Integration On Rough Paths∗
The material on Stieltjes integrals has a very important generalization called integration on
rough functions. This chapter gives an introduction to this topic. In order to show this,
we need a simple inequality called the triangle inequality. First here is a useful
As in the case of Stieltjes integrals all of this has generalizations to integrator functions
which have values in various normed linear spaces but this is a book on single variable
advanced calculus and so this level of generality is avoided. However, it is just a matter
regarding F has having values in a vector space. All the details are the same other than
Lemma 10.0.1 If a,b ≥ 0 and p′ is defined by
Proof: Let p′ = q to save on notation and consider the following picture:
Note equality occurs when ap = bq. ■
The following is a case of Holder’s inequality.
Lemma 10.0.2 Let ai,bi ≥ 0. Then
Proof: From the above inequality,
Hence the inequality follows from multiplying both sides by
Then with this lemma, here is the triangle inequality.
Theorem 10.0.3 Let ai,bi ∈ ℝ. Then
Proof: First note that from the definition, p − 1 = p∕p′.
Now from Lemma 10.0.2,
In case ∑
= 0 there is nothing to show in the inequality. It is obviously true. If
this is nonzero, then divide both sides of the above inequality by