in book's Bernt.Øks SDE i read that book and i have some serious issues :( page 21 Example 7.4.2 )
Consider n-dimensional Brownian motion $W=(W_1, \ldots ,W_n)$ starting at $a=(a_1,\ldots,a_n) \in \mathbb{R}^{n}(n ≥ 1)$ and assume $|a|>R , R\in \mathbb{R}^{n}$ Let $\tau_{k}$ be the first exit time from the annulus $$A_{k}=\{x,R<|x|<2^{k}R\}, k=1,2,\ldots$$ and put $$T_{k}=\inf\{t>0,R<|x|<2^{k}R\} $$ let $f=f_{n,k}$ be a $C^{2}(\mathbb{R}^{n})$ function with compact support such that, if $R\leq |x|\leq 2^{k}R$ $$f(x):=\begin{cases} -\log|x| & \text{when } n=2 , \\ |x|^{2-n} & \text{when } n>2. \end{cases}$$ Then,since $\Delta f=0$ in $A_{k}=\{x,R<|x|<2^{k}R\},$ we have by dynkin's formula $$ \mathbb{E}^{x}[f(W_{\tau_k})]=f(a) \forall k $$ my question how we can find
$$\mathbb{E}^{a}[f(W_{\tau_{k}})]=f(R)P^{a}|W_{\tau_{k}}|=R]+f(2^{k}R)P^{a}|W_{\tau_{k}}|=2^{k}R]$$
indeed we know that $f(x)=f(x)1_{R\leq |x|\leq 2^{k}R}$ then $\mathbb{E}^{a}[f(x)]=\mathbb{E}^{a}[f(x)1_{R\leq |x|\leq 2^{k}R}]=\int_{-\infty}^{\infty}f(x)1_{R\leq |x|\leq 2^{k}R} \, dP=\int_{{R\leq |x|\leq 2^{k}R}}f(x) \, dP= ?? $
Please respond I'll be grateful for any help offered!
The equality
$$\mathbb{E}^a[f(W_{\tau_k})] = f(R) \cdot \mathbb{P}^a[|W_{\tau_k}|=R] + f(2^k \cdot R) \cdot \mathbb{P}^a[|W_{\tau_k}|=2^k \cdot R]$$
follows from the fact that
$$|W_{\tau_k}| = R \cdot 1_{\{|W_{\tau_k}|=R\}} + 2^k \cdot R \cdot 1_{\{|W_{\tau_k}|=2^k \cdot R\}}$$
by the continuity of the sample paths and the definition of $\tau_k$ (in particular, $\tau_k<\infty$ a.s.). Note that this equality implies
$$\begin{align*} \mathbb{E}^a[f(W_{\tau_k})] &= \mathbb{E}^a(f(R) \cdot 1_{\{|W_{\tau_k}|=R\}} +f(2^k \cdot R) \cdot 1_{\{|W_{\tau_k}|=2^k \cdot R\}}) \\ &= f(R) \cdot \mathbb{P}^a[|W_{\tau_k}|=R] + f(2^k \cdot R) \cdot \mathbb{P}^a[|W_{\tau_k}|=2^k \cdot R] \end{align*} $$