Let $A,B$ be two right-continuous functions of finite variation. Then the integration by parts formula states:
$$ A_{t}B_{t} = A_0B_0+\int_{0}^{t} A_{-s}dB_s \, + \int_{0}^{t} B_{-s}dA_s + \sum_{ s\leq t}\triangle A_s \triangle B_s $$
This question The general integration by parts formula and the general change of variables formula does most of the work. The last step is showing $\int_0^t \triangle A_s dB_s = \sum_{s\leq t}\triangle A_s \triangle B_s.$ Now, $\triangle A_s$ is $0$ everywhere except on at most a countable set of discontinuities of $A_s$. How does one approximate $\triangle A_s$ by simple functions and subsequently compute its integral wrt $B_s$? My intuition was the integral would be $0$ since $\triangle A_s$ is $0$ a.e., but I think the problem is it is not $0$ a.e. with respect to the measure $dB_s$.
For left continuous functions there is a change of sign for the for the sum of jumps. Pl read the equation with the appropriate lower and upper scripts. I have a problem formatting the equation.
D is the set of points where both u and v are discontinuous. ∫ [a,b] u + d λ v + ∫ [a,b] v + d λ u = λ uv ( [a,b] )+ ∑ x∈D λ u (x) λ v (x)(Preserved post a user deleted to before answering it again.)