Solution of Dirichlet Problem with Itô's formula with Brownian Motion

Question

I have two questions to the application of Itô's formula characterizing the solution to the Dirichlet problem? (Here I put a picture with the two underlined points I'm referring to.)

1. Why is the term $\sum_{i=1}^d \int_0^t \partial_i u(B_s)dB_s^i$ a bounded martingale starting in 0?
2. Why is $u(x)=E_x[\phi(B_{\tau_G})]$?

I would be really, really thankful for answers!!

Answer 1

1. A stochastic integral (from $0$ to $t$) of a local martingale with respect to a Brownian Motion is, by construction, a local martingale starting in $0$. Now, a bounded local martingale $X_t$is an actual martingale by dominated convergence (pick a localising sequence $(T_n)_{n\in \mathbb{N}},$ then $X_{t\wedge T_n}\to X_t$ in $L^1$ as $n\to\infty$, and the corresponding Markov Property of $t\mapsto X_{t\wedge T_n}$ then ascends to $t\mapsto X_t$).

   Now, you can apply the Optional Sampling Theorem to get 2.

Answer 2

Hint: There is a typo in these notes. Instead of $t$, it should say $\tau_G$ in the limit of integration. The stochastic integral (with $t \wedge \tau_G$ as the limit of integration, that is) is automatically a local martingale starting in 0. It is bounded by the definition of $\tau_G$ and the facts that $G$ is bounded and $u \in C^2(\overline{G})$. Hence, it is also a global martingale. Altogether this yields that its expectation is always 0. For your second question, note that $E_x(u(B_0))=u(x)$ and simply rearrange the equation.