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Conditional Laws and Disintegration · Tutorial 275 of 946

Conditional Laws on Path Spaces

Build conditional path laws from measurable evaluation maps, and see how endpoint and finite-time information restricts paths to the corresponding fibers.

Advanced 11 min read

What You'll Learn

  • State the regular conditional distribution of a path given a measurable endpoint or evaluation map.
  • Identify the fiber on which a conditional path law is concentrated almost surely.
  • Construct a Brownian bridge kernel by separating the endpoint from an independent Gaussian residual.
  • Condition Brownian paths on finitely many evaluations using independent segment bridges.
  • Interpret degenerate conditional path laws and distinguish supported fibers from null-set versions.

Conditioning a Whole Path

A path is a single random element, even though it records values at many times. Conditioning on its endpoint or on a finite set of evaluations therefore means conditioning a random element of a path space, not conditioning each coordinate separately. The appropriate object is a probability law on paths, indexed by the observed value.

For example, if a continuous path is observed to end at \(y\), its conditional law should assign probability one to paths that end at \(y\). This restriction describes the relevant fiber of the endpoint map. It does not, by itself, specify how probability is distributed among the paths in that fiber.

Path Spaces and Evaluation Maps

A measurable space is standard Borel if it is measurably isomorphic to a Borel subset of a complete separable metric space. Standard Borel spaces include Polish spaces with their Borel sets. The standard Borel assumption is important: it ensures that regular conditional distributions exist for random elements taking values in these spaces.

Examples of standard Borel path spaces include \(C([0,T],\mathbb{R}^d)\), equipped with its Borel sets for the uniform topology, and the usual Skorokhod path space \(D([0,T],\mathbb{R}^d)\). Evaluation at a fixed time \(t\), the map \(\omega\mapsto\omega(t)\), is measurable on either space. A finite collection of evaluations is also measurable. We will use \(\mathsf P\) for a standard Borel path space and \(\mathsf Y\) for a standard Borel observation space.

Definition: A probability kernel \(K\) from \(\mathsf Y\) to \(\mathsf P\) assigns a probability measure \(K(y,\cdot)\) on \(\mathsf P\) to each \(y\in\mathsf Y\), with \(y\mapsto K(y,A)\) measurable for every Borel set \(A\subseteq\mathsf P\). If \(X\) is a \(\mathsf P\)-valued random element and \(Y=e(X)\) for a measurable map \(e:\mathsf P\to\mathsf Y\), a regular conditional distribution of \(X\) given \(Y\) is such a kernel satisfying
$$ \mathbb{P}(X\in A,\ Y\in B) = \int_B K(y,A)\,\mathbb{P}_Y(dy) $$

for every Borel \(A\subseteq\mathsf P\) and \(B\subseteq\mathsf Y\). Here \(\mathbb{P}_Y\) denotes the law of \(Y\). Equivalently, \(K(Y,A)\) is a version of the conditional expectation of \(\mathbf{1}_{\{X\in A\}}\) given \(Y\).

The fiber associated with \(y\) is \(F_y=e^{-1}(\{y\})\). For endpoint conditioning on continuous paths, \(e(\omega)=\omega(T)\), so \(F_y\) is the set of paths ending at \(y\). For finite evaluations at times \(t_1,\ldots,t_m\), the observation map is \(e(\omega)=(\omega(t_1),\ldots,\omega(t_m))\), and its fiber imposes all \(m\) observed values at once.

Existence and Concentration on Fibers

Theorem: Let \(\mathsf P\) and \(\mathsf Y\) be standard Borel spaces, let \(X\) have a probability law on \(\mathsf P\), and let \(e:\mathsf P\to\mathsf Y\) be measurable. There exists a regular conditional distribution \(K\) of \(X\) given \(e(X)\). Moreover,
$$ K(y,F_y)=1 \quad\text{for }\mathbb{P}_{e(X)}\text{-almost every }y. $$
Proof. Existence is the regular conditional distribution theorem for standard Borel spaces. To prove the fiber assertion, choose a countable collection \((C_n)_{n\in\mathbb{N}}\) of Borel subsets of \(\mathsf Y\) that separates points: whenever \(y\ne y'\), some \(C_n\) contains exactly one of them. Standard Borel spaces admit such a collection.

