by Varadhan has made a tremendous impact in a variety of fields such as mathematical physics, control theory, and statistics, to name a few. After a brief discussion of the general theory and examples, the large deviations principle (LDP) is shown to be equivalent to the Laplace principle in our context. The rate function for the LVP is obtained, in general, via relative entropy. Next, the Boué-Dupuis representation theorem for positive functionals of a Wiener process is established. Using the representation theorem, the Laplace principle is proved for diffusions.
Keywords: large deviation principle, Laplace principle, rate function, relative entropy, representation theorem for functionals of a Wiener process, Wentzell-Freidlin type large deviation principle
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