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I know that to a semimartingale $X_t$ one can associate the characteristic triplet $(a, B, \nu)$ of predictable processes such that the canonical representation (Jacod and Shiryaev, Theorem 2.34) holds true. This would seem to suggest that the semimartingale characteristic (for some given truncation function) uniquely determines $X_t$ up to evanescence.

However, I recall that in the book of Kuechler and Sorensen "Exponential Families of Stochastic Processes" (which I no longer can consult), an example is provided of two stochastic processes, with same characteristics, one of them being strongly Markovian and the other one not.

How can these two facts come together? Am I just remembering incorrectly the nature of such an example? Would it be possible for somebody to check?

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