Controlling Local Martingales¶

Guiding Principle¶

Everything in this section is about controlling local martingales.

The pages that follow develop a single unifying idea:

\[ \text{Financial modeling} = \text{constructing, transforming, and controlling local martingales} \]

Local martingales arise naturally from stochastic calculus, but they are often too weak for financial applications. The central problem is therefore:

When and how can we upgrade a local martingale into a true martingale?

This question underlies every page in this section:

Local Martingales — the raw, uncontrolled object
Stochastic Exponential — the transformation enabling control
Novikov and Kazamaki Conditions — technical guarantees that control is valid
Martingales and No-Arbitrage — the economic requirement demanding control
Risk-Neutral Valuation — the payoff of achieving control
Martingale Representation Theorem — full control (completeness and hedging)

The Structural Pipeline¶

The entire section can be understood as the following pipeline:

flowchart LR

A[Stochastic calculus] --> B[Local martingales]
B --> C[Stochastic exponential]
C --> D[Measure change]
D --> E[Risk-neutral measure Q]
E --> F[Pricing as expectation]
F --> G[Martingale representation]
G --> H[Hedging strategies]

B --> X[Failure: strict local martingale]
X --> Y[Bubbles / pricing anomalies]

The Guiding Principle¶

Informal Statement¶

A financial model is well-behaved if all relevant local martingales can be controlled so that they become true martingales.

Formal Perspective¶

Let \(M\) be a nonnegative local martingale. Then \(M\) is a supermartingale, and:

True martingale case. If \(M\) is a true martingale:

\[ \mathbb{E}[M_t] = \mathbb{E}[M_0] \quad \text{for all } t \geq 0 \]

Strict local martingale case. If \(M\) is not a true martingale:

\[ \mathbb{E}[M_t] < \mathbb{E}[M_0] \]

which signals loss of mass, instability, or economic inconsistency.

Why Local Martingales Are Not Enough¶

Local martingales arise automatically from Ito calculus:

Ito integrals \(\to\) local martingales
Drift-free SDEs \(\to\) local martingales
Stochastic exponentials \(\to\) local martingales

But finance requires more:

Requirement	Why needed	Local martingale sufficient?
No-arbitrage	Fair pricing	Not always
Measure change	Define \(\mathbb{Q}\)	Needs true martingale
Pricing formula	Expectations well-defined	Needs integrability
Hedging	Stability of value process	Needs stronger control

Tools for Controlling Local Martingales¶

This section introduces several mechanisms to upgrade local martingales.

1. Integrability Conditions¶

\(L^p\) bounds
Finite quadratic variation

These ensure:

\[ \text{Local martingale} \;\Longrightarrow\; \text{True martingale} \]

2. Stochastic Exponential¶

Transforms additive noise into multiplicative structure:

\[ \mathcal{E}(M)_t = \exp\!\left(M_t - \tfrac{1}{2}\langle M \rangle_t\right) \]

Key role: candidate density for measure change.

3. Novikov and Kazamaki Conditions¶

Ensure:

\[ \mathcal{E}(M) \text{ is a true martingale} \]

This is critical for:

Girsanov's theorem
Existence of the risk-neutral measure

4. Measure Change (Girsanov)¶

Reweights probability so that discounted prices become martingales under the new measure \(\mathbb{Q}\).

5. Martingale Representation¶

Ensures that every \(\mathbb{Q}\)-martingale can be written as a stochastic integral against the driving Brownian motion. Financial meaning: every contingent claim can be hedged.

When Control Fails¶

If a local martingale cannot be upgraded:

1. Strict Local Martingale¶

\[ \mathbb{E}[M_t] < M_0 \]

Interpretation: mass escapes to infinity and the model loses probability weight.

2. Financial Consequences¶

Asset price bubbles
Failure of put-call parity
Invalid measure change

3. Example Phenomena¶

CEV model with \(\beta > 1\)
Inverse Bessel process
Exploding stochastic exponentials

Big Picture Connections¶

Concept	Role in the control problem
Local martingale	Raw output of stochastic calculus
True martingale	Controlled, usable object
Stochastic exponential	Tool for transformation
Novikov/Kazamaki	Control conditions
ELMM (\(\mathbb{Q}\))	Controlled probability measure
MRT	Completeness (full control)

Section Takeaway¶

A model is arbitrage-free, priceable, and hedgeable if and only if its key local martingales can be controlled into true martingales.

