Prove Theorem 5.6 using It's formula. Data From Theorem 5.6 5.6 Theorem. Let (Bt, Ft)to be a

Prove Theorem 5.6 using Itô's formula.

Data From Theorem 5.6

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5.6 Theorem. Let (Bt, Ft)to be a d-dimensional Brownian motion and assume that the Ex. 5.10 function fee.2 ((0, co) x Rd, R) ne([0, o) x Rd, R) satisfies (5.5). Then - M{ := f(t, B,) f(0, Bo) | Lf(r, B,) dr, t0, is an F, martingale. Here Lf(t, x) = f(t, x)+4xf(t, x). (5.6) Proof. Observe that M-M-f(t, B.)-f(s, Bs)- Lfir, Br) dr for Os 0, and observe that B -B - B-s. Since fee.2 ((0, co) x Rd), the shifted function satisfies (5.5) on [0, T] x Rd where we can take C = c(t)=y and the constant y depends only on s, T and z. We have to show that for all z R E(MP-M) = E ((t-s, B.-)-(0, Bo) - [L(r, B.,) dr) = 0.