Stochastic Processes – Problem 43: Prove that

This rubric strictly follows the official solution approach, with a total of 7 points. Please score according to the proof path used by the student, choosing Chain A or Chain B; do not mix chains.

1. Checkpoints (max 7 pts)

Select exactly one scoring path | If both are partially addressed, take the single-chain maximum; do not accumulate across chains.

Chain A: Event Decomposition and Reflection Principle (Official Approach)

• Total probability decomposition [additive]
• Use the first hitting time $\tau_0$ to decompose $P_x(B(t)\in A)$ into two parts: $\{ \tau_0 \le t \}$ (hit the origin) and $\{ \tau_0 > t \}$ (did not hit the origin).
• +1 pt: Write a decomposition similar to $P_x(B(t) \in A) = P_x(\tau_0 \le t, B(t) \in A) + P_x(\tau_0 > t, B(t) \in A)$.
• Identify the target term (meaning of not hitting the origin) [additive]
• +1 pt: State that for $x>0$, the event $\{ \tau_0 > t \}$ is equivalent to the path staying strictly positive (or non-negative) on $[0,t]$, thereby confirming this term is the left-hand side of the identity to be proved.
• Apply the reflection principle (core step) [additive]
• +3 pts: Apply the reflection principle to establish symmetry, stating that the probability of starting from $x$, hitting the origin, and eventually reaching $A$ equals the probability of starting from $-x$ (hitting the origin) and reaching $A$.
• *Note: If the student writes $P_x(\tau_0 \le t, B(t) \in A) = P_{-x}(\tau_0 \le t, B(t) \in A)$ or directly writes $= P_{-x}(B(t) \in A)$, full 3 pts are awarded.*
• Argue "must pass through the origin" (geometric property) [additive]
• +1 pt: Explicitly explain that since $-x < 0$ and $A \subset [0, \infty)$, a continuous path from $-x$ to $A$ must pass through the origin, so the condition $\tau_0 \le t$ is redundant (has probability 1) for the event $\{B(0)=-x, B(t) \in A\}$.
• *Note: If the student writes the result directly without mentioning this geometric intuition or implying this logic, this 1 pt is not awarded.*
• Algebraic rearrangement and conclusion [additive]
• +1 pt: Substitute all terms back into the decomposition and rearrange to obtain the final conclusion.

Chain B: Density Function Computation (Integral Method)

• Cite the reflection principle density formula [additive]
• +4 pts: Directly write (or derive) the transition probability density for starting from $x>0$, not touching 0 on $[0,t]$, and $B(t)=y$ (i.e., the absorbing-barrier Brownian motion density):

$p^*(t, x, y) = \frac{1}{\sqrt{2\pi t}} \left( e^{-\frac{(y-x)^2}{2t}} - e^{-\frac{(y+x)^2}{2t}} \right)$.

• *Note: This step contains the core application of the reflection principle and carries higher weight.*
• Identify the first integral [additive]
• +1 pt: Identify and state that $\int_A \frac{1}{\sqrt{2\pi t}} e^{-\frac{(y-x)^2}{2t}} dy$ equals the unconditional probability $P_x(B(t) \in A)$.
• Identify the second integral [additive]
• +1 pt: Identify and state that $\int_A \frac{1}{\sqrt{2\pi t}} e^{-\frac{(y+x)^2}{2t}} dy$ equals $P_{-x}(B(t) \in A)$ (or $P_x(B(t) \in -A)$).
• Conclusion [additive]
• +1 pt: Combine the integral terms and explain that this equals the required probability difference, completing the proof.

Total (max 7)

2. Zero-credit items

• Merely copying the problem: Copying the identity to be proved verbatim with no intermediate steps.
• Circular reasoning: Starting from the identity to be proved, performing algebraic manipulations to arrive at $0=0$ or a tautology, without establishing the reversibility of the logic or with incorrect reasoning direction.
• Empty citation: Writing only "by the reflection principle" without any specific probability decomposition, density formula, or geometric correspondence.
• Incorrect premise: Assuming $A$ must be symmetric (e.g., $[-a, a]$) or assuming $A$ contains a negative part (the problem specifies $A \subset [0, \infty)$), causing the argument to deviate entirely from the problem.

3. Deductions

Apply at most the single largest deduction (score cannot go below 0).

• Logical gap (missing geometric argument): [-1]
• In Chain A, the student correctly applies the reflection principle to obtain the $-x$ term, but completely fails to explain why $P_{-x}(\tau_0 \le t, \dots)$ simplifies to $P_{-x}(\dots)$ (i.e., does not mention the fact that "from negative to positive one must cross zero").
• Notation confusion: [-1]
• Seriously confusing probability $P(\cdot)$ with probability density $p(\cdot)$, or confusing the random variable $B(t)$ with specific values $x, y$, rendering expressions mathematically meaningless (but if the core idea is clear, deduct only 1 pt).
• Missing measure-theoretic details: [no deduction]
• If the student does not explicitly mention "$A$ is a measurable set" or use measure-theoretic language to extend from density integrals to general sets, no deduction; this is assumed understood at the undergraduate level.