How to model discontinuities in sovereign debt interest rates?

For some emerging economies, the interest rates on sovereign debt are sometimes highly volatile: undergoing periods of stability then sharp upward/downward jumps.

In continuous-time models, however, it becomes very difficult to model that kind of pattern.

My question: are there theory models of sovereign debt which try to address these discontinuity features in sovereign spreads?

Any thoughts much appreciated.

The general idea is to look at the distribution of $$\Delta V$$ for a fixed time step $$\Delta t$$. The usual continuous stochastic process model is the Wiener process, so the distribution of values is normal. Even if the most granular (tiny) transitions are not normally distributed, the sum of many such small transitions should look like a normal, by the Central Limit Theorem. But if your distribution of $$\Delta V$$ does not look normal (e.g., say it has 'fat tails'), then you can argue that there are larger jumps interspersed among the small transitions that make up the usual random process. (Hint: The CLT fails for what reasons?)