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Could it be that your problem is only due to the $t^-$ notation convention? Think of it that way, it is only worth distinguishing $S_{t^-}$ from $S_t$ at a jump time. Elsewhere, knowing that Brownian motion paths are continuous, you'll always have $S_t = S_{t^-}$. Thus you could also write the SDE: $$\frac {dS_t}{S_{t^-}} = \alpha dt+\sigma dW_t+ ...

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