Kenneth Kuttler

378 APPENDIX B. INTEGRATION ON ROUGH PATHS∗

where a≡ F(t j+1)−F(t j)t j+1−t j

. Then this sum equals

∣∣F (t j+1)−F (t j)

∣∣p(k−1

∑l=i

∣∣∣∣ul+1−ul

t j+1− t j

∣∣∣∣p + ∣∣∣∣ t j+1−uk

t j+1− t j

∣∣∣∣p + ∣∣∣∣ ui− t j

t j+1− t j

∣∣∣∣p).

If all the ur were deleted from Pε for i < r < k, this results in∣∣F (t j+1

)−F (t j)

∣∣p which

must be at least as large from the above observation because ∑k−1l=i

∣∣∣ ul+1−ult j+1−t j

∣∣∣+ ∣∣∣ ui−t jt j+1−t j

∣∣∣+∣∣∣ t j+1−ukt j+1−t j

∣∣∣= 1and p≥ 1 and all these quotients are positive and smaller than 1. Thus we geta larger sum on the left in 2.1 by modifying Pε as just described. This shows that it can beassumed that every point of Pε is in P . Therefore, since each ui ∈Pε

∥F∥pp,[0,T ] ≥ ∑

ui∈Pε

|F (ui+1)−F (ui)|p = ∑ui∈Pε

∣∣∣FP (ui+1)−FP (ui)∣∣∣p > ∥∥∥FP

∥∥∥p

p,[0,T ]− ε

Since ε is arbitrary, this shows the conclusion of the lemma.Next is to show that these piecewise linear functions approximate F in terms of p

variation. In general, when you have a continuous function defined on a closed interval,the above piecewise continuous approximations always converge to the function uniformly.This is left to the reader to show. It is an exercise in uniform continuity of F .

Theorem B.2.3 Let F ∈V p ([0,T ]) , p≥ 1. Then lim|P|→0∥∥FP −F

∥∥q,[0,T ] = 0 for

every q > p.

Proof: Let ε > 0 be given and let P be a dissection of [0,T ].Using what was just observed about uniform continuity of piecewise linear approxima-

tions,

∑ti∈P

∣∣∣FP (ti+1)−F (ti+1)−(

FP (ti)−F (ti))∣∣∣q

≤ maxti∈P

(∣∣∣FP (ti+1)−F (ti+1)−(

FP (ti)−F (ti))∣∣∣q−p

)·

∑ti∈P

∣∣∣FP (ti+1)−F (ti+1)−(

FP (ti)−F (ti))∣∣∣p

≤ maxti∈P

(∣∣∣FP (ti+1)−F (ti+1)−(

FP (ti)−F (ti))∣∣∣q−p

)∥∥∥FP −F∥∥∥p

p,[0,T ]

≤ maxti∈P

((∣∣∣FP (ti+1)−F (ti+1)∣∣∣+ ∣∣∣FP (ti)−F (ti)

∣∣∣)q−p)·(∥∥∥FP

∥∥∥p,[0,T ]

+∥F∥p,[0,T ]

≤ 2q−p maxt∈[0,T ]

∣∣∣FP (t)−F (t)∣∣∣q−p(

2∥F∥p,[0,T ]

from Lemma B.2.2 since∥∥FP

∥∥p,[0,T ] ≤ ∥F∥p,[0,T ]. Now it follows that if |P| is small

enough, the above is no more than ε .The method of proof of the above yields the following useful lemma. To save space,

∥F∥∞≡ supt |F (t)| .

378 APPENDIX B. INTEGRATION ON ROUGH PATHS*"If all the u, were deleted from Ye for i <r < k, this results in |F (t;41) — F (t;) |’ whichF(ti1)-F (4)where a =tipi. Then this sum equalsF (at) -FODIP (x[=ipm4i1—uy |? |tirr—uK |? uj — 1;tiga tjtig. — tj tig. —tjmust be at least as large from the above observation because Y—; oh a_ J J J Ja co . *,°a = land p > | and all these quotients are positive and smaller than 1. Thus we getJ Ja larger sum on the left in 2.1 by modifying Ae as just described. This shows that it can beassumed that every point of “, is in #. Therefore, since each uj; € PeIFieon= DL Fwn)-F wl = YL |F? wn) —F? (wi)ujePe ujePeTP Tenn?p,(0,7|Since € is arbitrary, this shows the conclusion of the lemma. JNext is to show that these piecewise linear functions approximate F in terms of pvariation. In general, when you have a continuous function defined on a closed interval,the above piecewise continuous approximations always converge to the function uniformly.This is left to the reader to show. It is an exercise in uniform continuity of F’.Theorem B.2.3 Let F € Vv? ((0,7]),p > 1. Then lim)y\0 ||F” — F |every q > p.4.l0.7) = 0 forProof: Let € > 0 be given and let Y be a dissection of [0,7].Using what was just observed about uniform continuity of piecewise linear approxima-tions,E har) -(ePt0-rw)P< her (\F” (ti) —F Gist) (F? (ti) —F (1) "") .& pay ([P 0) —Ftosd—(F200-FE0) DF< max ((|F? (tis1) —F (tixs)| 4 iF? (1;) — F (ti) )"")(lion tHloon)”7 q-P P< 29-? max |F” ()-F(o| (2IIFllpjo.r))te [0,7]from Lemma B.2.2 since FIL .0.7) < |IFllpJ0,7)- Now it follows that if | #| is smallenough, the above is no more than €. JJThe method of proof of the above yields the following useful lemma. To save space,I|F||.. = sup, |F (¢)|.