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The integration by parts formula for the Itō integral is

If $X$ and $Y$ are semimartingales then $$ X_tY_t = X_0Y_0+\int_0^t X_{s-}\,dY_s + \int_0^t Y_{s-}\,dX_s + [X,Y]_t $$ where $[X, Y]$ is the quadratic covariation process.

I was wondering what $X_{s-}$ means? Thanks and regards!

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up vote 3 down vote accepted

It's defined pointwise by $$ X_{s-}=\lim_{t\to s,t<s} X_t $$

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