Kenneth Kuttler

384 APPENDIX B. INTEGRATION ON ROUGH PATHS∗

but that on the inside equals∣∣∣∫ a

0 Y dF +∫ T

a Y dF−∫ T

0 Y dF∣∣∣ because this property of the

integral holds for Stieltjes integrals. Thus the claim holds. Other properties of the integralcan also be inferred from this approximation with the Stieltjes integrals for piecewise linearFn. It follows that∣∣∣∣∫ t

sY dF

∣∣∣∣= ∣∣∣∣∫ t

0Y dF−

∫ s

0Y dF

∣∣∣∣≤Cpq ∥Y∥V q([0,T ]) ∥F∥p,[s,t]

What of continuity of t→∫ t

0 Y dF? Letting Fn be the sequence of piecewise linear func-tions described above, t→

∫ t0 Y dFn is continuous and letting p′ > p∣∣∣∣∫ t

0Y dF−

∫ t

0Y dFn

∣∣∣∣= ∣∣∣∣∫ t

0Y d (F−Fn)

∣∣∣∣≤Cp′q ∥Y∥V q([0,T ]) ∥F−Fn∥p′,[0,T ]

Thus, as n→ ∞, one has uniform convergence of the continuous functions t →∫ t

0 Y dFn tot→

∫ t0 Y dF thanks to Theorem B.2.3 which implies that the right side converges to 0 since

p′ > p.Consider the other estimate 2.4. Let P be a dissection 0 = t0 < t1 < · · ·< tn = T . Let

Ψ(t)≡∫ t

0 Y dF . From the definition of the integral in terms of a limit as |P|→ 0, it followsthat, since p≥ 1,

n−1

∑i=0|Ψ(ti+1)−Ψ(ti)|p ≤

n−1

∑i=0

∣∣∣∣∫ ti+1

tiY dF

∣∣∣∣p ≤ n−1

∑i=0

Cppq ∥Y∥

pV q([0,T ]) ∥F∥

pp,[ti,ti+1]

≤ Cppq ∥Y∥

pV q([0,T ])

n−1

∑i=0∥F∥p

p,[ti,ti+1]≤Cp

pq ∥Y∥pV q([0,T ])

(n−1

∑i=0∥F∥p,[ti,ti+1]

≤ Cppq ∥Y∥

pV q([0,T ]) ∥F∥

pp,[0,T ]

Recall that since p ≥ 1, if each ai ≥ 0,∑i api ≤ (∑i ai)

p. Then taking the sup over all such

dissections, it follows that∥∥∥∫ (·)0 Y dF

∥∥∥p,[0,T ]

≤ Cpq ∥Y∥V q([0,T ]) ∥F∥p,[0,T ]. Also, from the

estimate, ∣∣∣∣∫ t

0Y dF

∣∣∣∣≤Cpq ∥Y∥V q([0,T ]) ∥F∥p,[0,t] ≤Cpq ∥Y∥V q([0,T ]) ∥F∥p,[0,T ]

Taking sup for all t,∥∥∥∫ (·)0 Y dF

∥∥∥∞

≤Cpq ∥Y∥V q([0,T ]) ∥F∥p,[0,T ] and so∥∥∥∥∫ (·)

0Y dF

∥∥∥∥V p([0,T ])

≤ 2Cpq ∥Y∥V q([0,T ]) ∥F∥p,[0,T ]

384 APPENDIX B. INTEGRATION ON ROUGH PATHS*but that on the inside equals | Jo YaF + fr YdF — Io. YdF | because this property of theintegral holds for Stieltjes integrals. Thus the claim holds. Other properties of the integralcan also be inferred from this approximation with the Stieltjes integrals for piecewise linearF,,. It follows that[var|-What of continuity of t > fo YdF? Letting F,, be the sequence of piecewise linear func-tions described above, t > fo YdF, is continuous and letting p’ > pt t[ YdF — [ YdF,0 0Thus, as n —> oo, one has uniform convergence of the continuous functions t + fo YdF,, tot— fo YdF thanks to Theorem B.2.3 which implies that the right side converges to 0 sincep'>p.Consider the other estimate 2.4. Let Y be a dissection 0 = fo < ty <---<t, =T. LetW (t) = Jy YdF. From the definition of the integral in terms of a limit as | P| — 0, it followsthat, since p > 1,Ti41/ YdFtjt AY[ var — | var| S Coq |I¥ Ilva qo,ry) IF lpsS CyqlI¥ llvacory FP — Fall yor[vamPp n—1P Ps 2 Cha I¥ llvaco.r Fl tec£n—1 n—1View) -P (HPs VYi=0 i=0IAn—1 n—1 PCha IY \lyaqo,ry) d FU tictivs] Coa |I¥ llvaqo.r) (s Flot)i= i=Sha MI¥llyaqo,ry |IF lp 0,7)Recall that since p > 1, if each a; > 0,Y; a; < (¥;a;)”. Then taking the sup over all suchdissections, it follows that |." vaP| or < Cpq|I¥ llvaco,7)) | llp,o,7)-. Also. from theestimate,t/ var| S Cpq|I¥ Ilva qo,ryy IF lp fo. S Cpa lI¥ llvaco.ry |Fllp.to,7)Taking sup for all ¢, |fy vaF | < Coq |I¥ llvaqo,ry IF llpjo,r) and so()[ YdF0S 2Cpq|I¥ llvaqo.ry IF llpjo.7)ve((0,7))