Questions: Martingale Representation Theorem

Question 1

The martingale representation theorem says every ℱ_t^W-martingale is an Itô integral against W. What does this imply about the sources of randomness in the Brownian filtration?

• A. There are infinitely many independent sources of randomness, one for each time increment dW(t)
• B. All randomness in the filtration ℱ_t^W is generated by the single process W — there are no other independent sources of noise
• C. The filtration contains no randomness at all, since martingales have zero expected change
• D. The randomness can be decomposed into drift and diffusion components independently

Answer: (B)

The theorem says the Brownian motion W generates all the randomness in its filtration: every adapted martingale is expressible as a stochastic integral against W. If there were an independent source of noise (say, an independent Brownian motion B), then B itself would be a martingale in its own enlarged filtration that cannot be written as ∫H dW. The theorem fails in filtrations larger than the natural Brownian filtration — additional sources of randomness require additional driving processes.

Question 2

In the Black-Scholes model, the martingale representation theorem guarantees that every contingent claim (like a European option) can be perfectly replicated by dynamically trading the stock and bond. This property is called:

• A. Arbitrage-freeness
• B. Market completeness
• C. Risk neutrality
• D. The efficient market hypothesis

Answer: (B)

Market completeness means every contingent claim is attainable — it can be replicated by a self-financing trading strategy. The martingale representation theorem provides the mathematical mechanism: the discounted claim price is a martingale (under the risk-neutral measure) adapted to the Brownian filtration, so by the theorem it equals M(0) + ∫H dW for some H. The process H gives the hedging strategy — how many shares to hold at each time. Without the representation theorem, some claims might not be hedgeable, and the market would be incomplete (as it is with stochastic volatility models).

Question 3

Does the martingale representation theorem hold if the filtration is generated by two independent Brownian motions W₁ and W₂?

• T. True
• F. False

Answer: False

In the filtration generated by (W₁, W₂), the process W₂ is a martingale that cannot be written as ∫H dW₁ — it is independent of W₁ and carries genuinely new randomness. The correct generalization is that every martingale in the filtration of (W₁, W₂) can be written as ∫H₁ dW₁ + ∫H₂ dW₂ — a representation using both driving Brownian motions. This is precisely why models with multiple sources of risk (stochastic volatility, multi-asset models) require multiple hedging instruments for completeness.

Question 4

Explain why the martingale representation theorem is fundamentally a result about the structure of Brownian filtrations, not about martingales in general.

Model answer: Not every martingale on every probability space can be written as a stochastic integral against a Brownian motion. The theorem is specific to the filtration generated by W: in this filtration, W is the only source of randomness, so every adapted martingale must be 'built from' W via stochastic integration. In larger filtrations (with jumps, additional noise, or non-Brownian features), the representation fails — there are martingales orthogonal to W that require additional integrators. The key structural fact is that the Brownian filtration is generated by a single continuous martingale, and this characterizes it as 'simple' in a precise sense.

The theorem is sometimes stated as 'the Brownian filtration has the predictable representation property.' This is a property of the filtration, not of individual martingales. Filtrations generated by Poisson processes or more general Levy processes have their own representation theorems with additional compensated jump terms.