| The first part of this thesis deals with weak convergence of a sequence of stochastic differential equations (SDE) driven by Hilbert space-valued semimartingales, for which the sequence of driving semimartingales fails to satisfy the uniform tightness condition. A theorem, similar to the one developed by Kurtz and Protter ('91) for finite dimensional semimartingales, has been proved. The result has been used to study the limit of the solution of SDE of the type Xnt=Xn 0+ Rd×&sqbl0;0,t &parr0;gX ns,x Wn&d2;x,s dxds, where the Wn are smooth approximations of the space-time Gaussian white noise W..;The second part 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 {Xn− · Yn}, where {Yn} is a sequence of infinite dimensional semimartingales indexed by H × [0, ∞) and the Xn are H valued cadlag processes. Assuming that {(Xn, Y n)} satisfy large deviation principle, a uniform exponential tightness condition is described under which a large deviation principle holds for {(Xn, Yn, Xn− · Yn)}. A simplified expression of the rate function for the sequence of stochastic integrals {Xn− · Yn} has been given in terms of the rate function for {( Xn, Yn)} under the assumption that H has a Schauder basis. A similar result for stochastic differential equations has also been proved. |
