Kenneth Kuttler

870 CHAPTER 32. QUADRATIC VARIATION

By 32.9, ≤ 2−nE(∥M (T )∥2

)1/2which shows {Pn} is a Cauchy sequence in M 2

T (R).Therefore, by Proposition 31.7.2, there exists {N (t)} ∈M 2

T (R) such that Pn → N inM 2

T (H) . That is

limn→∞

(sup

t∈[0,T ]|Pn (t)−N (t)|2

)1/2

= 0.

Since {N (t)} ∈M 2T (R) , it is a continuous martingale and N (t) ∈ L2 (Ω) , and N (0) = 0

because this is true of each Pn (0) . From the above 32.5,

∥M (t)∥2 = Qn (t)+Pn (t) (32.10)

whereQn (t) = ∑

k≥0

∥∥M(t ∧ τ

nk+1)−M (t ∧ τ

nk)∥∥2

and Pn (t) is a martingale. Then from 32.10, Qn (t) is a submartingale and converges foreach t to something, denoted as [M] (t) in L1 (Ω) uniformly in t ∈ [0,T ]. This is becausePn (t) converges uniformly on [0,T ] to N (t) in L2 (Ω) and ∥M (t)∥2 does not depend on n.Then also [M] is a submartingale which equals 0 at 0 because this is true of Qn and becauseif A ∈Fs where s < t,∫

AE ([M] (t) |Fs)dP≡

∫A[M] (t)dP = lim

n→∞

∫A

(∥M (t)∥2−Pn (t)

)dP

= limn→∞

∫A

E(∥M (t)∥2−Pn (t) |Fs

)dP≥ lim inf

n→∞

∫A∥M (s)∥2−Pn (s)dP

= lim infn→∞

∫A

Qn (s)dP≥∫

A[M] (s)dP.

Note that Qn (t) is increasing because as t increases, the definition allows for the pos-sibility of more nonzero terms in the sum. Therefore, [M] (t) is also increasing in t. Thefunction t→ [M] (t) is continuous because ∥M (t)∥2 = [M] (t)+N (t) and t→ N (t) is con-tinuous as is t → ∥M (t)∥2 . That is, off a set of measure zero, these are both continuousfunctions of t and so the same is true of [M] .

Now put back in Mα p∧δ in place of M where δ is a stopping time. From the above, thishas shown ∥∥∥Mα p∧δ (t)

∥∥∥2=[Mα p∧δ

](t)+Np (t)

where Np is a martingale and[Mα p∧δ

](t) = lim

n→∞∑k≥0

∥∥∥Mα p∧δ(t ∧ τ

nk+1)−Mα p∧δ (t ∧ τ

nk)∥∥∥2

= limn→∞

∑k≥0

∥∥M(t ∧ τ

nk+1∧α p∧δ

)−M (t ∧ τ

nk ∧α p∧δ )

∥∥2 in L1 (Ω) , (32.11)

the convergence being uniform on [0,T ] . The above formula shows that[Mα p∧δ

](t) is a

Ft∧δ∧α p measurable random variable which depends on t ∧ δ ∧α p.(Note that t ∧ δ is areal valued stopping time even if δ = ∞.) Therefore, by Proposition 32.3.3, there exists arandom variable, denoted as

[Mδ](t) which is the pointwise limit as p→∞ of these random

870 CHAPTER 32. QUADRATIC VARIATION1/2By 32.9, <2"E (im (7)\7) which shows {P, } is a Cauchy sequence in .47 (R).Therefore, by Proposition 31.7.2, there exists {N (t)} € .@ (IR) such that P, > N inM7 (H). That is1/2limE| sup |P,(t)—N()|? } =O.ne \ te(0,7]Since {N(t)} € 47 (R), it is a continuous martingale and N (t) € L?(Q), and N(0) =0because this is true of each P, (0). From the above 32.5,\|M (t)||> = Qn (t) + Pa (t) (32.10)where 5On(t) = Ye M(H th) MOAT)k>0and P,,(t) is a martingale. Then from 32.10, Q, (t) is a submartingale and converges foreach ¢ to something, denoted as [M] (t) in L' (Q) uniformly in t € [0,7]. This is becauseP, (t) converges uniformly on [0,7] to V(t) in L? (Q) and ||M(t)||? does not depend on n.Then also [M] is a submartingale which equals 0 at 0 because this is true of Q, and becauseif A € F, wheres <1,[eno Fars [moar = tim | (|? —P.(0) aP= lim fE (\|M (|? —Pa (0)|F) dP > lim int ||M(s)||? Pa (s)dPn+ JA n—yoo JA=lim inf | Q,(s)dP> | [M] (s) dP.n—-oo A ANote that Q, (t) is increasing because as f increases, the definition allows for the pos-sibility of more nonzero terms in the sum. Therefore, [M] (+) is also increasing in t. Thefunction t + [M] (t) is continuous because ||M (t) ||? = [M] (t) +N (t) and t + N(t) is con-tinuous as is t + ||M(t)||*. That is, off a set of measure zero, these are both continuousfunctions of t and so the same is true of [M].Now put back in M @p6 in place of M where 6 is a stopping time. From the above, thishas shown|mavr (t) i = iMar'| (t) ENp (t)where N, is a martingale and2mar?) (t) _ lim y marr’ (tA thy) — M&A (t \ tr)k>0_: n _ n 2. 1= firm De IM (eh A Gp \8)—M (tt; AG, AS)|| inL'(Q), (32.11)the convergence being uniform on [0,7]. The above formula shows that [M@?/°] (r) is aFn8xq, Measurable random variable which depends on ¢/\ 6 A @).(Note that ¢\ 6 is areal valued stopping time even if 6 = co.) Therefore, by Proposition 32.3.3, there exists arandom variable, denoted as [M 5] (t) which is the pointwise limit as p + © of these random