The Strong Markov Property of Itô Diffusions

Proof.

The switched-off integrand is admissible.
The set \(\{ (s, \omega) : s \lt \tau(\omega) \}\), intersected with \([0, t] \times \Omega\), can be written as

\[ \Bigl( \bigcup_{q \in \mathbb{Q},\, q \leq t} \bigl( [0, q) \cap [0, t] \bigr) \times \{ \tau \geq q \} \Bigr) \cup \bigl( [0, t] \times \{ \tau \gt t \} \bigr) , \]

because \(s \lt \tau\) holds exactly when some rational \(q\) satisfies \(s \lt q \leq \tau\), and any such \(q\) exceeding \(t\) forces \(\tau \gt t\) while \(s \leq t\) already gives \(s \lt q\). Each set \(\{ \tau \geq q \} = \Omega \setminus \{\tau \lt q\}\) lies in \(\mathcal{F}_q \subseteq \mathcal{F}_t\), and \(\{\tau \gt t\} \in \mathcal{F}_t\), so the displayed set lies in \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\): the indicator \(\mathbf{1}_{[0, \tau)}\) is progressively measurable. The product \(\mathbf{1}_{[0, \tau)} f\) therefore satisfies (V1) and (V2), and (V3) holds because the product is dominated by \(|f|\). So \(\mathbf{1}_{[0, \tau)} f \in \mathcal{V}(0, T)\), and the right-hand integral is defined.

A stopping time with finitely many values.
Let \(\rho\) be a stopping time taking finitely many values \(0 \leq a_1 \lt \cdots \lt a_N \leq T\). Pointwise on \([0, T] \times \Omega\),

\[ \mathbf{1}_{[0, \rho)}(s) = 1 - \sum_{i=1}^{N} \mathbf{1}_{\{\rho = a_i\}}\, \mathbf{1}_{[a_i, T]}(s) \quad \text{for } s \in [0, T) , \]

since exactly one of the events \(\{\rho = a_i\}\) occurs and, on it, \(s \geq \rho\) reads \(s \geq a_i\). Each summand lies in \(\mathcal{V}(0, T)\) when multiplied by \(f\): the level \(\mathbf{1}_{\{\rho = a_i\}}\) is \(\mathcal{F}_{a_i}\)-measurable, the factor vanishes before \(a_i\), and the product is dominated by \(|f|\). By linearity,

\[ \int_0^T \mathbf{1}_{[0, \rho)} f \, d w = J_T - \sum_{i=1}^{N} \int_0^T \mathbf{1}_{\{\rho = a_i\}}\, \mathbf{1}_{[a_i, T]}\, f \, d w . \]

Fix \(i\) with \(a_i \lt T\). The integrand of the \(i\)-th term vanishes on \([0, a_i)\), so additivity over intervals reduces its integral to \([a_i, T]\), where the clock-shift lemma converts it to an integral from \(0\) to \(T - a_i\) against the shifted motion \(w_{a_i + v} - w_{a_i}\). Over the shifted filtration \(\{\mathcal{F}_{a_i + v}\}\) the factor \(\mathbf{1}_{\{\rho = a_i\}}\) is measurable at time zero, so the take-out lemma extracts it, and shifting the clock back,

\[ \int_0^T \mathbf{1}_{\{\rho = a_i\}}\, \mathbf{1}_{[a_i, T]}\, f \, d w = \mathbf{1}_{\{\rho = a_i\}} \int_{a_i}^T f \, d w = \mathbf{1}_{\{\rho = a_i\}} \bigl( J_T - J_{a_i} \bigr) , \]

the last step by additivity again. If \(a_i = T\), the integrand is null for the time integral and both sides vanish, so the display holds for every \(i\). Summing and using \(\sum_i \mathbf{1}_{\{\rho = a_i\}} = 1\),

\[ \int_0^T \mathbf{1}_{[0, \rho)} f \, d w = J_T - J_T + \sum_{i=1}^{N} \mathbf{1}_{\{\rho = a_i\}}\, J_{a_i} = J_\rho \quad \text{almost surely.} \]