For each \(n\), the defining identity for \(K\) says that \(K(y,e^{-1}(C_n))\) is a version of the conditional expectation of \(\mathbf{1}_{\{e(X)\in C_n\}}\) given \(e(X)\). Since this indicator is already determined by \(e(X)\), it follows that

$$ K(y,e^{-1}(C_n))=\mathbf{1}_{C_n}(y) \quad\text{for }\mathbb{P}_{e(X)}\text{-almost every }y. $$

There are only countably many \(n\), so there is one set of full \(\mathbb{P}_{e(X)}\)-measure on which all these identities hold. Fix \(y\) in that set. For every \(n\), the conditional measure gives probability one to paths \(x\) for which \(e(x)\in C_n\) exactly when \(y\in C_n\). Because the sets separate points, the only possible value of \(e(x)\) satisfying all these equivalences is \(y\). Thus \(K(y,F_y)=1\), as claimed.

\(\square\)

This concentration statement is an almost-everywhere assertion under the observation law. At values \(y\) that have probability zero under that law, a regular conditional distribution can be changed without affecting its defining integral identity. The theorem does not require a meaningful or unique conditional law at every possible observation value.

Worked Example: The Brownian Bridge Given Its Endpoint

Worked Example: The Brownian Bridge Given Its Endpoint

Let \((B_t)_{0\le t\le T}\) be standard Brownian motion in \(\mathbb{R}\), with \(T>0\), and regard \(B\) as a random element of \(C([0,T],\mathbb{R})\). Define the residual path

$$ R_t=B_t-\frac{t}{T}B_T,\qquad 0\le t\le T. $$

For each \(t\), its covariance with the endpoint is

$$ \operatorname{Cov}(R_t,B_T) =\operatorname{Cov}(B_t,B_T)-\frac{t}{T}\operatorname{Var}(B_T) =t-\frac{t}{T}T=0. $$

Every finite vector of residual values together with \(B_T\) is jointly Gaussian. Zero cross-covariances therefore imply that each such residual vector is independent of \(B_T\). The Borel sigma-field of continuous paths is generated by evaluations at a countable dense set of times; consequently the residual path \(R\) is independent of \(B_T\).

For \(y\in\mathbb{R}\), define \(K(y,\cdot)\) to be the law of the path \(t\mapsto R_t+(t/T)y\). The map that sends a path \(r\) to \(r+(t/T)y\) is measurable, and in fact continuous in the uniform topology, so this defines a probability kernel. Since \(R_T=0\), every path under \(K(y,\cdot)\) ends at \(y\). Also, the identity \(B_t=R_t+(t/T)B_T\), together with independence of \(R\) and \(B_T\), gives the disintegration identity for conditioning on \(B_T\). Thus \(K\) is a regular conditional distribution of the full Brownian path given its endpoint. It is the Brownian bridge from \(0\) to \(y\) over \([0,T]\).

Worked Example: Conditioning on Several Brownian Evaluations

Worked Example: Conditioning on Several Brownian Evaluations

Let \(B\) again be standard Brownian motion on \([0,T]\), and fix \(0=t_0<t_1<\cdots<t_m=T\). Write \(\Delta_i=t_i-t_{i-1}\). Suppose the observed values are \(x_0=0,x_1,\ldots,x_m\). On interval \(i\), let \(W^i_u=B_{t_{i-1}+u}-B_{t_{i-1}}\) for \(0\le u\le\Delta_i\), and set

$$ R^i_u=W^i_u-\frac{u}{\Delta_i}W^i_{\Delta_i}. $$

The same covariance calculation as in the endpoint example gives \(\operatorname{Cov}(R^i_u,W^i_{\Delta_i})=u-(u/\Delta_i)\Delta_i=0\). Gaussianity makes the residual path on interval \(i\) independent of its increment \(W^i_{\Delta_i}\). Brownian increments on disjoint intervals are independent, so the collection of segment residuals is independent of the vector of interval increments.

