Brownian Bridge™

1. A filtration $( \mathcal{F}_t )_{t \geq 0}$ on $(\Omega, \mathcal{F}, \mathbb{P})$ is defined by which core property?

ℱ_s ⊆ ℱ_t for all s ≤ t (increasing family of sub-σ-algebras)ℱ_s = ℱ_t for all s, t (all σ-algebras are the same)ℱ_t = ℱ for all t (each element is the full σ-algebra)ℱ_s ⊇ ℱ_t for all s ≤ t (decreasing family)

2. Let $\Omega = \{HH, HT, TH, TT\}$ with the two-flip filtration $\mathcal{F}_1 = \sigma(\{HH,HT\}, \{TH,TT\})$ . How many events does $\mathcal{F}_1$ contain?

24816

3. A process $(X_t)_{t \geq 0}$ is adapted to $(\mathcal{F}_t)$ if and only if:

X_t is ℱ_t-measurable for every t ≥ 0X_t is ℱ-measurable for every t ≥ 0X_t is independent of ℱ_t for every t ≥ 0X_t is constant on each ℱ_t-atom for every t ≥ 0

4. Which of the following is a stopping time with respect to the natural filtration $(\mathcal{F}_t^B)$ of a Brownian motion $(B_t)$ ?

τ = inf{t ≥ 0 : B_t = a} for a constant a > 0τ = T − inf{t ≥ 0 : B_t = a} (time-reversed hitting time)τ = argmax_{0 ≤ t ≤ T} B_t (time of the maximum)τ = inf{t ≥ 0 : B_{t+1} = a} (uses future path)

5. An adapted integrable process $(M_t)$ is a martingale if and only if:

E[M_t | ℱ_s] = M_s a.s. for all 0 ≤ s ≤ tE[M_t | ℱ_s] ≥ M_s a.s. for all 0 ≤ s ≤ tE[M_t] = 0 for all t ≥ 0M_t − M_s is independent of ℱ_s for all s ≤ t

6. Under the risk-neutral measure $\mathbb{Q}$ , why is the discounted stock price $\tilde{S}_t = e^{-rt} S_t$ a martingale but not under the real-world measure $\mathbb{P}$ ?

Under ℚ the drift of d(e^{−rt}S_t) is zero; under ℙ it is (μ − r)·S_t dt ≠ 0 when μ ≠ rUnder ℚ the volatility σ = 0; under ℙ volatility makes S_t a submartingaleUnder ℙ, S_t = 0 so the discounted price is trivially a martingaleThe Girsanov theorem adds a drift μ under ℚ that exactly cancels e^{−rt}

7. Doob's Optional Stopping Theorem states $\mathbb{E}[M_\tau] = \mathbb{E}[M_0]$ for a discrete martingale and stopping time $\tau$ . Which integrability condition is sufficient when $\tau$ may be unbounded?

The stopped process (M_{τ ∧ n}) is uniformly integrableτ has finite variance: Var(τ) < ∞M_n is non-negative for all nτ is almost surely finite, with no further condition needed

8. A rates quant is pricing a knock-out barrier swaption. She checks numerically that the discounted NPV process is a martingale under the risk-neutral measure, but her OST calculation gives $\mathbb{E}[\text{NPV}_\tau] \neq \mathbb{E}[\text{NPV}_0]$ . Which is the most likely explanation?

The stopping time τ (first knock-out) is unbounded and the stopped process is not uniformly integrableThe martingale property fails under the swap measure but not under the risk-neutral measureThe optional stopping theorem does not apply to interest rate modelsThe discounted NPV is a submartingale, not a martingale, under the risk-neutral measure

Quiz: Filtrations, Adapted Processes, and Martingales

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