How to Read This Section¶

When reading each subsection, keep asking:

What is the local martingale here?
Why is it not automatically a true martingale?
What condition ensures it becomes one?
What breaks if it fails?

If you track these four questions, the entire chapter becomes a single coherent story rather than a collection of technical results.

Exercises¶

Exercise 1. Let \(W_t\) be a standard Brownian motion and define \(M_t = \int_0^t W_s^2\, dW_s\). Show that \(M_t\) is a local martingale. Compute its quadratic variation \(\langle M \rangle_t\) and verify that \(\mathbb{E}[\langle M \rangle_t] < \infty\) for each \(t\), concluding that \(M_t\) is in fact a true martingale.

Solution to Exercise 1

Local martingale. By definition, \(M_t = \int_0^t W_s^2\, dW_s\) is an Ito integral with respect to Brownian motion. Every Ito integral against Brownian motion with a progressively measurable integrand is a local martingale (with localizing sequence given by stopping times \(\tau_n = \inf\{t : |M_t| \geq n\}\)).

Quadratic variation. For the Ito integral \(M_t = \int_0^t f_s\, dW_s\) with \(f_s = W_s^2\):

\[ \langle M \rangle_t = \int_0^t f_s^2\, ds = \int_0^t W_s^4\, ds \]

Expectation of quadratic variation. Since \(\mathbb{E}[W_s^4] = 3s^2\) (fourth moment of Gaussian):

\[ \mathbb{E}[\langle M \rangle_t] = \int_0^t \mathbb{E}[W_s^4]\, ds = \int_0^t 3s^2\, ds = t^3 < \infty \]

Conclusion. Since \(\mathbb{E}[\langle M \rangle_t] < \infty\) for each \(t\), the Ito isometry implies \(\mathbb{E}[M_t^2] = \mathbb{E}[\langle M \rangle_t] = t^3 < \infty\). An \(L^2\)-bounded local martingale is a true martingale (by the martingale convergence theorem applied to the localizing sequence). Therefore \(M_t\) is a true martingale. \(\square\)

Exercise 2. Let \(M_t\) be a continuous local martingale with localizing sequence \((\tau_n)\). Prove that if \(M\) is uniformly integrable (i.e., the family \(\{M_t : t \geq 0\}\) is uniformly integrable), then \(M\) is a true martingale.

Solution to Exercise 2

Let \(0 \leq s < t\). Since \((\tau_n)\) is a localizing sequence, \(M^{\tau_n}\) is a true martingale for each \(n\), so:

\[ \mathbb{E}[M_{t \wedge \tau_n} \mid \mathcal{F}_s] = M_{s \wedge \tau_n} \]

As \(n \to \infty\), \(\tau_n \to \infty\) a.s., so \(M_{t \wedge \tau_n} \to M_t\) a.s. and \(M_{s \wedge \tau_n} \to M_s\) a.s.

By uniform integrability of \(\{M_t\}\), the family \(\{M_{t \wedge \tau_n}\}_n\) is also uniformly integrable (since \(|M_{t \wedge \tau_n}| \leq \sup_{u \leq t} |M_u|\) and continuous local martingales on \([0,t]\) inherit UI from the endpoint). Therefore convergence in \(L^1\) holds:

\[ M_{t \wedge \tau_n} \xrightarrow{L^1} M_t \]

Taking \(n \to \infty\) in the conditional expectation (using \(L^1\) convergence on both sides):

\[ \mathbb{E}[M_t \mid \mathcal{F}_s] = M_s \]

This is the martingale property. \(\square\)

Exercise 3. Consider the stochastic exponential \(\mathcal{E}(\theta W)_t = \exp\!\left(\theta W_t - \frac{1}{2}\theta^2 t\right)\) where \(\theta\) is a constant. Verify the Novikov condition \(\mathbb{E}\!\left[\exp\!\left(\frac{1}{2}\int_0^T \theta^2\, ds\right)\right] < \infty\) and conclude that \(\mathcal{E}(\theta W)\) is a true martingale on \([0, T]\). Then explain why this result is essential for the Girsanov construction of the risk-neutral measure in the Black-Scholes model.

