2306 CHAPTER 67. THE EASY ITO FORMULA

dY = g(X)

(hT dW− 1

2|h|2 dt

)+

12

g(X)

(hT dW− 1

2|h|2 dt

)(hT dW− 1

2|h|2 dt

)= Y

(hT dW− 1

2|h|2 dt

)+

12

Y[(

hT dW)(

hT dW)−hT dW |h|2 dt +

14|h|2 dt2

]Then neglecting the terms of the form dWdt,dt2 and so forth,

dY = Y hT dW−12

Y |h|2 dt +12

Y(hT dW

)(hT dW

)Now the dW occurs twice in the last term so it leads to a dt and you get

dY = Y hT dW−12

Y |h|2 dt +12(Y hT ,hT )dt

dY = Y hT dW−12

Y |h|2 dt +12

Y |h|2 dt

dY = Y hT dW

Note that∥∥hT

∥∥L2(Rn,R) ≡ ∑

nk=1(hT ek

)2= |h|2Rn . Place an

∫ t0 in place of both sides to

obtain

Y (t)−Y (0) =∫ t

0Y hT dW

Y (t) = 1+∫ t

0Y hT dW (67.9.18)

Now here is the interesting part of this formula.

E(∫ t

0Y hT dW

)= 0

because the stochastic integral is a martingale and equals 0 at t = 0.

E(∫ t

0Y hT dW

)= E

(E(∫ t

0Y hT dW|F0

))= 0

ThusE (Y (t)) = 1

and for Y one obtains

Y (t) = E (Y (t))+∫ t

0Y hT dW

≡ E (Y (t))+∫ t

0fT dW

where fT is adapted and square integrable. It is just Y hT where h does not depend on ω

and Y is a function of an adapted function.Does such a function f exist for all F ∈ L2 (Ω,Gt ,P)? The answer is yes and this is the

content of the next theorem which is called the Itô representation theorem.