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

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

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 \(\mathcal{F}_q \subseteq \mathcal{F}_t\), the set \(\{\tau \lt q\} = \bigcup_m \{\tau \leq q - 1/m\}\) being a countable union of sets in \(\mathcal{F}_q\), and \(\{\tau \gt t\} \in \mathcal{F}_t\), so the displayed set lies in \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\). Hence the indicator \(\mathbf{1}_{[0, \tau)}\) is progressively measurable.

The product \(\mathbf{1}_{[0, \tau)} f\) is therefore adapted, and measurable on \([0, \infty) \times \Omega\) as the union over \(t \in \mathbb{N}\) of its restrictions, so (V1) and (V2) hold. Condition (V3) holds because the product is dominated by \(|f|\). So \(\mathbf{1}_{[0, \tau)} f\) lies in the class \(\mathcal{V}(0, T)\), 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 \(f\), because 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 \([a_i, T]\). There the clock-shift lemma converts it to an integral from \(0\) to \(T - a_i\) against the shifted motion \(w_{a_i + v} - w_{a_i}\), the same lemma placing the shifted integrand in \(\mathcal{V}(0, T - a_i)\) over the shifted filtration \(\{\mathcal{F}_{a_i + v}\}\). Over that filtration the factor \(\mathbf{1}_{\{\rho = a_i\}}\) is measurable at time zero, and its product with the shifted \(f\), itself admissible by the same lemma applied to \(f\) alone, is that same integrand. The take-out lemma, available over the shifted pair like the rest of the integral theory, therefore extracts the factor, and shifting the clock back, we obtain

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

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 \(\sum_i \mathbf{1}_{\{\rho = a_i\}} = 1\),

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

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 integration windows decrease to a set of at most one point, which has Lebesgue measure zero, while \(f(\cdot, \omega)^2\) is integrable on \([0, T]\) almost surely, so the inner integral tends to zero pathwise, and it is dominated by \(\int_0^T f^2\, ds\), which is integrable because \(f\) belongs to the class. 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 \(X := \int_0^T \mathbf{1}_{[0, \tau \wedge T)} f\, dw\).

Since \(J_{\rho_n} \to J_{\tau \wedge T}\) almost surely, Fatou's lemma applied to the nonnegative variables \(( J_{\rho_n} - X )^2\) gives \[ \begin{align*} \mathbb{E} \bigl[ ( J_{\tau \wedge T} - X )^2 \bigr] &\leq \liminf_n \mathbb{E} \bigl[ ( J_{\rho_n} - X )^2 \bigr] \\\\ &= 0 , \end{align*} \] so \(J_{\tau \wedge T} = X\) 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 identity of the theorem follows.

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

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

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 property of Brownian motion, 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 clock-shift lemma already ran on.

Admissibility.
\(Z\) has continuous paths because \(X^x\) does. The diffusion is progressively measurable, being continuous and adapted, by the criteria for progressive measurability, 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 the process \(u \mapsto \sigma( Z_u )\) is continuous and adapted, hence progressively measurable by the same criteria. 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 \(C_i\). 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 \(\mathcal{F}_{a_i}\). Taking \(A = \Omega\) puts \(C_i\) itself in \(\mathcal{G}_u\), so \(\mathbf{1}_{C_i}\) is \(\mathcal{G}_u\)-measurable, and 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 measurable with respect to the corresponding \(\mathcal{G}_u\) 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, because 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 \(i\). We take out the initially known factor \(\mathbf{1}_{C_i}\) twice, first over \(( \tilde{B}, \{\mathcal{G}_u\} )\), where \(C_i \in \mathcal{F}_\rho = \mathcal{G}_0\), then over \(( \tilde{w}^{(i)}, \{\mathcal{F}_{a_i+u}\} )\), where \(C_i \in \mathcal{F}_{a_i}\), and use the claim in between. The products below lie in the relevant class because they are dominated by admissible integrands and the factor is measurable at time zero. Together these steps give

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

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. Each fixed \(v\) excludes a null event, and countably many rational \(v\) exclude their union. Adding the null event where the continuous version fails to be continuous leaves one full-probability event on which agreement holds at every rational \(v\) and extends by continuity to 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 again at \(\rho\), each of its three inputs being available in the restarted setting. First, every \(\mathcal{F}_\rho\)-measurable start is \(\mathcal{G}_0\)-measurable, since \(\mathcal{G}_0 = \mathcal{F}_\rho\), and the existence proof and the uniqueness half, which rest only on the two properties of the carrying pair and on a square-integrable initial value known at time zero, run over \(( \tilde{B}, \{\mathcal{G}_v\} )\) for every such start.

Second, we run 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. It 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 \(h\). The \(\sigma\)-algebra \(\mathcal{H}\) is independent of \(\mathcal{F}_\rho\). Clause (ii) of the strong Markov property of Brownian motion gives independence of the restarted noise itself, and adjoining the null sets changes no probabilities, since any set of the join differs from a set of \(\sigma( \tilde{B}_v : v \geq 0 )\) by a null symmetric difference, which no probability sees.

Third, pathwise uniqueness over the restarted data identifies \(Z\) with the solution from \(X^x_\rho\) of the substitution lemma, and the lemma, applied verbatim there with independence and the measurable version in hand, gives \(Z_h = \tilde{\Phi}_h( X^x_\rho, \cdot )\) 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

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

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 as a weak solution with initial law \(\delta_y\), 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, its constant \(A\) read on a horizon \(T \geq h\) with \(T \gt 0\), gives \[ \mathbb{E} \bigl[ | X^{y_j}_h - X^y_h |^2 \bigr] \leq 3\, | y_j - y |^2\, e^{A h} \to 0 . \]

From any subsequence, select a further subsequence along which the bound is summable. Along it the series \(\sum_j | X^{y_j}_h - X^y_h |^2\) has finite expectation, the sum and the expectation exchanging by monotone convergence, so the series is finite almost surely and its terms tend to zero. Thus \(X^{y_j}_h \to X^y_h\) almost surely along the subsequence, and 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. Hence \(u\) 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 \(t \lt m\), its 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 applied along that increasing family removes the truncation. The display of the previous step therefore 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 already used to extend the restart of the noise beyond continuous tests. 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 \(\{\tau \lt \infty\}\) is an event of full probability lying in \(\mathcal{F}_\tau\), on which \(X^x_\tau\) is measurable by the calculus, the diffusion being progressively measurable. Setting the variable to zero off that event changes nothing almost surely, so the variable \(u( X^x_\tau )\) is \(\mathcal{F}_\tau\)-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 \(\tau + h_i\) is a stopping time by the calculus, and \(X^x_{\tau + h_i}\) is measurable with respect to \(\mathcal{F}_{\tau + h_i} \subseteq \mathcal{F}_{\tau + h_{k-1}}\) on the full-probability event \(\{\tau \lt \infty\}\) as before, 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 the product over \(i \leq k-2\) is empty when \(k = 2\),

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

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.