Let be a piecewise continuous path and . It is well-known that we can integrate against by using the Riemann–Stieltjes integral which is a natural extension of the Riemann integral. The idea is to use the Riemann sums
where . It is easy to prove that, when the mesh of the subdivision goes to 0, the Riemann sums converge to a limit which is independent from the sequence of subdivisions that was chosen. The limit is then denoted and called the Riemann-Stieltjes integral of against . Since has a bounded variation, it is easy to see that, more generally,
with would also converge to . If
is an absolutely continuous path, then it is not difficult to prove that we have
where the integral on the right hand side is understood in Riemann’s sense.
Thus, by taking the limit when the mesh of the subdivision goes to 0, we obtain the estimate
where is the notation for the Riemann-Stieltjes integral of against the bounded variation path . We can also estimate the Riemann-Stieltjes integral in the 1-variation distance. We collect the following estimate for later use:
Proposition: Let be a piecewise continuous path and . We have
The Riemann-Stieltjes satisfies the usual rules of calculus, for instance the integration by parts formula takes the following form
Proposition: Let and .
We also have the following change of variable formula:
Proposition: Let and let be a map. We have
Proof: From the mean value theorem
with . The result is then obtained by taking the limit when the mesh of the subdivision goes to 0
We finally state a classical analysis lemma, Gronwall’s lemma, which provides a wonderful tool to estimate solutions of differential equations.
Proposition: Let and let be a bounded measurable function. If,
for some , then
Proof: Iterating the inequality
times, we get
where is a remainder term that goes to 0 when . Observing that
and sending to finishes the proof