884 CHAPTER 33. QUADRATIC VARIATION AND STOCHASTIC INTEGRATION

because it is a finite sum of continuous random variables. In the argument to get 33.2, I willwrite M rather than Mτ l . Then it is understood that you can let l→ ∞ to obtain the desiredformula for a local martingale. Thus, in what follows, M is a bounded martingale.

E

(∥∥∥∥∫ t

0f dM

∥∥∥∥2

U

)

= E

(mn−1

∑k=0

fk (M (t ∧ tk+1)−M (t ∧ tk)) ,mn−1

∑j=0

f j(M(t ∧ t j+1

)−M (t ∧ t j)

))

Let M (t ∧ tk+1)−M (t ∧ tk) = ∆Mk. Thus ∆Mk is Ftk+1 measurable. Consider a mixed termin the above in which j < k

E ( fk∆Mk, f j∆M j) = E(E(

fk∆Mk, f j∆M j|Ftk

))= E

(fk, f j∆M jE

(∆Mk|Ftk

))= 0

because E(M (t ∧ tk+1)−M (t ∧ tk) |Ftk

)= M (t ∧ tk+1∧ tk)−M (t ∧ tk) . Thus

E

(∥∥∥∥∫ t

0f dM

∥∥∥∥2

U

)=

mn−1

∑k=0

E ( fk∆Mk, fk∆Mk)

=mn−1

∑k=0

E(E(( fk∆Mk, fk∆Mk) |Ftk

))(33.4)

=mn−1

∑k=0

E(

f 2k E((∆Mk,∆Mk) |Ftk

))(33.5)

now

E((M (t ∧ tk+1) ,M (t ∧ tk)) |Ftk

)=

(E(M (t ∧ tk+1) |Ftk

),M (t ∧ tk)

)= ∥M (t ∧ tk)∥2

and so

E((∆Mk,∆Mk) |Ftk

)= E

((M (t ∧ tk+1)−M (t ∧ tk) ,M (t ∧ tk+1)−M (t ∧ tk)

)|Ftk

)=

E(∥M (t ∧ tk+1)∥2 |Ftk

)+E

(∥M (t ∧ tk)∥2 |Ftk

)−2E

((M (t ∧ tk+1) ,M (t ∧ tk)) |Ftk

)= E

(∥M (t ∧ tk+1)∥2 |Ftk

)−∥M (t ∧ tk)∥2

Therefore, the right side of 33.5 is

mn−1

∑k=0

E(

f 2k ∥M (t ∧ tk+1)∥2

)−E

(f 2k ∥M (t ∧ tk)∥2

).

Now recall that ∥M (t)∥2 = [M] (t)+N (t) where N (t) is a martingale. It then reduces to

mn−1

∑k=0

E(

f 2k ([M (t ∧ tk+1)]+N (t ∧ tk+1))

)−E

(f 2k ([M] (t ∧ tk)+N (t ∧ tk))

)

884 CHAPTER 33. QUADRATIC VARIATION AND STOCHASTIC INTEGRATIONbecause it is a finite sum of continuous random variables. In the argument to get 33.2, I willwrite M rather than M*’. Then it is understood that you can let / + ce to obtain the desiredformula for a local martingale. Thus, in what follows, M is a bounded martingale.7mn—1 mn—|= (S fi (M (tte) -M(tAte)), YY ft (earin) ter)j=0tLet M(t \te41) —M (t Ath) = AMg. Thus AM, is ¥;,,, measurable. Consider a mixed termin the above in which j < kE (feAMy, f;AM;) = E(E (fiAMg, f;AMj|Fr,))= E (fk, fiAMjE (AM:|F;,)) =0because E (M(t Atey1) —M(t Ate) | Fy.) =M (tA tea Ath) — M (t At). Thuse (| aw2 mn—1YE (FAME, fe AMg)Uk=0mn—1Y EE ((FcAMe, fiAMi) |Fx,)) (33.4)k=0myn—1= Y E(fgE ((AM:,AM;) | Fy.) (33.5)k=0nowE ((M(tAtey1) .M(tAt,)) | Fy) = (E (M(t Atey1)|-Fy,) M(t Atk)= ||M(tAn) |?and so. M(tAtes1)-M(tAt), \) >E ((AMx, AMx) | Fi.) =E(( MGM uM nt) ) Fx) ~E (IM (Ata)? Fu) +E (IM At) |)—2E (M(t \tey1),M (tte) | Fy)= E(IM(Ate)IP| Fy.) = IM (eA) |PTherefore, the right side of 33.5 ismo! 2 2Y E(B MO Atesi)I?) —E (WIM (An) I7)k=0Now recall that || (t) ||? = [M] (t) +N (t) where N (t) is a martingale. It then reduces tomy,—1YE (fe (Mt A test] +N (tA te1))) —E (fe (IM) (tte) +N (tAt)))k=0