1
$\begingroup$

$N_{t}$ 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.

$\endgroup$
2
$\begingroup$

$ \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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.