Show that Dynkin's formula (7.30) in Proposition 7.31 follows from Theorem 7.30. Data From Theorem 7.30 Ex.

Show that Dynkin's formula (7.30) in Proposition 7.31 follows from Theorem 7.30.

Data From  Theorem 7.30

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Ex. 7.18 7.30 Theorem. Let (Xt, F)to be a Feller process on Rd with transition semigroup (P+)+20 and generator (A, D(A)). Then is an M := u(x,) = u(x) - Au(x,) dr_ for all u D(A) martingale. Proof. Let u = D(A), xe Rd and s, t> 0. By the Markov property (6.4c), s+t s+t | E (Mt | Ft) = E ( u(Xs,t) = u(x) [ Au(x,) dr | Ft) - s+t 0 () = Ex u(Xs) - u(x) - Au(x,) dr - Ex ( - AMX ) dr - B ("|" Au(x) dr | 5) = Psu(x) = u(x) - - t 0 0 Au(x,) dr - EX -E ( S 0 Au(X,) dr = Psu(x) = u(x) - Au(x,) dr - [EX (Au(x,)) dr 0 t 0 S = Psu(X+) - u(x) - [Au(x,) dr-[P,Au(X+) dr Pu(X)-u(X) by (7.11c)