380 APPENDIX B. INTEGRATION ON ROUGH PATHS∗

Definition B.3.2 Let P be a dissection of [0,T ] and let Y,F be continuous. Then∫P

Y dF ≡∑P

Y (ti)(F (ti+1)−F (ti))

where P = {t0, · · · , tn}. This is like a Riemann Stieltjes sum except that you don’t have abounded variation integrator function.

∫ T0 Y dF is said to exist if there is I ∈ R such that

lim|P|→0

∣∣∣∣∫P

Y dF− I∣∣∣∣= 0

meaning that for every ε > 0 there exists δ > 0 such that whenever |P|< δ , it follows that|∫P Y dF− I|< ε . It suffices to show that for every ε there exists δ such that if |P| , |P ′|<

δ , then ∣∣∣∣∫P

Y dF−∫

P ′Y dF

∣∣∣∣< ε

This last condition says that the set of all these∫P for |P| sufficiently small has small

diameter.

Note how this looks just like the Stieltjes integral except here one is considering onesided sums just like Cauchy did in the 1820’s. The following theorem is from Young.

Theorem B.3.3 Let 1≤ p,q, 1p +

1q > 1. Also suppose that F ∈V p ([0,T ]) and Y ∈

V q ([0,T ]). Then for all sub interval [a,b] of [0,T ] there exists I[a,b] such that

lim|P|→0

∣∣∣∣∫P

Y dF− I∣∣∣∣= 0

exists where here P ⊆ [a,b] is a dissection of [a,b] . Also there exist estimates of the form∣∣∣∣∫ t

sY dF

∣∣∣∣≤Cpq ∥Y∥V q([0,T ]) ∥F∥p,[s,t] (2.2)

∣∣∣∣∫ (·)

0Y dF

∣∣∣∣p,[0,T ]

≤Cpq ∥Y∥V q([0,T ]) ∥F∥p,[0,T ] (2.3)∣∣∣∣∫ t

sY dF

∣∣∣∣= ∣∣∣∣∫ t

0Y dF−

∫ s

0Y dF

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

0Y dF

∣∣∣∣V p([0,T ])

≤ 2Cpq ∥Y∥V q([0,T ]) ∥F∥p,[0,T ] (2.4)

where Cpq depends only on p,q. Also t→∫ t

0 Y dF is continuous.

Proof: Let Z ∈ V q ([0,T ]) ,1≤ p,q, 1p +

1q > 1. Define for s≤ t

ω (s, t)≡ ∥F∥pp,[s,t]+∥Z∥

qq,[s,t]

Here Z will end up being Y −Y (0). From Lemma B.1.5,

ω (ti, ti+1)+ω (ti+1, ti+2)+ · · ·+ω (ti+r, ti+r+1)≤ ω (ti, ti+r+1)