On the weak compactness of continuous semi-martingales

The aim of this paper is to find conditions for weak compactness of measures in $C[0,\infty)$ corresponding to continuous local semi-martingales.
By a continuous local semi-martingale we mean here any real-valued stochastic process with parameter in $[0,\infty)$ which can be represented as the sum of a continuous local martingale and a continuous process with trajectories of bounded variation on each segment.
Conditions of compactness are sought in terms of the natural characteristics of semimartingales: drift and diffusion.
These characteristics are mappings of $C[0,\infty)$ into itself which transform the semimartingale in question into its part of bounded variation and into the squared variation of its martingale part respectively. In the particular case of semi-martingales which are solutions of stochastic differential equations, the drift and diffusion are, respectively, the integrals in time of the drift coefficient and the squared diffusion coefficient.
Typical conditions of the weak compactness we obtain are: the drift and diffusion must well behave themselves on bounded functions and satisfy some growth restrictions when the supremum of functions tends to infinity. The most illustrative application is, perhaps, Theorem 5.2. It asserts that, in the problem of solving stochastic differential equations, the classical condition of linear growth in the phase variable of the coefficients can be weakened by allowing a logarithmic factor.