The dyadic squeeze.
For a general stopping time \(\tau\), the time \(\rho_n = \tau_n \wedge T\) is a stopping time with finitely many values in \([0, T]\): the dyadic discretization \(\tau_n\) of the calculus of stopping times takes grid values, the minimum with the constant \(T\) is a stopping time by the same calculus, and only the grid values below \(T\), together with \(T\) itself, survive. The previous step gives \(J_{\rho_n} = \int_0^T \mathbf{1}_{[0, \rho_n)} f\, dw\) for every \(n\). On the left, \(\rho_n \downarrow \tau \wedge T\) and \(J\) has continuous paths, so \(J_{\rho_n} \to J_{\tau \wedge T}\) almost surely. On the right, the Itô isometry gives

\[ \mathbb{E} \Bigl[ \Bigl( \int_0^T \bigl( \mathbf{1}_{[0, \rho_n)} - \mathbf{1}_{[0, \tau \wedge T)} \bigr) f \, d w \Bigr)^2 \Bigr] = \mathbb{E} \Bigl[ \int_{\tau \wedge T}^{\rho_n} f(s, \omega)^2 \, ds \Bigr] , \]

the two indicators differing only for \(s \in [\tau \wedge T, \rho_n)\). For each \(\omega\) the window shrinks to a point while \(\int_0^T f^2\, ds \lt \infty\) almost surely, so the inner integral tends to zero pathwise by absolute continuity of the time integral, and it is dominated by \(\int_0^T f^2\, ds\), which is integrable by (V3); dominated convergence sends the right side to zero. The integrals on the right of the discrete identity therefore converge in \(L^2(\mathbb{P})\) to \(\int_0^T \mathbf{1}_{[0, \tau \wedge T)} f\, dw\), and along a subsequence almost surely; since \(\mathbf{1}_{[0, \tau \wedge T)}\) and \(\mathbf{1}_{[0, \tau)}\) agree at every \(s \lt T\), the two integrands define the same integral, and the limit identity is the theorem's display.

Mean and variance.
The display represents \(J_{\tau \wedge T}\) as an Itô integral of a member of \(\mathcal{V}(0, T)\), so the zero-mean property applies, and the isometry gives

\[ \mathbb{E} \bigl[ J_{\tau \wedge T}^2 \bigr] = \mathbb{E} \Bigl[ \int_0^T \mathbf{1}_{[0, \tau)}(s)\, f(s, \omega)^2 \, ds \Bigr] = \mathbb{E} \Bigl[ \int_0^{\tau \wedge T} f(s, \omega)^2 \, ds \Bigr] . \]

Proof.

Write \(C_i = \{\rho = a_i\}\), a finite partition of \(\Omega\), with \(C_i \in \mathcal{F}_{a_i}\) and \(C_i \in \mathcal{F}_\rho\), the latter because \(\rho\) is \(\mathcal{F}_\rho\)-measurable. Write \(\tilde{w}^{(i)}_u = w_{a_i + u} - w_{a_i}\) for the motion restarted at the deterministic time \(a_i\); on \(C_i\) the processes \(\tilde{B}\) and \(\tilde{w}^{(i)}\) coincide. All integrals against \(\tilde{B}\) below are Itô integrals over \(( \tilde{B}, \{\mathcal{G}_v\} )\), legitimate because the strong Markov theorem, applied to the stopping time \(\rho\), delivers the two properties of the enlargement lemma for this pair, and the integral theory runs verbatim over any data with these properties; the same applies to \(( \tilde{w}^{(i)}, \{\mathcal{F}_{a_i + u}\} )\), the case the previous page already used.

