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Why a Process’s Left Limit Is Measurable Before a Stopping Time

Article Quant Q&A · Author: An old man in the sea.
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Summary

The note explains why the value just before a stopping time, X_{T-}, is measurable with respect to the information available strictly before T. It uses a standard result from stochastic-process theory: evaluating a predictable process at a stopping time gives a random variable measurable with respect to the pre-stopping-time sigma-algebra. Since the left-limit process is left-continuous and adapted, it is predictable, so the result applies.

The cited proof relies on a monotone class argument, but the note does not reproduce that proof or define all the underlying filtration concepts. Its contribution is therefore a concise justification by identifying the relevant theorem and checking the process’s properties. This is a mathematical result that can support work involving stochastic models and stopping times; it does not describe a trading strategy, empirical test, or market application.

Key ideas

  • Evaluating a predictable process at a stopping time yields a value measurable with respect to the information available just before that time.
  • A left-continuous adapted process is predictable.
  • The left-limit process therefore has the required measurability property at the stopping time.
  • The cited proof uses a monotone class argument, which the note does not spell out.

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# $X_{T-}$ is $\mathcal{F}_{T-}$ measurable


# $X_{T-}$ is $\mathcal{F}_{T-}$ measurable












By definition, $\mathcal{F}_{T-}=\mathcal{F}_0 \vee \sigma(A\cap \{ t<T\}, A \in \mathcal{F}_t, t \in [0,\infty[)$.

Why is $X_{T-}$ is $\mathcal{F}_{T-}$ measurable?

## Answer by ir7 (score 1, accepted)

https://quant.stackexchange.com/a/65690

If $Y$ is predictable, then $Y_T$ is ${\cal F}_{T-}$-measurable. The left-limit process $Y$, as defined above, is left-continuous and adapted, hence predictable.

See this source, Lemma 1, for a proof using a monotone class argument.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.