Brownian Bridge™

1. For $p = 1$ and $p = 2$ , the $L^p$ norms of a random variable $X$ on a probability space satisfy which ordering?

‖X‖₁ ≤ ‖X‖₂ always‖X‖₂ ≤ ‖X‖₁ alwaysNeither is always larger; the ordering depends on the distribution‖X‖₁ = ‖X‖₂ on any probability space

2. Jensen's inequality states that for a convex function $\varphi$ and $X \in L^1$ : $\varphi(\mathbb{E}[X]) \leq \mathbb{E}[\varphi(X)]$ . Which of the following is a direct application in options pricing?

The price of a call option exceeds the payoff at the forward priceThe price of a call option equals the expected payoff under the real-world measureThe delta of an option is always between 0 and 1The implied volatility smile is convex in log-strike

3. Which pairs of convergence modes have an implication going in exactly one direction (A ⟹ B but B ⇏ A)?

Almost sure convergence ⟹ convergence in probability (but not vice versa)Convergence in probability ⟹ almost sure convergence (but not vice versa)Convergence in distribution ⟹ convergence in probability (but not vice versa)L¹ convergence ⟹ almost sure convergence (but not vice versa)

4. Let $X_n = n \cdot \mathbf{1}_{[0,1/n]}$ on $[0,1]$ with Lebesgue measure. Which statement is correct?

X_n → 0 almost surely, but E[X_n] = 1 for all nX_n → 0 in L¹ and almost surelyX_n does not converge almost surely but does converge in L²X_n → 0 in L² and almost surely

5. A family of random variables $\{X_\alpha\}$ is uniformly integrable (UI) if and only if:

sup_α E[|X_α| · 1_{|X_α| > M}] → 0 as M → ∞sup_α E[|X_α|²] < ∞E[|X_α|] → 0 for all αX_α → 0 in probability as α → ∞

6. Hölder's inequality with conjugates $p$ and $q$ ( $1/p + 1/q = 1$ ) states $\mathbb{E}[|XY|] \leq \|X\|_p \|Y\|_q$ . For $p = q = 2$ , this becomes the Cauchy-Schwarz inequality. If $\|X\|_2 = 3$ and $\|Y\|_2 = 4$ , what is the tightest upper bound on $\mathbb{E}[|XY|]$ ?

127625

7. The Vitali convergence theorem says: if $X_n \xrightarrow{\mathbb{P}} X$ and $\{X_n\}$ is uniformly integrable, then $X_n \to X$ in $L^1$ . The Dominated Convergence Theorem is a special case because:

Domination |X_n| ≤ Y with Y ∈ L¹ implies uniform integrability of {X_n}Domination implies convergence in probability automaticallyDomination implies L² convergence, which is stronger than L¹The DCT requires Y ∈ L² while Vitali only needs Y ∈ L¹

8. A quant is running a Monte Carlo simulation for an exotic payoff $g(S_T)$ that is unbounded. She finds that her sample mean converges as she adds paths, but she notices the sample variance of the payoff is growing with the number of paths rather than stabilising. What is the most likely explanation and the appropriate remedy?

The payoff g(S_T) is not in L², so the CLT does not apply; she should verify whether E[g(S_T)²] < ∞ and consider payoff truncation or importance samplingThe random number generator is not properly seeded; she should use a fixed seedMonte Carlo always has growing variance; she needs to add antithetic variatesThe model drift is incorrect; she should recalibrate the drift to match market forwards

Quiz: Lp Spaces and Modes of Convergence

Quick Quiz