Apply part (a)'s square formula, extended by the <Jordan decomposition of a function of bounded variation> from nondecreasing functions to arbitrary càdlàg functions of <bounded variation>, to $f+g$, $f$, and $g$. Since
$$
fg=\frac12\bigl((f+g)^2-f^2-g^2\bigr),
$$
linearity of the <Lebesgue-Stieltjes integral> leaves $\int_0^tf\,dg+\int_0^tg\,df$. At each time $s$, polarization of the jump correction gives
$$
\frac12\left((\Delta f(s)+\Delta g(s))^2-\Delta f(s)^2-\Delta g(s)^2\right)
=\Delta f(s)\Delta g(s).
$$
Only common jump times contribute, and hence
$$
f(t)g(t)=f(0)g(0)+\int_0^tf\,dg+\int_0^tg\,df
-\sum_{s\in A_f\cap A_g\cap(0,t]}\Delta f(s)\Delta g(s).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-202/1/c/solution