Probability Theory – Problem 61: Prove that is also a stopping time with respect to

The following is the complete marking scheme for this probability theory problem.

1. Checkpoints (max 7 pts total)

Core idea: Use the definition of $\mathcal{F}_T$ to decompose $\{\tau \le n\}$ and convert it into events in $\mathcal{F}_k$.

*The following steps form a single logical chain; points are awarded cumulatively:*

• Event decomposition [additive] (1 pt)
• Decompose the event $\{\tau \le n\}$ by the values $k$ of $T$, writing a union of the form:

$$\{\tau \le n\} = \bigcup_{k} \left( \{\tau \le n\} \cap \{T = k\} \right)$$

• *Note: The upper limit of summation may be $n$ or $\infty$ (as long as subsequent handling is correct). If the union formula is not explicitly written but each term for each $k$ is correctly discussed subsequently, credit may still be awarded.*
• Applying the $\mathcal{F}_T$-measurability definition [additive] (3 pts)
• Identifying the measurable set (1 pt): State that $\{\tau \le n\}$ (as the preimage of a Borel set under the random variable $\tau$) belongs to $\mathcal{F}_T$.
• Applying the definition (2 pts): Using the definition $A \in \mathcal{F}_T \iff A \cap \{T=k\} \in \mathcal{F}_k$ for all $k$, conclude:

$$\{\tau \le n\} \cap \{T = k\} \in \mathcal{F}_k$$

• *Scoring note: This is the most critical technical step. If the student only vaguely writes "since it is measurable, it belongs to $\mathcal{F}_k$" without explicitly invoking the structural definition of $\mathcal{F}_T$, deduct 1 point.*
• Monotonicity of the filtration and index restriction [additive] (2 pts)
• Index control (1 pt): State that since $\tau \ge T$, when $\{\tau \le n\}$ occurs we necessarily have $T \le n$ (i.e., $k \le n$), or state that when $k > n$ the intersection is the empty set (which belongs to $\mathcal{F}_n$).
• Subset property (1 pt): Use $k \le n \implies \mathcal{F}_k \subseteq \mathcal{F}_n$ to argue that the above non-empty intersections belong to $\mathcal{F}_n$.
• Conclusion [additive] (1 pt)
• State that $\sigma$-algebras are closed under finite unions, therefore $\{\tau \le n\} \in \mathcal{F}_n$, hence $\tau$ is a stopping time.
• *Note: This must be based on the preceding proof; stating only the conclusion without justification receives no credit.*

Total (max 7)

2. Zero-credit items

• Merely copying the problem conditions (e.g., listing "$\tau \ge T$", "$\tau$ is $\mathcal{F}_T$-measurable").
• Merely listing the general definition of stopping time or $\sigma$-algebra without applying it to the specific variables $\tau$ and $T$.
• Citing a theorem: Directly citing "if $\tau \ge T$ and is $\mathcal{F}_T$-measurable then $\tau$ is a stopping time" as a known result without providing a proof.
• Circular reasoning: Using the conclusion "$\tau$ is a stopping time" within the proof (e.g., using properties of $\mathcal{F}_\tau$).

3. Deductions

*Only deduct for the most significant single error; deductions should not reduce the total below 0.*

• Index logic error (flat -2):
• Summing over $k$ up to $\infty$ in the decomposition and claiming $\mathcal{F}_k \subseteq \mathcal{F}_n$ for all $k$ (ignoring the case $k > n$).
• *Exception: If the student correctly identifies that the set is empty when $k > n$, no deduction applies.*
• Symbol confusion or unclear definitions (flat -1):
• Confusing events (sets) with random variables, or confusing the relationship between $\mathcal{F}_T$ and $\mathcal{F}_n$.
• Failing to specify the range of $k$ or handling $T=k$ ambiguously.
• Logical gap (flat -1):
• After obtaining $\{\tau \le n\} \cap \{T=k\} \in \mathcal{F}_k$, jumping directly to the conclusion $\{\tau \le n\} \in \mathcal{F}_n$ without mentioning $\mathcal{F}_k \subset \mathcal{F}_n$.