Exercise: Equivalent Measures and Lebesgue's Decomposition

Prerequisites: Radon-Nikodym Theorem

Problem

1. Equivalent measures from normal shifts. For the normal-shift Radon-Nikodym derivative $Z = \exp(\theta X - \theta^2/2)$ with $X \sim \mathcal{N}(0, 1)$ under $\mathbb{P}$, show that:
   • $Z > 0$ everywhere, so $\mathbb{P} \ll \mathbb{Q}$ as well, hence $\mathbb{P} \sim \mathbb{Q}$.
   • Compute $d\mathbb{P}/d\mathbb{Q} = 1/Z$ and verify $\mathbb{E}^{\mathbb{Q}}[1/Z] = 1$.
2. Non-equivalent example. Let $\mathbb{P}$ be the uniform distribution on $[0, 1]$ and $\mathbb{Q}$ be the uniform distribution on $[0.5, 1.5]$ (both supported on subsets of $\mathbb{R}$). Is $\mathbb{Q} \ll \mathbb{P}$? Is $\mathbb{P} \ll \mathbb{Q}$? Is either absolutely continuous with respect to the other?
3. Lebesgue decomposition. For $\mathbb{P}$ and $\mathbb{Q}$ as in part 2, write Lebesgue's decomposition $\mathbb{Q} = \mathbb{Q}_{\text{ac}} + \mathbb{Q}_{\text{sing}}$ where $\mathbb{Q}_{\text{ac}} \ll \mathbb{P}$ and $\mathbb{Q}_{\text{sing}} \perp \mathbb{P}$. Identify the two components explicitly.
4. Quant-finance interpretation. In the framework of risk-neutral pricing, the measure $\mathbb{P}$ (real-world) and $\mathbb{Q}$ (risk-neutral) must be equivalent (same null sets) — not just absolutely continuous one way. Explain in one or two sentences why the equivalence condition matters: what financial catastrophe would it prevent if $\mathbb{Q} \ll \mathbb{P}$ but not $\mathbb{P} \ll \mathbb{Q}$?

Hint

For part 4, think about "impossible events" under $\mathbb{Q}$ that are possible under $\mathbb{P}$ — or vice versa. Would a model that assigns zero probability to a possible adverse event be a good pricing model?