I have the following integral that I am trying to evaluate with integration by parts.
$\int_{[0,T]} P_{t^{-}} d\gamma_t$
Where $P_{t^{-}}$ is the left limit of $P_t$ for a simple random walk,
$dP_t= \mu dt + \sigma dW_t$
And a function consisting of the difference of two non-decreasing, Right-Continuous functions with Left Limits,
$\gamma_t = \gamma_t^{+} - \gamma_t^{-}$
I have having a difficult time understanding how I should treat Integration by Parts with these RCLL functions. According to 3.12 on http://math4tune.com/stats%205-11-2007.pdf (page 33), I can express my integral as:
$ \int_{[0,T] } P_{t^{-}} d\gamma_t = P_T \gamma_T - P_0\gamma_0 - \int_{[0,T] } \gamma_{t} dP_t$
However, this goes against what I intuitively think is correct: Might the following work?
$ \int_{[0,T] } P_{t^{-}} d\gamma_t = P_{T^{-}} \gamma_{T^{-}} - P_{0^{-}}\gamma_{0^{-}} - \int_{[0,T] } \gamma_{t^{-}} dP_t$
Conceptually, how should I be thinking about this kind of Integration by Parts?
Thanks!
M