Appendix B

Integration on Rough Paths∗The material on Stieltjes integrals has a very important generalization called integration onrough functions. This chapter gives an introduction to this topic. In order to show this, weneed a simple inequality called the triangle inequality. First here is a useful lemma.

As in the case of Stieltjes integrals all of this has generalizations to integrator functionswhich have values in various normed linear spaces but this is a book on single variableadvanced calculus and so this level of generality is avoided.

Lemma B.0.1 If a,b≥ 0, p > 1 and p′ is defined by 1p +

1p′ = 1, then

ab≤ ap

p+

bp′

p′.

Proof: Let p′ = q to save on notation and consider the following picture:

b

a

x

t

x = t p−1

t = xq−1

ab≤∫ a

0t p−1dt +

∫ b

0xq−1dx =

ap

p+

bq

q.

Note equality occurs when ap = bq.The following is a case of Holder’s inequality.

Lemma B.0.2 Let ai,bi ≥ 0. Then for p≥ 1,

∑i

aibi ≤

(∑

iap

i

)1/p(∑

ibp′

i

)1/p′

Proof: From the above inequality,

n

∑i=1

ai(∑i ap

i

)1/p

bi(∑i bp′

i

)1/p′ ≤n

∑i=1

1p

(ap

i

∑i api

)+

1p′

(bp′

i

∑i bp′i

)

=1p

(∑i ap

i

∑i api

)+

1p′

(∑i bp′

i

∑i bp′i

)=

1p+

1p′

= 1

Hence the inequality follows from multiplying both sides by(∑i ap

i

)1/p(

∑i bp′i

)1/p′

.

Then with this lemma, here is the triangle inequality.

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