I have some problems on reading the proof of following theorem (From Le Gall's Stochastic Calculus Page 250):
First recall the notation of increasing process $L_t^a(X)$ derived from Tanaka's formula.
There exists a process $(L_t^a(B))_{a\in\mathbb{R},t\geqslant 0}$, whose sample paths are continuous functions of the pair $(a,t)$, such that, for every fixed $a\in\mathbb{R}$, $(L_t^a(B))_{t\geqslant 0}$ is an increasing process, and, a.s. for every nonnegative $t$, for every nonnegative measurable function $\phi$ on $\mathbb{R}$, $$\int_0^t\phi(B_s)\text{d}s=\int \phi(a)L_t^a(B)\text{d}a.$$ Furthermore, a.s for every $a\in\mathbb{R}$, $$\text{supp}(\text{d}_sL_s^a(B))\subset \left\{s\geqslant 0:B_s=a\right\},$$ and this inclusion is an equality with probability one if $a$ is fixed.
In the proof of this theorem; I cannot understand the following two parts:
- In one step, we have to verify the inclusion above is an a.s. equality if $a\in\mathbb{R}$ is fixed. So let us fix $a\in\mathbb{R}$, and for every rational $q\geqslant 0$, set $$H_q:=\inf\left\{t\geqslant q:B_t=a\right\}.$$ Our claim will follow if we can verify that a.s. for every $\epsilon>0,L_{H_q+\epsilon}^a(B)> L_{H_q}^a(B)$.
Then I have doubts in the following:
"We can use strong Markov Property at time $H_q$, so it suffices to prove that, if $B'$ is a real Brownian motion started from $a$, we have $L_{\epsilon}^{a}(B')>0$, for every $\epsilon>0$, a.s. "
How did the above claim come out?
- Another question is comes out accordingly, after we take $a=0$, how can we observe that $$L_\epsilon^0(B)\stackrel{d}{=}\sqrt{\epsilon}L_1^0(B)?$$ by an easy scaling argument (quoted from the book)?