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A semimartingale is an adapted process of the form $X=X_0+M+A$, where $M$ is a local martingale and $A$ is an adapted finite-variation process. A sequence of processes $X^n$ converges to $X$ in uniform convergence on compacts in probability, abbreviated ucp, when for every $T<\infty$ and $\delta >0$,

The predictable sigma-algebra on $\Omega \times [0,\infty )$ is generated by $A\times \{0\}$ for $A\in {\mathcal{F}}_0$ and by $A\times (s,t]$ for $A\in {\mathcal{F}}_s$. Equivalently, it is the smallest sigma-algebra making every left-continuous adapted process measurable.