Admissibility.
\(Z\) has continuous paths because \(X^x\) does. The diffusion is progressively measurable, being continuous and adapted, so item (v) of the calculus of stopping times, applied at the stopping time \(\rho + v\), makes \(Z_v\) measurable with respect to \(\mathcal{G}_v\). For the moments, \(\mathbb{E}[ Z_0^2 ] = \sum_i \mathbb{E}[ \mathbf{1}_{C_i} ( X^x_{a_i} )^2 ] \leq \sum_i \mathbb{E}[ ( X^x_{a_i} )^2 ] \lt \infty\) by the second-moment bound, and for \(u \leq h\) the same decomposition gives \(\mathbb{E}[ Z_u^2 ] \leq \sum_i \mathbb{E}[ ( X^x_{a_i + u} )^2 ]\), bounded uniformly over \(u \in [0, h]\) by the moment bound on the horizon \(a_N + h\). Since \(\sigma\) is Lipschitz, it grows at most linearly, so \(\mathbb{E} \int_0^h \sigma( Z_u )^2\, du \lt \infty\), and \(\sigma( Z_\cdot )\) is continuous and adapted, hence progressively measurable. This proves (i), and the stochastic integral in (ii) is defined.

A trace of measurability.
One bookkeeping fact is used twice below: if \(A \in \mathcal{F}_{a_i + u}\), then \(A \cap C_i \in \mathcal{G}_u = \mathcal{F}_{\rho + u}\). Indeed, for \(t \geq 0\),

\[ ( A \cap C_i ) \cap \{ \rho + u \leq t \} = \begin{cases} A \cap C_i , & a_i + u \leq t , \\\\ \emptyset , & a_i + u \gt t , \end{cases} \]

because \(\rho = a_i\) on \(C_i\); and in the first case \(A \cap C_i \in \mathcal{F}_{a_i + u} \subseteq \mathcal{F}_t\), the factor \(C_i\) lying already in \(\mathcal{F}_{a_i}\). In particular, a process of the form \(\mathbf{1}_{C_i} p\) with \(p\) adapted to \(\{\mathcal{F}_{a_i + u}\}\) is adapted to \(\{\mathcal{G}_u\}\) as well.

Two integrators, one integral.
Fix \(v \gt 0\) and \(i\), and abbreviate \(q_u = \mathbf{1}_{C_i}\, \sigma( X^x_{a_i + u} )\), a member of \(\mathcal{V}(0, v)\) over \(\{\mathcal{F}_{a_i + u}\}\) by the argument of the previous step, and over \(\{\mathcal{G}_u\}\) by the trace fact, the two readings being the same function of \((u, \omega)\). We claim

\[ \int_0^v q_u \, d \tilde{B}_u = \int_0^v q_u \, d \tilde{w}^{(i)}_u \quad \text{almost surely.} \]

By the definition of the Itô integral, the left side is the \(L^2\) limit of the elementary integrals of any sequence of bounded elementary processes converging to \(q\) in the mean-square sense, and likewise the right side over its own filtration. Choose bounded elementary \(p_j\) over \(\{\mathcal{F}_{a_i + u}\}\) with \(p_j \to q\), and set \(q_j = \mathbf{1}_{C_i} p_j\). Then \(q_j \to \mathbf{1}_{C_i} q = q\) in the same sense, each \(q_j\) is bounded and elementary over \(\{\mathcal{F}_{a_i + u}\}\), and by the trace fact its levels are \(\mathcal{G}_{u_j}\)-measurable as well, so \(q_j\) is elementary over \(\{\mathcal{G}_u\}\) with the same partition and levels. The elementary integral of \(q_j\) against \(\tilde{B}\) and against \(\tilde{w}^{(i)}\) is the same random variable: off \(C_i\) every level vanishes and both sums are zero, while on \(C_i\) the increments of the two integrators coincide. The two sides of the claim are therefore \(L^2\) limits of one and the same sequence, and they agree almost surely.

