Strong Markov property of Brownian motion

For $n\ge 1$ define the dyadic ceiling ${\tau }_n:=2^{-n}\lceil 2^n\tau \rceil$, with ${\tau }_n:=\mathrm{\infty }$ when $\tau =\mathrm{\infty }$. Then $\tau \le {\tau }_n\le \tau +2^{-n}$ wherever $\tau <\mathrm{\infty }$, so ${\tau }_n\downarrow \tau$ almost surely, each ${\tau }_n$ takes values in the countable set $2^{-n}{\mathbb{Z}}_{\ge 0}\cup \{\mathrm{\infty }\}$, and each ${\tau }_n$ is a stopping time for $({\mathcal{F}}_t)$: for $t\ge 0$, $\{{\tau }_n\le t\}=\{\tau \le 2^{-n}\lfloor 2^{n}t\rfloor \}\in {\mathcal{F}}_{2^{-n}\lfloor 2^{n}t\rfloor }\subseteq {\mathcal{F}}_t$. Moreover ${\mathcal{F}}_{\tau }\subseteq {\mathcal{F}}_{{\tau }_n}$ because for $A\in {\mathcal{F}}_{\tau }$ one has $A\cap \{{\tau }_n\le t\}=A\cap \{\tau \le t\}\cap \{{\tau }_n\le t\}\in {\mathcal{F}}_t$.

For a countably valued stopping time $\rho$, the variable $B_{\rho }$ set to zero at infinity is ${\mathcal{F}}_{\rho }$-measurable: its Borel inverse image, intersected with $\{\rho \le t\}$, is ${\bigcup }_{r\le t}(\{\rho =r\}\cap \{B_r\in D\})\in {\mathcal{F}}_t$. For the dyadic ceilings, ${\bigcap }_n{\mathcal{F}}_{{\tau }_n}={\mathcal{F}}_{\tau }$. Indeed, if $A$ belongs to the intersection, then $A\cap \{\tau <t\}={\bigcup }_n(A\cap \{{\tau }_n<t\})\in {\mathcal{F}}_t$; the right-continuous strict-test criterion [F1] proves membership in ${\mathcal{F}}_{\tau }$. Each tail $(B_{{\tau }_n}{)}_{n\ge m}$ is measurable for ${\mathcal{F}}_{{\tau }_m}$ by the inclusion in step 1.1. The normalized finite limit is therefore measurable for every ${\mathcal{F}}_{{\tau }_m}$ by [F9], hence for ${\mathcal{F}}_{\tau }$. The event $\{\tau <\mathrm{\infty }\}$ belongs to each of these stopped sigma-algebras, so the stated zero convention preserves this conclusion. For $u\ge 0$, $B_{{\tau }_n+u}$ is an ambient-measurable countable sum over the values of ${\tau }_n$, and its normalized limit $V_u$ is ambient measurable by [F9]. Thus $V$ and $Z=(V_u-V_0{)}_u$ are random elements of the product measurable space. On the common event of path continuity and finite $\tau$, these limits equal $B_{\tau +u}$ simultaneously for every $u$.

Let $\rho$ be a stopping time taking values in a countable set $D\subseteq [0,\mathrm{\infty })$ with $\rho <\mathrm{\infty }$ almost surely; by the argument of step 2.1 with $\rho$ in place of $\tau$ the values $B_{\rho }$ are ${\mathcal{F}}_{\rho }$-measurable and ${\mathrm{\Psi }}_{\mathrm{\Phi }}(B_{\rho })$ is bounded and ${\mathcal{F}}_{\rho }$-measurable for every bounded Borel functional $\mathrm{\Phi }$. For every $A\in {\mathcal{F}}_{\rho }$, ${\int }_A\mathrm{\Phi }((B_{\rho +u}{)}_{u\ge 0})dP={\int }_A{\mathrm{\Psi }}_{\mathrm{\Phi }}(B_{\rho })dP$: for each $r\in D$ the event $A_r:=A\cap \{\rho =r\}$ lies in ${\mathcal{F}}_r$, and [F3] at time $r$ gives ${\int }_{A_r}\mathrm{\Phi }((B_{r+u}{)}_{u\ge 0})dP={\int }_{A_r}{\mathrm{\Psi }}_{\mathrm{\Phi }}(B_r)dP$; on $A_r$ one has $(B_{\rho +u})=(B_{r+u})$ and $B_{\rho }=B_r$, and summing over the countable set $D$ gives the identity. By [F5]'s uniqueness, $E[\mathrm{\Phi }((B_{\rho +u}))\mathrm{\mid }{\mathcal{F}}_{\rho }]={\mathrm{\Psi }}_{\mathrm{\Phi }}(B_{\rho })$ almost surely.

