Description
- Abstract:
- We prove a moderate deviations principle for the continuous time linear interpolation of discrete time recursive stochastic processes, and then investigate importance sampling schemes based on subsolutions to the Hamilton-Jacobi-Bellman equation associated with the moderate deviations structure. Both the proof of the moderate deviations principle itself as well as the proof of the asymptotic performance of importance sampling schemes based on moderate deviations rely on proving tightness of the empirical measures of the conditional means of the controlled noises as well as the controlled processes themselves. This is more complex than what is needed in the large deviation setting primarily because of the moderate deviations scaling which amplifies the noise and the weaker assumptions imposed on the moment generating function. The main tools used are the relative entropy representation of exponential integrals and the weak convergence of probability measures. The resulting moderate deviations structure is essentially a linear approximation of the large deviations dynamics and a quadratic approximation of the large deviations costs, both centered around the law of large numbers limit. Consequently importance sampling schemes based on moderate deviations subsolutions generally don't perform as well as their large deviations counterparts, but the subsolutions themselves are typically easier to find. For this reason we recommend using importance sampling based on moderate deviations when finding large deviation subsolutions is prohibitively difficult, which can occur even in simple situations, or when considering events that are “rare but not too rare” so that the moderate deviation approximation centered around the law of large numbers limit captures the distributional properties which are important for determining the probability.
- Notes:
- Thesis (Ph.D. -- Brown University (2015)
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Citation
Johnson, Dane Michael,
"Moderate Deviations and Subsolution-Based Importance Sampling for Recursive Stochastic Algorithms"
(2015).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0M32T5M
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....