Splicing on one cell.
Fix \(v \gt 0\) and \(i\). Taking out the initially known factor \(\mathbf{1}_{C_i}\), first over \(( \tilde{B}, \{\mathcal{G}_u\} )\) where it is measurable at time zero because \(C_i \in \mathcal{F}_\rho = \mathcal{G}_0\), then over \(( \tilde{w}^{(i)}, \{\mathcal{F}_{a_i+u}\} )\) where \(C_i \in \mathcal{F}_{a_i}\), and using the claim in between:

\[ \begin{align*} \mathbf{1}_{C_i} \int_0^v \sigma( Z_u )\, d \tilde{B}_u &= \int_0^v \mathbf{1}_{C_i}\, \sigma( Z_u )\, d \tilde{B}_u \\\\ &= \int_0^v \mathbf{1}_{C_i}\, \sigma( X^x_{a_i + u} )\, d \tilde{w}^{(i)}_u \\\\ &= \mathbf{1}_{C_i} \int_0^v \sigma( X^x_{a_i + u} )\, d \tilde{w}^{(i)}_u , \end{align*} \]

where the middle equality also used \(\mathbf{1}_{C_i} \sigma( Z_u ) = \mathbf{1}_{C_i} \sigma( X^x_{a_i + u} )\), the two processes agreeing on \(C_i\). The clock-shift lemma, read from the shifted side back to the original clock, turns the last integral into \(\int_{a_i}^{a_i + v} \sigma( X^x_s )\, d w_s\), and the defining identity of the diffusion, evaluated at \(a_i + v\) and at \(a_i\) and subtracted, computes it pathwise:

\[ \int_{a_i}^{a_i + v} \sigma( X^x_s )\, d w_s = X^x_{a_i + v} - X^x_{a_i} - \int_{a_i}^{a_i + v} b( X^x_s )\, ds . \]

On \(C_i\) the right side reads \(Z_v - Z_0 - \int_0^v b( Z_u )\, du\), the time integral rewritten by the deterministic substitution \(s = a_i + u\), path by path. Combining,

\[ \mathbf{1}_{C_i} \int_0^v \sigma( Z_u )\, d \tilde{B}_u = \mathbf{1}_{C_i} \Bigl( Z_v - Z_0 - \int_0^v b( Z_u )\, du \Bigr) . \]

Pasting the cells.
Summing the last display over the finitely many \(i\) and using \(\sum_i \mathbf{1}_{C_i} = 1\) gives, for each fixed \(v\),

\[ Z_v = Z_0 + \int_0^v b( Z_u )\, du + \int_0^v \sigma( Z_u )\, d \tilde{B}_u \quad \text{almost surely,} \]

and \(Z_0 = X^x_\rho\). Every term is continuous in \(v\): the left side and the time integral pathwise, the stochastic integral by taking its continuous version, which the theory over \(( \tilde{B}, \{\mathcal{G}_v\} )\) provides. Agreement at every rational \(v\) on a common full-probability event therefore upgrades to agreement simultaneously for every \(v \geq 0\). This proves (ii).

Proof.