Solution to Exercise 3

Novikov condition. With constant integrand \(\theta\):

\[ \frac{1}{2}\int_0^T \theta^2\, ds = \frac{1}{2}\theta^2 T \]

This is a deterministic constant, so:

\[ \mathbb{E}\!\left[\exp\!\left(\frac{1}{2}\theta^2 T\right)\right] = \exp\!\left(\frac{1}{2}\theta^2 T\right) < \infty \]

The Novikov condition is satisfied.

Conclusion. By Novikov's theorem, \(\mathcal{E}(\theta W)_t\) is a true martingale on \([0, T]\). In particular, \(\mathbb{E}[\mathcal{E}(\theta W)_T] = \mathcal{E}(\theta W)_0 = 1\), so \(\mathcal{E}(\theta W)_T\) is a valid Radon-Nikodym density.

Application to Black-Scholes. In the Black-Scholes model, the market price of risk is \(\theta = (\mu - r)/\sigma\), a constant. The Girsanov density is:

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_T} = \mathcal{E}(-\theta W)_T = \exp\!\left(-\theta W_T - \frac{1}{2}\theta^2 T\right) \]

The Novikov condition guarantees this is a true martingale, which ensures that \(\mathbb{Q}\) is a well-defined probability measure equivalent to \(\mathbb{P}\). Without this, the Girsanov measure change would be invalid and the risk-neutral pricing framework would collapse. The fact that \(\theta\) is constant (hence the Novikov condition is trivially satisfied) is one reason the Black-Scholes model is so tractable. \(\square\)

Exercise 4. The inverse Bessel(3) process \(X_t = 1/R_t\), where \(R_t\) is a 3-dimensional Bessel process started at \(R_0 = 1\), is a strict local martingale. Explain why \(\mathbb{E}[X_t] < X_0 = 1\) for \(t > 0\). If \(X_t\) were used as the discounted price of a risky asset, explain which of the following would fail: (a) the FTAP, (b) put-call parity, (c) the pricing formula \(V_0 = \mathbb{E}^{\mathbb{Q}}[e^{-rT}\Phi(X_T)]\) for a European call.

Solution to Exercise 4

Why the expectation drops. The 3-dimensional Bessel process \(R_t\) satisfies \(R_t \to \infty\) a.s. as \(t \to \infty\), so \(X_t = 1/R_t \to 0\) a.s. Since \(X_t \geq 0\) and \(X_t\) is a continuous local martingale, Fatou's lemma gives:

\[ \mathbb{E}[X_t] \leq X_0 = 1 \]

If \(X_t\) were a true martingale, equality would hold. But the explicit computation (using the known density of the Bessel process) shows \(\mathbb{E}[X_t] = 1 - 2\mathcal{N}(-1/\sqrt{t}) + (2/\sqrt{2\pi t})e^{-1/(2t)} < 1\) for \(t > 0\). Mass is lost because \(X_t\) can approach large values (when \(R_t\) is near zero) that contribute to the expectation under stopping but escape to the boundary.

(a) FTAP. The fundamental theorem of asset pricing requires the existence of an equivalent local martingale measure (ELMM). Since \(X_t\) is already a local martingale under the given measure, the FTAP is not violated — it only requires a local martingale measure, not a true martingale measure. No-arbitrage (in the NFLVR sense) still holds.

(b) Put-call parity. Standard put-call parity states \(C - P = S_0 - Ke^{-rT}\), which relies on \(\mathbb{E}^{\mathbb{Q}}[e^{-rT}X_T] = X_0\). Since \(\mathbb{E}[X_t] < X_0\), the forward price is strictly less than \(X_0 e^{rT}\), and put-call parity fails in its standard form. The correction involves the "bubble" term \(X_0 - \mathbb{E}^{\mathbb{Q}}[e^{-rT}X_T] > 0\).

(c) Call pricing formula. For a European call with payoff \((X_T - K)^+\), the formula \(V_0 = \mathbb{E}^{\mathbb{Q}}[e^{-rT}(X_T - K)^+]\) gives a value strictly less than the superreplication price. The call pricing formula undervalues the option because it misses the bubble component. The correct superreplication price exceeds the risk-neutral expectation. \(\square\)

Exercise 5. Suppose \(\theta_t\) is a deterministic function of time satisfying \(\int_0^T \theta_t^2\, dt < \infty\). Show that the Novikov condition \(\mathbb{E}\!\left[\exp\!\left(\frac{1}{2}\int_0^T \theta_t^2\, dt\right)\right] < \infty\) is automatically satisfied, and conclude that the stochastic exponential \(\mathcal{E}\!\left(\int_0^\cdot \theta_s\, dW_s\right)\) is a true martingale on \([0, T]\). Give an example of a random (adapted) process \(\theta_t\) for which the Novikov condition fails.