First let $\mathrm{\Phi }(z)=g(z(u_1),\dots ,z(u_k))$, where $g$ is a bounded continuous function on ${\mathbb{R}}^k$. These cylinder functionals are product-measurable. On the common continuity event, the coordinates $B_{{\tau }_n+u_i}$ tend to $V_{u_i}$, so $\mathrm{\Phi }((B_{{\tau }_n+u}))\to \mathrm{\Phi }(V)$. Further, ${\mathrm{\Psi }}_{\mathrm{\Phi }}$ is continuous: if $x_n\to x$, its integrand $g(x_n+w(u_1),\dots ,x_n+w(u_k))$ converges pointwise and is bounded uniformly, so dominated convergence applies. For $A\in {\mathcal{F}}_{\tau }\subseteq {\mathcal{F}}_{{\tau }_n}$, step 3.1 at $\rho ={\tau }_n$ and dominated convergence therefore give ${\int }_A\mathrm{\Phi }(V)dP={\int }_A{\mathrm{\Psi }}_{\mathrm{\Phi }}(V_0)dP$. Null exceptional events contribute zero to these integrals; no assertion that they belong to the past filtration is used.

For $c\in \mathbb{R}$ and $k\ge 1$ let $g_{k,c}(y):=\max \{0,\min \{1,k(c-y)+1\}\}$; then $g_{k,c}$ is continuous and bounded with $g_{k,c}\downarrow 1_{(-\mathrm{\infty },c]}$ pointwise as $k\to \mathrm{\infty }$. Consequently, for a half-line cylinder $\mathrm{\Gamma }=\{z:z(t_i)\le c_i,i\le m\}$ the functions $G_k(z):={\prod }_{i\le m}{g}_{k,c_i}(z(t_i))$ are bounded, continuous on the product space, and decrease pointwise to $1_{\mathrm{\Gamma }}$. Applying step 4.1 to $G_k$ and passing to the limit with dominated convergence [F5] on both sides, using $G_k((B_{\tau +u}))\downarrow 1_{\mathrm{\Gamma }}((B_{\tau +u}))$ and ${\mathrm{\Psi }}_{G_k}(B_{\tau })\to {\mathrm{\Psi }}_{1_{\mathrm{\Gamma }}}(B_{\tau })$ pointwise, gives ${\int }_A{1}_{\mathrm{\Gamma }}((B_{\tau +u}))dP={\int }_A{\mathrm{\Psi }}_{1_{\mathrm{\Gamma }}}(B_{\tau })dP$ for every $A\in {\mathcal{F}}_{\tau }$.

Let $\mathcal{D}$ be the class of product-measurable sets $\mathrm{\Gamma }$ for which ${\int }_A{1}_{\mathrm{\Gamma }}((B_{\tau +u}))dP={\int }_A{\mathrm{\Psi }}_{1_{\mathrm{\Gamma }}}(B_{\tau })dP$ for every $A\in {\mathcal{F}}_{\tau }$. Then $\mathcal{D}$ is a lambda-system: it contains the whole product space because ${\mathrm{\Psi }}_1={\mathrm{\Psi }}_{\mathrm{\Phi }}$ for $\mathrm{\Phi }\equiv 1$ is the constant $1$; it is closed under complements by subtracting the two finite identities; and it is closed under countable disjoint unions: first add the identities for the first $m$ disjoint sets, then use [F5] to pass to their union by monotone convergence on both sides. By step 5.1 it contains every half-line cylinder, which together with the empty cylinder form a generating pi-system, so [F6] gives $\mathcal{D}$ equal to the whole product sigma-algebra.

