Stopped Process is Adapted Stochastic Process

Theorem

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.

Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

Let $\sequence {X_n}_{n \ge 0}$ be a $\sequence {\FF_n}_{n \ge 0}$-adapted process.

Let $\sequence {X_n^T}_{n \ge 0}$ be the stopped process.

Then $\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-adapted process.

Proof

Let $n \in \Z_{\ge 0}$.

From Constant Function is Stopping Time, $n$ is a stopping time.

From Pointwise Minimum of Stopping Times is Stopping Time, $n \wedge T$ is a stopping time, where $\wedge$ is the pointwise minimum.

From Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra, we have:

$X_{n \wedge T}$ is $\FF_{n \wedge T}$-measurable

where $\FF_{n \wedge T}$ is the stopped $\sigma$-algebra associated with $n \wedge T$.

We have by the definition of pointwise minimum:

$n \wedge T \le n$

From Stopped Sigma-Algebra preserves Inequality between Stopping Times and Stopped Sigma-Algebra of Constant Stopping Time coincides with Filtration, we have:

$\FF_{n \wedge T} \subseteq \FF_n$

So:

$X_{n \wedge T}$ is $\FF_n$-measurable.

So:

$\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-adapted process.

$\blacksquare$