$N_{t}$ represents Poisson process on filtered probability space.

Calculate $ \int \limits_{0}^{t} N_{s-} dN_{s} $ ? I am trying to learn it, and I have the solution but can not understand a step

$ = \sum \limits_{0<s \leq t} N_{s-} \Delta N_{s} = \sum \limits_{k=1}^{N_{t} - 1}k $

I do not understand the last step of equality.


$ \sum \limits_{0<s \leq t} N_{s-} \Delta N_{s}$ has terms at the points of jumps of $N_s$. At each such point, the jump is by one i.e. $\Delta N_{s} = 1$ a.s., and the value of $N_{s_k-} = k-1$ where $s_k$ is the $k$-the jump point.

That's it.


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