The following exercise contains an alternative proof of It's formula (18.1) for a one-dimensional Brownian motion (left(B_{t}

The following exercise contains an alternative proof of Itô's formula (18.1) for a one-dimensional Brownian motion \(\left(B_{t}\right)_{t \geqslant 0}\).

a) Let \(f \in \mathcal{C}^{1}(\mathbb{R})\). Show that for \(\Pi=\left\{t_{0}=0\}\) and suitable intermediate values \(\xi_{l}\)\[\begin{gathered}B_{t} f\left(B_{t}\right)=\sum_{l} B_{t_{l}}\left(f\left(B_{t_{l}}\right)-f\left(B_{t_{l1}}\right)\right)+\sum_{l} f\left(B_{t_{l-1}}\right)\left(B_{t_{l}}-B_{t_{l-1}}\right) \\+\sum_{l} f^{\prime}\left(\xi_{l}\right)\left(B_{t_{l}}-B_{t_{l-1}}\right)^{2} .\end{gathered}\]

Conclude from this that the differential expression \(B_{t} d f\left(B_{t}\right)\) is well-defined.

b) Show that \(d\left(B_{t} f\left(B_{t}\right)\right)=B_{t} d f\left(B_{t}\right)+f\left(B_{t}\right) d B_{t}+f^{\prime}\left(B_{t}\right) d t\).

c) Use (b) to find a formula for \(d B_{t}^{n}, n \geqslant 2\), and show that for every polynomial \(p(x)\) Itô's formula holds:

\[d p\left(B_{t}\right)=p^{\prime}\left(B_{t}\right) d B_{t}+\frac{1}{2} p^{\prime \prime}\left(B_{t}\right) d t .\]

d) Show the following stability result for Itô integrals: \[L^{2}(\mathbb{P})-\lim _{n \rightarrow \infty} \int_{0}^{T} g_{n}(s) d B_{s}=\int_{0}^{T} g(s) d B_{s}\]

for any sequence \(\left(g_{n}\right)_{n \geqslant 1}\) which converges in \(\mathcal{L}_{T}^{2}\).

e) Use a suitable version of Weierstraß' approximation theorem and the stopping technique from the proof of Theorem 18.1, Step \(4^{\circ}\), to deduce from c) Itô's formula for \(f \in \mathcal{C}^{2}\).

Data From 18.1 Theorem 

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18.1 Theorem (It 1942). Let (Bt)to be a BM and let f: RR be a e-function. Then we have for all t > 0 almost surely f(Bt) f(Bo) = [ f'(Bs) dBs + ["(Bs) ds. (18.1) Before we prove Theorem 18.1, we need to discuss It processes and It differentials.