Continuous And Left-Continuous Adapted Processes

Path Regularity and Measurability in Time

Adaptedness says that the value \(X_t\) at each fixed time \(t\) is measurable with respect to the information \(\mathcal{F}_t\). By itself, this is a statement about separate time sections; it does not assert the joint measurability needed for many constructions involving an entire process. Progressive and predictable measurability supply stronger, process-level conditions. Path continuity or left continuity provides a useful route from adaptedness to these conditions, but the approximations must respect the direction of time.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\), with an increasing filtration, carrying a Brownian motion \(B_t\). The measurability results below do not require Brownian motion. A real-valued process \(X=(X_t)_{t\ge0}\) is adapted if \(X_t\) is \(\mathcal{F}_t\)-measurable for every \(t\). We take continuity or left continuity to hold for every path; the same statements apply to a chosen adapted version with that path property.

The half-open intervals in the predictable generators encode past information: on \((s,t]\), the event \(A\) is already known at time \(s\). The value at time \(0\) is handled separately by \(\mathcal{F}_0\). We now show that continuity and left continuity lead to the corresponding measurability properties.

Continuous Adapted Processes Are Progressive

Proof. Fix \(T>0\); the case \(T=0\) follows directly from adaptedness. For each integer \(n\ge1\), partition \([0,T]\) into \(2^n\) equal intervals of length \(\Delta_n=T2^{-n}\), with endpoints \(t_k=k\Delta_n\). Define \(X^n\) on \([0,T]\) by setting \(X^n_t=X_{t_k}\) for \(t_k\le t<t_{k+1}\), and \(X^n_T=X_T\). On any one of these intervals, the random value \(X_{t_k}\) is \(\mathcal{F}_{t_k}\)-measurable and therefore \(\mathcal{F}_T\)-measurable. The interval is Borel, so \(X^n\) is \(\mathcal{F}_T\otimes\mathcal{B}([0,T])\)-measurable. For every \((\omega,t)\), continuity of the path gives \(X^n_t(\omega)\to X_t(\omega)\): for \(t<T\), the left endpoint \(t_k\) of the interval containing \(t\) converges to \(t\), and at \(T\) the approximant equals \(X_T\). A pointwise limit of measurable real-valued functions is measurable. Thus \(X\) is measurable with respect to \(\mathcal{F}_T\otimes\mathcal{B}([0,T])\). Since \(T\) was arbitrary, \(X\) is progressive. \(\square\)

The proof uses a separate partition for each finite horizon. Its approximating value at time \(t\) comes from a time no later than \(t\), so it is compatible with the information available by the end of the horizon. The definition of progressive measurability asks for \(\mathcal{F}_T\)-measurability on each whole interval \([0,T]\); it does not require each approximant to be measurable with respect to \(\mathcal{F}_t\) at every individual \(t\).

Left-Continuous Adapted Processes Are Predictable

Proof. Let \(X\) be left-continuous and adapted. For each integer \(n\ge1\), set \(\delta_n=2^{-n}\) and define \(X^n_0=X_0\), while \(X^n_t=X_{k\delta_n}\) on each interval \((k\delta_n,(k+1)\delta_n]\), for \(k=0,1,2,\ldots\). To verify predictability, first take an interval \((k\delta_n,(k+1)\delta_n]\). For any Borel set \(C\subseteq\mathbb{R}\), its inverse image under the process that equals \(X_{k\delta_n}\) on this interval is \(\{\omega:X_{k\delta_n}(\omega)\in C\}\times(k\delta_n,(k+1)\delta_n]\). The event in the first factor belongs to \(\mathcal{F}_{k\delta_n}\) by adaptedness, so this rectangle is a predictable generator. Taking the countable union over \(k\), and adding the time-zero process \(X_0\), which is measurable on \(\{0\}\times\Omega\) by \(\mathcal{F}_0\)-measurability, shows that \(X^n\) is predictable. For \(t>0\), the left endpoint \(k\delta_n\) of the interval containing \(t\) is strictly less than or equal to \(t\) and converges to \(t\) from below as \(n\to\infty\). Left continuity gives \(X^n_t\to X_t\); at \(t=0\), equality holds for every \(n\). Hence \(X\), a pointwise limit of predictable processes, is predictable. \(\square\)

The intervals are deliberately \((k\delta_n,(k+1)\delta_n]\), not intervals that include their left endpoint. This convention makes the value on each interval known at its left endpoint, as required by the predictable generators. At a grid point \(t>0\), the approximating value comes from the preceding grid point, which still converges to \(X_t\) by left continuity.

Progressive Step Processes and the Direction of Approximation

A process can be progressive without being left-continuous. For example, fix \(s>0\), take \(A\in\mathcal{F}_s\), and set \(J_t=\mathbf{1}_A\mathbf{1}_{[s,\infty)}(t)\). It is adapted: before \(s\) it is zero, and at and after \(s\) the event \(A\) is known. For any horizon \(u\), if \(u<s\) its restriction is zero; if \(u\ge s\), the only nonzero part is \([s,u]\times A\), which is measurable in \(\mathcal{F}_u\otimes\mathcal{B}([0,u])\). Thus \(J\) is progressive. On \(A\), however, its value at \(s\) is \(1\), while its left limit is \(0\), so its paths need not be left-continuous.

This example highlights a distinction in the results: progressive measurability is obtained from continuity and adaptedness by approximating with values at earlier times, whereas predictability of a left-continuous process is obtained by step processes whose value on \((s,t]\) is already \(\mathcal{F}_s\)-measurable. In particular, every continuous adapted process is also left-continuous, so it is predictable as well as progressive.

Common Pitfall: Using Future Values

A tempting approximation for proving predictability is to put \(X_{(k+1)\delta_n}\) on \((k\delta_n,(k+1)\delta_n]\). Although this may converge to \(X_t\) for a continuous process, the coefficient is measurable with respect to \(\mathcal{F}_{(k+1)\delta_n}\), not necessarily \(\mathcal{F}_{k\delta_n}\). Thus the resulting rectangle need not be one of the predictable generators. For instance, with \(X=B\), the value \(B_{(k+1)\delta_n}\) uses an increment after \(k\delta_n\); it is not justified to treat that value as known at \(k\delta_n\). The correction is to use \(X_{k\delta_n}\) on \((k\delta_n,(k+1)\delta_n]\). Left continuity still gives convergence, and the coefficient now uses only information available at the left endpoint.

The time-zero convention is another point to check. The interval generators \((s,t]\times A\) do not include time \(0\). A predictable step approximation must therefore specify its value at \(0\) separately and use an \(\mathcal{F}_0\)-measurable value there. In the theorem, that value is \(X_0\), which is \(\mathcal{F}_0\)-measurable by adaptedness.