Brownian reflection principle

By [F1] the time ${\tau }_a$ is a stopping time of the usual augmentation and $S={\tau }_a\wedge T$ is a stopping time bounded by the deterministic time $T$, hence a.s. finite; on the event $\{{\tau }_a\le T\}$ one has $S={\tau }_a$ and $B_S=a$ by continuity, while on $\{{\tau }_a>T\}$ one has $S=T$. Also $M_T\ge a⟺{\tau }_a\le T$ by continuity of the path.

The law of any finite-dimensional marginal of $B$ is unchanged by the substitution $B\mapsto -B$, and the cylinder pi-system determines a law on ${\mathbb{R}}^{[0,T]}$; hence the law of $(B_u{)}_{u\le T}$ is invariant under negation, and the same holds for every process whose finite-dimensional marginals are those of Brownian motion, by [F5] and [F3].

Define ${\tilde{B}}_u:=B_u$ for $u\le S$ and ${\tilde{B}}_u:=2a-B_u$ for $S<u\le T$. Every coordinate is measurable by the stopping-time tests. The path is continuous: if $S<T$, then $B_S=a$, and if $S=T$ it is unchanged. For finitely many $u_i\in [0,T]$, put $R=(S,B_{u_1\wedge S},\dots ,B_{u_k\wedge S})$. This is ${\mathcal{F}}_S$-measurable: $S$ is measurable there by its test events, and [F2]'s stopped-value measurability at $u_i\wedge S$, followed by the stopped-sigma-algebra inclusion for $u_i\wedge S\le S$, gives the other coordinates. Since paths are everywhere continuous and these times finite, the version convention in [F2] gives the literal values. The path $W(r)=B_{S+r}-B_S$ is everywhere continuous, has Wiener law and is independent of ${\mathcal{F}}_S$ by [F2], [F7] and coordinate generation. Random evaluation $(s,w)\mapsto w((u_i-s{)}^{+})$ is measurable: replace its time by the ceiling on the dyadic mesh, use the countable coordinate formula, then pass to the pointwise limit by continuity. Hence the vectors $B_{u_i}=B_{u_i\wedge S}+W((u_i-S{)}^{+})$ and ${\tilde{B}}_{u_i}=B_{u_i\wedge S}-W((u_i-S{)}^{+})$ are product-measurable functions of $(R,W)$. These formulas also hold when $S=T$ because $W(0)=0$. For every bounded Borel test of this vector, [F7] and invariance of Wiener measure under $w\mapsto -w$ from step 1.2 give equal conditional expectations. Thus the finite-dimensional laws agree, and [F5] gives equality on the entire cylinder sigma-algebra.

On the event $\{{\tau }_a\le T\}$, the reflected path coincides with $B$ on $[0,{\tau }_a]$, so ${\tau }_a(\tilde{B})={\tau }_a$; moreover ${\tilde{B}}_T=2a-B_T$ there. Hence $\{{\tau }_a\le T,B_T\le b\}=\{{\tau }_a(\tilde{B})\le T,{\tilde{B}}_T\ge 2a-b\}$ as events. These are cylinder-measurable events on the continuous-path representatives: use the rational-distance formula of [F1] for the hit event and the terminal coordinate for the inequality. Thus by step 2.1 the probability equals $P({\tau }_a\le T,B_T\ge 2a-b)$. Since $b\le a$ implies $2a-b\ge a$, the event $\{B_T\ge 2a-b\}$ is contained in $\{B_T\ge a\}\subseteq \{{\tau }_a\le T\}$, so the last probability is $P(B_T\ge 2a-b)$. Together with $M_T\ge a⟺{\tau }_a\le T$ from step 1.1, this is the first identity of assertion 2.

Taking $b=a$ in step 3.1 gives $P(M_T\ge a,B_T\le a)=P(B_T\ge a)$. Since $\{B_T>a\}\subseteq \{M_T\ge a\}$ and $\{B_T>a\}$ differs from $\{B_T\ge a\}$ by the null event $\{B_T=a\}$ by [F4], adding $P(B_T>a)=P(B_T\ge a)$ to both sides gives $P(M_T\ge a)=2P(B_T\ge a)$, the second identity of assertion 2. Assertion 1 is step 2.1.

The endpoint and degenerate cases are covered: the statement requires $a>0$ and $T>0$; the value $b=a$ is used in step 4.1 and is the boundary case of the constraint $b\le a$; the case $b<a$ is step 3.1 unchanged; the case ${\tau }_a>T$ contributes to neither event in the first probability identity, by the containment in step 3.1; the case ${\tau }_a\le T$ with equality $B_T=a$ is the null event excluded by [F4]. AC is used through [F6] for the strong Markov and conditional-expectation interfaces.