The discrete case.
Let \(\rho\) be a stopping time with finitely many values, and let \(\tilde{B}\), \(\{\mathcal{G}_v\}\), and \(Z_v = X^x_{\rho+v}\) be as in the restart theorem, which exhibits \(Z\) as a solution over \(( \tilde{B}, \{\mathcal{G}_v\} )\) with the \(\mathcal{G}_0\)-measurable, square-integrable initial value \(X^x_\rho\). The assembly that the Markov property page ran at a deterministic time now runs verbatim at \(\rho\), because each of its three inputs is available. First, every \(\mathcal{F}_\rho\)-measurable start is independent of the entire restarted motion, by clause (ii) of the strong Markov theorem for the noise, so the existence and uniqueness theory over \(( \tilde{B}, \{\mathcal{G}_v\} )\) applies in its literal reading. Second, running the dyadic-snapping construction of the measurable version over the restarted noise, on solutions built over the smaller filtration generated by \(\tilde{B}\) and the null sets, produces a map \(\tilde{\Phi}_h\) measurable with respect to \(\mathcal{B}(\mathbb{R}) \otimes \mathcal{H}\), where \(\mathcal{H} = \sigma( \tilde{B}_v : v \geq 0 ) \vee \mathcal{N}\) is the restarted noise augmented by the null sets, with \(\tilde{\Phi}_h( y, \cdot )\) a version of the solution from the constant start \(y\) at time \(h\); and \(\mathcal{H}\) is independent of \(\mathcal{F}_\rho\): clause (ii) gives independence of the restarted noise itself, and adjoining the null sets changes no probabilities, by the same symmetric-difference argument the earlier track pages ran for their augmented \(\sigma\)-algebras. Third, with independence and the measurable version in hand, the substitution lemma applies verbatim over the restarted data and identifies \(Z_h = \tilde{\Phi}_h( X^x_\rho, \cdot )\) almost surely.

The freezing lemma, applied with \(\mathcal{G} = \mathcal{F}_\rho\), the independent \(\mathcal{H}\) above, the \(\mathcal{F}_\rho\)-measurable variable \(X^x_\rho\), and the bounded \(\mathcal{B}(\mathbb{R}) \otimes \mathcal{H}\)-measurable function \(\varphi( y, \omega ) = f( \tilde{\Phi}_h( y, \omega ) )\), gives

\[ \mathbb{E} \bigl[ f( X^x_{\rho + h} ) \bigm| \mathcal{F}_\rho \bigr] = g( X^x_\rho ) , \quad g( y ) = \mathbb{E} \bigl[ f( \tilde{\Phi}_h( y, \cdot ) ) \bigr] . \]

It remains to identify \(g\) with \(u\). For fixed \(y\), the variable \(\tilde{\Phi}_h( y, \cdot )\) is almost surely the time-\(h\) value of the solution from the constant start \(y\) over \(( \tilde{B}, \{\mathcal{G}_v\} )\). That solution and the diffusion \(X^y\) are both solutions of the same equation from the same deterministic start, each over driving data with the two properties of the enlargement lemma, so each qualifies under the weak-solution checklist, and uniqueness in law makes their time-\(h\) values equal in distribution. Hence \(g( y ) = \mathbb{E}[ f( X^y_h ) ] = u( y )\), and the discrete case is proved.

Continuity of \(u\) for continuous \(f\).
Let \(f\) be bounded and continuous and let \(y_j \to y\). The mean-square stability bound of the Markov property page gives \(\mathbb{E}[ | X^{y_j}_h - X^y_h |^2 ] \leq 3\, | y_j - y |^2\, e^{A h} \to 0\). From any subsequence, select a further subsequence along which the bound is summable. Then Chebyshev's inequality makes the probability of a deviation beyond any fixed tolerance summable along it, and Borel–Cantelli, applied at a countable family of tolerances shrinking to zero, gives \(X^{y_j}_h \to X^y_h\) almost surely along it, so continuity of \(f\), boundedness, and dominated convergence give \(u( y_j ) \to u( y )\) along it. Every subsequence of the bounded real sequence \(u( y_j )\) therefore has a further subsequence converging to \(u( y )\), which forces convergence of the full sequence: \(u\) is continuous.

