Kenneth Kuttler

2302 CHAPTER 67. THE EASY ITO FORMULA

Then

νn (B) =m

∑k=1

∫Ω

XDkXB (Y)dP =m

∑k=1

dkµk (B)

Hence ∫Ω

g(Y)sndP =∫

Ω

g(Y)m

∑k=1

dkXDk dP =m

∑k=1

∫Rp

g(y)dµk

=∫Rp

g(y)m

∑k=1

dkdµk =∫Rp

g(y)dνn

Then let sn (ω) ↑ X (ω) . Clearly νn ≪ ν and so by the Radon Nikodym theorem dνn =hndν . Then by the monotone convergence theorem, for any B Borel in Rp,∫

Bhndν = νn (B)≡

∫Ω

sn (ω)XB (Y(ω))dP ↑∫

Ω

X (ω)XB (Y(ω))dP≡ ν (B)

Thus for each B Borel, 0≤ hn ≤ 1 and∫B

hndν → ν (B)

and so hn ↑ 1 ν a.e. Thus, from the above,∫Ω

g(Y)sndP =∫Rp

g(y)dνn =∫Rp

g(y)hn (y)dν

It follows from the monotone convergence theorem that one can pass to a limit in theabove and obtain ∫

Ω

g(Y)XdP =∫Rp

g(y)dν

Note that the same conclusion will hold if the functions are suitably integrable withoutany restriction on the sign. In particular, this will hold if g(y) is bounded. One just con-siders positive and negative parts of real and imaginary parts of g and applies the abovelemma.

LetGt ≡ σ (W(s) : s≤ t)

thus the normal filtration for the Wiener process and the Ito integral and so forth is

Ft = ∩s>tGs

Lemma 67.9.3 Let h be a deterministic step function of the form

h =m−1

∑i=0

aiX[ti,ti+1), tm = t

Then for h of this form, linear combinations of functions of the form

exp(∫ t

0hT dW− 1

∫ t

0h ·hdτ

)(67.9.14)

are dense in L2 (Ω,Gt ,P) for each t.

2302 CHAPTER 67. THE EASY ITO FORMULAThen in mnvn (B) = Y¥ dy L 2p, Be (W)dP = ¥° dybty (B)k=l k=lHence[eCOndr = [2 Yd dr= Yi ad [,2(9) die[8X deta = [904%Then let s,(@) + X (@). Clearly v, < v and so by the Radon Nikodym theorem dv, =h,dv. Then by the monotone convergence theorem, for any B Borel in R’,[imav =vn(B) = [50 () 2a (¥())aPt [x (o) 29 (¥ (w)) dP = vB)Thus for each B Borel, 0 < h, < 1 and| h,dv > v(B)and so h, + 1 v a.e. Thus, from the above,[soar =f svdvn= [ e(y) hn y)avIt follows from the monotone convergence theorem that one can pass to a limit in theabove and obtain[emxar= | eoav tNote that the same conclusion will hold if the functions are suitably integrable withoutany restriction on the sign. In particular, this will hold if g(y) is bounded. One just con-siders positive and negative parts of real and imaginary parts of g and applies the abovelemma.LetG =o (W(s):5<t)thus the normal filtration for the Wiener process and the Ito integral and so forth isF; = sot 4%Lemma 67.9.3 Leth be a deterministic step function of the formm—1h= L aj 2irjt.41) tm =Fi=0UThen for h of this form, linear combinations of functions of the formt 1 texp ( | h’ dW - = [ h-hat) (67.9.14)0 2 Joare dense in L? (Q,4,,P) for each t.