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Dadashi-Arani, Hassan | 2010

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 40544 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Zohori Zangeneh, Bijan; Zamani, Shiva
  7. Abstract:
  8. Large deviation principle (LDP) for stochastic differential equation is one of the interesting and modern topics in stochastic analysis. Principally, this theory gives the rate of convergence to the solution of the corresponding deterministic equation when the noise tends to zero. The study of LDP for SDE’s has been initiated by M. Freidlin and A. Wentzell and then has been considered by many other researchers. Freidlin andWentzell divided the interval [0, T] to small subintervals and considered the diffusion coefficient as a constant on any small subintervals. Then the problem is reduced to the additive noise case. But using the contraction principle, the study of LDP for an equation with additive noise is reduced to the study of the LDP for Brownian motion which has been studied by Schilder in 1966. This method was the only tool for different equation for many years. But applying this method to SPDE’s has severe technical difficulties. In their theses S. Peszat and R. Sowers studied the large deviation principle for some type of SPDE’s, using this method. The outcomes of P. Dupuis and his colleagues in variational representations offered a new approach in dealing with the large deviations for SDE’s. Using this approach we can study the Laplace principle, which is equivalent to large deviation principle, easier. In this thesis we apply this method to some type of stochastic evolution equation with non-Lipchitz (monotone) diffusion coefficient.
  9. Keywords:
  10. Stochastic Evolution Equation ; Large Deviation Principle ; Monotone Property of Coefficients ; Multiplicative Noise ; Variational Representation

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