The dyadic squeeze, for bounded \(\tau\) and continuous \(f\).
Suppose \(\tau \leq m\) for a constant \(m\), and let \(\tau_n\) be its dyadic discretization, a stopping time with finitely many values below \(m + 1\). Fix \(A \in \mathcal{F}_\tau\); since \(\tau \leq \tau_n\), the monotonicity clause of the calculus places \(A\) in \(\mathcal{F}_{\tau_n}\). Integrating the discrete case at \(\tau_n\) over \(A\),

\[ \mathbb{E} \bigl[ \mathbf{1}_A\, f( X^x_{\tau_n + h} ) \bigr] = \mathbb{E} \bigl[ \mathbf{1}_A\, u( X^x_{\tau_n} ) \bigr] . \]

As \(n \to \infty\), \(\tau_n \downarrow \tau\) and the paths of the diffusion are continuous, so \(X^x_{\tau_n + h} \to X^x_{\tau + h}\) and \(X^x_{\tau_n} \to X^x_\tau\) almost surely. Both \(f\) and \(u\) are bounded and continuous, so dominated convergence passes the display to the limit:

\[ \mathbb{E} \bigl[ \mathbf{1}_A\, f( X^x_{\tau + h} ) \bigr] = \mathbb{E} \bigl[ \mathbf{1}_A\, u( X^x_\tau ) \bigr] \quad \text{for every } A \in \mathcal{F}_\tau . \]

The truncation squeeze, for general \(\tau\) and continuous \(f\).
For general \(\tau \lt \infty\) almost surely, the truncation \(\tau \wedge m\) is a bounded stopping time. If \(A \in \mathcal{F}_\tau\), then \(A \cap \{ \tau \leq m \} \in \mathcal{F}_{\tau \wedge m}\): for \(t \lt m\) the intersection with \(\{ \tau \wedge m \leq t \}\) is \(A \cap \{ \tau \leq t \} \in \mathcal{F}_t\), and for \(t \geq m\) it is \(A \cap \{ \tau \leq m \}\) itself, which lies in \(\mathcal{F}_m \subseteq \mathcal{F}_t\) by the definition of \(\mathcal{F}_\tau\). On \(\{ \tau \leq m \}\) the times \(\tau\) and \(\tau \wedge m\) coincide, so the previous step, applied at \(\tau \wedge m\) with the event \(A \cap \{ \tau \leq m \}\), reads

\[ \mathbb{E} \bigl[ \mathbf{1}_{A \cap \{\tau \leq m\}}\, f( X^x_{\tau + h} ) \bigr] = \mathbb{E} \bigl[ \mathbf{1}_{A \cap \{\tau \leq m\}}\, u( X^x_\tau ) \bigr] . \]

Since \(\tau \lt \infty\) almost surely, the indicators increase to \(\mathbf{1}_A\) as \(m \to \infty\), and both integrands are bounded, so dominated convergence removes the truncation: the display of the previous step holds for every almost surely finite stopping time and every bounded continuous \(f\).

From continuous to Borel \(f\).
Fix \(A \in \mathcal{F}_\tau\) and consider the two set functions

\[ \begin{align*} \mu_1( B ) &= \mathbb{E} \bigl[ \mathbf{1}_A\, \mathbf{1}_{\{ X^x_{\tau + h} \in B \}} \bigr] , \\\\ \mu_2( B ) &= \mathbb{E} \bigl[ \mathbf{1}_A\, u_{\mathbf{1}_B}( X^x_\tau ) \bigr] , \end{align*} \]

for Borel \(B \subseteq \mathbb{R}\), where \(u_{\mathbf{1}_B}( y ) = \mathbb{P}[ X^y_h \in B ]\). Both are finite measures: \(\mu_1\) visibly, and \(\mu_2\) because each \(u_{\mathbf{1}_B}\) is bounded Borel, countable additivity holding pointwise in \(y\) and passing through the outer expectation by monotone convergence. For a closed set \(F\) with distance function \(d_F\), the continuous approximants \(\varphi_j( z ) = \max( 0,\; 1 - j\, d_F( z ) )\) decrease to \(\mathbf{1}_F\), the device the previous page used for its own extension; dominated convergence on both sides of the display, and inside \(u\), shows \(\mu_1( F ) = \mu_2( F )\). Closed sets are stable under intersection and generate the Borel \(\sigma\)-algebra, so the two measures coincide, by the monotone-class step this track takes on faith. The display therefore holds for \(f = \mathbf{1}_B\), hence for simple \(f\) by linearity, hence for every bounded Borel \(f\) by uniform approximation with simple functions, the error on both sides being at most \(\| f - s \|_\infty\).

