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In this doctoral thesis, first we prove the continuous semimartingale local time Lt is of bounded p-variation in the space variable in the classical sense for any p > 2 a.s., and based on this fact we define the integral of local time in the sense of Young integral, and in the sense of Lyons' rough path integral, so that we obtain the new extensions to Tanaka-Meyer's formula for more classes of f. We also give new conditions to two-parameter Young integral and extend Elworthy-Truman-Zhao's formula. In the final part we define a new integral, i.e. stochastic Lebesgue-Stieltjes integral and extend Tanaka-Meyer's formula to two dimensions.

The long-time / large-scales behaviour of solutions to stochastic differentials equations (SDEs) describing the motion of a single particle in a velocity field subject to molecular diffusion is the central topic of this thesis. The particle can be considered mass-less (passive tracers) or not (inertia! particles). Under appropriate assumptions on the different velocity fields studied, we show, using homogenization theory, that the effective behaviour of passive tracers and inertial particles is described by a pure diffusion characterized by the effective diffusivity matrix K.

Implicit numerical methods such as the stochastic theta-method offer a practical way to approximate solutions of stochastic differential equations. The method involves a parameter, θ, which is freely chosen. In this thesis, we investigate strong convergence and linear stability, both mean-square and asymptotic, arising from the implementation of the theta-method when applied to ordinary stochastic differential equations incoroporating jumps. Such models are used in several disciplines; in particular, we note their use as models for various financial quantities such as asset prices, interest rates and volatility.

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In this thesis we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. For this, we prove the existence and uniqueness of the L2ρ (Rd; R1) × L2ρ (Rd; Rd) valued solutions of BDSDEs with Lipschitz nonlinear term on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary weak solutions (independent of any initial value) of SPDEs. Also the L2ρ (Rd; R1) × L2ρ (Rd; Rd) valued BDSDE with non-Lipschitz term is considered. Moreover, we verify the time and space continuity of solutions of real-valued BDSDEs, so obtain the stationary stochastic viscosity solutions of real-valued SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.

This thesis is concerned with statistical methodology for the analysis of stochastic SIR (Susceptible->Infective->Removed) epidemic models. We adopt the Bayesian paradigm and we develop suitably tailored Markov chain Monte Carlo (MCMC) algorithms. The focus is on methods that are easy to generalise in order to accomodate epidemic models with complex population structures. Additionally, the models are general enough to be applicable to a wide range of infectious diseases. We introduce the stochastic epidemic models of interest and the MCMC methods we shall use and we review existing methods of statistical inference for epidemic models. We develop algorithms that utilise multiple precision arithmetic to overcome the well-known numerical problems in the calculation of the final size distribution for the generalised stochastic epidemic. Consequently, we use these exact results to evaluate the precision of asymptotic theorems previously derived in the literature. We also use the exact final size probabilities to obtain the posterior distribution of the threshold parameter $R_0$. We proceed to develop methods of statistical inference for an epidemic model with two levels of mixing. This model assumes that the population is partitioned into subpopulations and permits infection on both local (within-group) and global (population-wide) scales. We adopt two different data augmentation algorithms. The first method introduces an appropriate latent variable, the \emph{final severity}, for which we have asymptotic information in the event of an outbreak among a population with a large number of groups. Hence, approximate inference can be performed conditional on a ``major'' outbreak, a common assumption for stochastic processes with threshold behaviour such as epidemics and branching processes. In the last part of this thesis we use a \emph{random graph} representation of the epidemic process and we impute more detailed information about the infection spread. The augmented state-space contains aspects of the infection spread that have been impossible to obtain before. Additionally, the method is exact in the sense that it works for any (finite) population and group sizes and it does not assume that the epidemic is above threshold. Potential uses of the extra information include the design and testing of appropriate prophylactic measures like different vaccination strategies. An attractive feature is that the two algorithms complement each other in the sense that when the number of groups is large the approximate method (which is faster) is almost as accurate as the exact one and can be used instead. Finally, it is straightforward to extend our methods to more complex population structures like overlapping groups, small-world and scale-free networks