The conditional path on interval \([t_{i-1},t_i]\) is therefore distributed as

$$ B_{t_{i-1}+u} = x_{i-1}+\frac{u}{\Delta_i}(x_i-x_{i-1})+R^i_u, \qquad 0\le u\le\Delta_i, $$

with independent Brownian bridge residuals \(R^i\) on the separate intervals. At \(u=0\), the formula gives \(x_{i-1}\); at \(u=\Delta_i\), it gives \(x_i\), since \(R^i_0=R^i_{\Delta_i}=0\). Concatenating these segments yields a path-valued conditional kernel given \((B_{t_1},\ldots,B_{t_m})=(x_1,\ldots,x_m)\). The increment vector has a positive Gaussian density on \(\mathbb{R}^m\), so these observation vectors have full support; the displayed construction supplies the conditional law for every such vector.

Worked Example: A Degenerate Conditional Path Law

Worked Example: A Degenerate Conditional Path Law

Let \(U\) be uniform on \([0,1]\), and define a random continuous path by \(X_t=tU\) for \(0\le t\le1\). Its endpoint is \(Y=X_1=U\). For \(y\in\mathbb{R}\), let \(\gamma_y(t)=ty\) and define \(K(y,\cdot)=\delta_{\gamma_y}\), the point mass at that path. The map \(y\mapsto\gamma_y\) is continuous in the uniform topology, so \(K\) is a measurable probability kernel.

For every Borel set \(A\) of paths and \(B\subseteq\mathbb{R}\),

$$ \mathbb{P}(X\in A,Y\in B) =\int_0^1 \mathbf{1}_{A}(\gamma_y)\mathbf{1}_{B}(y)\,dy =\int_B K(y,A)\,\mathbb{P}_Y(dy). $$

Thus this is a conditional path law given the endpoint. Unlike a Brownian bridge, it has no residual randomness: each endpoint determines exactly one path. For \(y\in[0,1]\), the path \(\gamma_y\) belongs to the support of the original path law. Values outside that interval have zero endpoint probability, so the chosen version there is permitted by the definition; here it still lies in the endpoint fiber.

What Fiber Constraints Do—and Do Not—Say

When conditioning on finitely many evaluations, the fiber is the set of paths satisfying all the observed coordinate constraints. The fiber theorem guarantees that the conditional path law is concentrated on that set for almost every observation vector. It does not say that the law is uniform on the fiber, nor does it provide a canonical geometric measure there. Brownian bridges, for instance, have substantial randomness within their endpoint fiber, while the straight-line example has a point mass.

There is also a useful limit to the phrase “path space.” An arbitrary uncountable product of state spaces need not be standard Borel, even when each individual state space is. The existence result applies when the chosen path space itself is standard Borel and the endpoint or evaluation map is measurable. Continuous-path and usual càdlàg-path spaces satisfy this requirement; one should not assume it for an unrestricted product without checking.

Common pitfall: A conditional law given an observation is determined only up to observation-law-null sets. The fiber concentration claim is therefore almost sure, not a promise of a unique conditional distribution at every value one might ask about.

Check Your Understanding

Use the definitions and constructions above to answer the following questions.

  1. What is the fiber for conditioning a continuous path on its values at three fixed times?
  2. Why does a standard Borel assumption matter for the existence of a regular conditional path law?
  3. In the Brownian endpoint example, which covariance calculation establishes independence of the residual and endpoint?
  4. Why is the conditional law in the straight-line example a point mass rather than a bridge-type law?
  5. Why can a conditional kernel be changed at an endpoint value that has probability zero?

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