The note contains a construction of (possibly explosive) diffusions whose coefficients satisfy Stroock-Varadhan-type regularity conditions, but are unbounded. The appropriate state space here is the one-point compactification of $R^k$, and the sample paths of the diffusions are shown to be a.s. continuous on this state space.
R. N. Bhattacharya. "Correction Note: Correction to "Criteria for Recurrence and Existence of Invariant Measures for Multidimensional Diffusions"." Ann. Probab. 8 (6) 1194 - 1195, December, 1980. https://doi.org/10.1214/aop/1176994584