Let (left(N_{t}, mathscr{F}_{t} ight)_{t geqslant 0}) be a continuous, real-valued local martingale and (u in mathcal{C}^{2}(mathbb{R})). Show

Let \(\left(N_{t}, \mathscr{F}_{t}\right)_{t \geqslant 0}\) be a continuous, real-valued local martingale and \(u \in \mathcal{C}^{2}(\mathbb{R})\). Show the following Itô formula \(d u\left(N_{t}\right)=u^{\prime}\left(N_{t}\right) d N_{t}+\frac{1}{2} u^{\prime \prime}\left(N_{t}\right) d\langle Nangle_{t}\).

Mimic the proof of Theorem 18.7, stopping everything at a suitable localizing sequence \(\tau_{n}\), and observing that \(\langle Nangle=\mathrm{ucp}-\lim _{\Pi} \sum_{t_{j}, t_{j-1} \in \Pi}\left(N_{t_{j}}-N_{t_{j-1}}\right)^{2}\), cf. Exercise 17.3.

Data From Theorem 18.7

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18.7 Theorem (It 1942). Let X, be a one-dimensional It process in the sense of Defint- tion 18.2 such that dX+= (t) dB+ + b(t) dt and let f: RR be a e-function. Then we have for all to almost surely = (X;) f(Xo) = ' (Xs)o(s) dBs + ' (Xs)b(s) ds + ;)b(s) ds + {" (X,)o(s) ds (X,) dX; + } {[{ (X,) o(s) ds. = f'(Xs) (18.13) Proof. 1 As in the proof of Theorem 18.1 it is enough to show (18.13) locally for t < T(e) where T(e) = inf{s>0: |xs|>l}. This means that we can assume, without loss of generality, that floo + f' loo+f" lo+ + b < C <00. 2 Lemma 18.5 shows that we can approximate and b by elementary processes and b. Denote the corresponding It process by X". Since fee, the following limits are uniform in probability: XII x, f(x). f(x).