For a bounded nonnegative Borel $\mathrm{\Phi }$ with simple functionals $s_m\uparrow \mathrm{\Phi }$, step 6.1 and linearity of the integral give ${\int }_A{s}_m((B_{\tau +u}))dP={\int }_A{\mathrm{\Psi }}_{s_m}(B_{\tau })dP$ for every $A\in {\mathcal{F}}_{\tau }$; monotone convergence [F5] on both sides, using ${\mathrm{\Psi }}_{s_m}(x)\uparrow {\mathrm{\Psi }}_{\mathrm{\Phi }}(x)$ pointwise and the Borel measurability of ${\mathrm{\Psi }}_{\mathrm{\Phi }}$ from [F7], gives ${\int }_A\mathrm{\Phi }((B_{\tau +u}))dP={\int }_A{\mathrm{\Psi }}_{\mathrm{\Phi }}(B_{\tau })dP$. Since ${\mathrm{\Psi }}_{\mathrm{\Phi }}(B_{\tau })$ is bounded and ${\mathcal{F}}_{\tau }$-measurable by step 2.1 and [F7], [F5]'s uniqueness identifies it with $E[\mathrm{\Phi }((B_{\tau +u}))\mathrm{\mid }{\mathcal{F}}_{\tau }]$; splitting a bounded real $\mathrm{\Phi }$ into positive and negative parts extends the identity to all bounded Borel $\mathrm{\Phi }$. This is assertion 2.

For assertion 1, fix a cylinder ${\mathrm{\Gamma }}_0$ and put ${\mathrm{\Phi }}_0(u):=1_{{\mathrm{\Gamma }}_0}((u(t_i)-u(0){)}_{i\le k})$ with ${\mathrm{\Gamma }}_0$ a Borel subset of the finite coordinate space; the coordinate $u(0)$ is the translation offset. For every $x$ one has ${\mathrm{\Phi }}_0(x+w)=1_{{\mathrm{\Gamma }}_0}((w(t_i)-w(0){)}_{i\le k})$, independent of $x$, so ${\mathrm{\Psi }}_{{\mathrm{\Phi }}_0}(x)={\mu }_{\mathrm{c}\mathrm{y}\mathrm{l}}({\mathrm{\Gamma }}_0):=\mu (\{w:(w(t_i)-w(0){)}_{i\le k}\in {\mathrm{\Gamma }}_0\})$ is a constant; step 7.1 gives $P(A\cap \{Z\in {\mathrm{\Gamma }}_0\})={\mu }_{\mathrm{c}\mathrm{y}\mathrm{l}}({\mathrm{\Gamma }}_0)P(A)$ for every $A\in {\mathcal{F}}_{\tau }$, and taking $A=\mathrm{\Omega }$ shows that the finite-dimensional marginals of $Z$ are those of Wiener measure. The class of product-measurable sets satisfying $P(A\cap \{Z\in \mathrm{\Gamma }\})=P(A)P(Z\in \mathrm{\Gamma })$ for all $A\in {\mathcal{F}}_{\tau }$ is a lambda-system containing the cylinder pi-system, hence by [F6] equals the product sigma-algebra, so $Z$ is independent of ${\mathcal{F}}_{\tau }$ and its law on the product sigma-algebra has the finite-dimensional marginals of Wiener measure.

The degenerate cases are consistent with the proof: if $\tau \equiv t$ is deterministic then ${\mathcal{F}}_{\tau }={\mathcal{F}}_t$ by [F1], and step 3.1 reduces to the deterministic future-path theorem [F3]; the null set $\{\tau =\mathrm{\infty }\}$ is handled by the convention $B_{\tau }=0$ there and all identities are asserted almost surely; if $\mathrm{\Phi }$ is constant, then ${\mathrm{\Psi }}_{\mathrm{\Phi }}$ is that constant and both sides agree; a coordinate at time $0$ is the deterministic offset $B_{\tau }$ and was covered in step 8.1; and the case $\mathrm{\Phi }\equiv 0$ gives $0=0$. AC is declared for the Brownian and conditional-expectation interfaces [F8]; no additional selections are made.