The conditional form.
For bounded Borel \(f\), the function \(u\) is bounded and Borel, and \(X^x_\tau\) is \(\mathcal{F}_\tau\)-measurable by the calculus, the diffusion being progressively measurable. The variable \(u( X^x_\tau )\) is therefore \(\mathcal{F}_\tau\)-measurable, bounded, and satisfies the defining integral identity of the conditional expectation of \(f( X^x_{\tau + h} )\) given \(\mathcal{F}_\tau\) against every \(A \in \mathcal{F}_\tau\). This is the theorem.

Proof.

Induction on \(k\), the case \(k = 1\) being the theorem above. Let \(k \geq 2\), abbreviate \(P = \prod_{i=1}^{k-1} f_i( X^x_{\tau + h_i} )\), and set \(u_k( y ) = E^y[ f_k( X_{h_k - h_{k-1}} ) ]\), bounded and Borel. Each \(X^x_{\tau + h_i}\) is measurable with respect to \(\mathcal{F}_{\tau + h_i} \subseteq \mathcal{F}_{\tau + h_{k-1}}\), so \(P\) is \(\mathcal{F}_{\tau + h_{k-1}}\)-measurable and bounded. The tower property over \(\mathcal{F}_\tau \subseteq \mathcal{F}_{\tau + h_{k-1}}\), then taking out what is known, then the single-time theorem at the almost surely finite stopping time \(\tau + h_{k-1}\) with the function \(f_k\) and the gap \(h_k - h_{k-1}\), give

\[ \begin{align*} \mathbb{E} \Bigl[ P\, f_k( X^x_{\tau + h_k} ) \Bigm| \mathcal{F}_\tau \Bigr] &= \mathbb{E} \Bigl[ P\; \mathbb{E} \bigl[ f_k( X^x_{\tau + h_k} ) \bigm| \mathcal{F}_{\tau + h_{k-1}} \bigr] \Bigm| \mathcal{F}_\tau \Bigr] \\\\ &= \mathbb{E} \Bigl[ P\; u_k( X^x_{\tau + h_{k-1}} ) \Bigm| \mathcal{F}_\tau \Bigr] . \end{align*} \]

The last conditional expectation involves \(k - 1\) bounded Borel functions of the positions at \(\tau + h_1, \ldots, \tau + h_{k-1}\), the final one being the product \(f_{k-1}\, u_k\). The induction hypothesis evaluates it as \(w( X^x_\tau )\), where

\[ w( y ) = \mathbb{E} \Bigl[ \prod_{i=1}^{k-2} f_i( X^y_{h_i} ) \cdot ( f_{k-1}\, u_k )( X^y_{h_{k-1}} ) \Bigr] . \]

Finally, the same two moves at the deterministic clock identify \(w\) with the right side of the theorem: for each \(y\), the Markov property at time \(h_{k-1}\) with gap \(h_k - h_{k-1}\) gives \(\mathbb{E}[ f_k( X^y_{h_k} ) \mid \mathcal{F}_{h_{k-1}} ] = u_k( X^y_{h_{k-1}} )\), and the tower property together with taking out the \(\mathcal{F}_{h_{k-1}}\)-measurable factor \(\prod_{i \leq k-1} f_i( X^y_{h_i} )\) turn \(w( y )\) into \(\mathbb{E} \bigl[ \prod_{i=1}^{k} f_i( X^y_{h_i} ) \bigr]\). Measurability of the right side in \(y\) comes from the same reduction, which exhibits the product expectation as finitely many nested applications of maps of the form \(f \mapsto u\), each preserving bounded Borel functions.