Solution to Exercise 5

Deterministic case. If \(\theta_t\) is deterministic, then \(\int_0^T \theta_t^2\, dt\) is a deterministic constant \(c < \infty\). Therefore:

\[ \mathbb{E}\!\left[\exp\!\left(\frac{1}{2}\int_0^T \theta_t^2\, dt\right)\right] = \exp\!\left(\frac{c}{2}\right) < \infty \]

The Novikov condition is trivially satisfied. By Novikov's theorem, \(\mathcal{E}\!\left(\int_0^\cdot \theta_s\, dW_s\right)\) is a true martingale on \([0, T]\).

This covers all models with deterministic volatility of the market price of risk, including the Black-Scholes model (\(\theta_t = (\mu - r)/\sigma\)) and the Ho-Lee interest rate model.

Random counterexample. Let \(\theta_t = W_t / \sqrt{T - t}\) for \(t \in [0, T)\). Then:

\[ \int_0^T \theta_t^2\, dt = \int_0^T \frac{W_t^2}{T - t}\, dt \]

This integral diverges a.s. as \(t \to T^-\) (since \(W_t^2\) does not vanish while \(1/(T-t) \to \infty\)), so:

\[ \mathbb{E}\!\left[\exp\!\left(\frac{1}{2}\int_0^T \frac{W_t^2}{T - t}\, dt\right)\right] = +\infty \]

The Novikov condition fails. The corresponding stochastic exponential may be a strict local martingale, meaning the Girsanov measure change is not valid on \([0, T]\). \(\square\)

Exercise 6. Let \(M_t\) be a continuous local martingale with \(M_0 = 0\). Apply Ito's formula to \(f(x) = e^x\) with \(X_t = M_t - \frac{1}{2}\langle M \rangle_t\) to verify that \(\mathcal{E}(M)_t = \exp\!\left(M_t - \frac{1}{2}\langle M \rangle_t\right)\) satisfies the SDE

\[ d\mathcal{E}(M)_t = \mathcal{E}(M)_t\, dM_t \]

Conclude that \(\mathcal{E}(M)\) is a nonnegative local martingale, and deduce that \(\mathbb{E}[\mathcal{E}(M)_t] \leq 1\) for all \(t \geq 0\).

Solution to Exercise 6

Define \(X_t = M_t - \frac{1}{2}\langle M \rangle_t\) so that \(\mathcal{E}(M)_t = e^{X_t}\). Since \(\langle M \rangle_t\) has continuous paths of bounded variation:

\[ dX_t = dM_t - \frac{1}{2}\, d\langle M \rangle_t, \qquad d\langle X \rangle_t = d\langle M \rangle_t \]

Applying Ito's formula with \(f(x) = e^x\) (so \(f' = f'' = e^x\)):

\[ d\mathcal{E}(M)_t = e^{X_t}\, dX_t + \frac{1}{2} e^{X_t}\, d\langle X \rangle_t \]

\[ = e^{X_t}\!\left(dM_t - \frac{1}{2}\, d\langle M \rangle_t\right) + \frac{1}{2} e^{X_t}\, d\langle M \rangle_t = e^{X_t}\, dM_t = \mathcal{E}(M)_t\, dM_t \]

Local martingale. The SDE \(d\mathcal{E}(M)_t = \mathcal{E}(M)_t\, dM_t\) has no drift term, so \(\mathcal{E}(M)\) is a stochastic integral against the local martingale \(M\), hence itself a local martingale.

Nonnegativity. Since \(\mathcal{E}(M)_t = \exp(M_t - \frac{1}{2}\langle M \rangle_t)\) is the exponential of a real number, \(\mathcal{E}(M)_t > 0\) a.s. for all \(t\).

Supermartingale bound. A nonnegative local martingale is a supermartingale (by Fatou's lemma applied to the localizing sequence). Therefore:

\[ \mathbb{E}[\mathcal{E}(M)_t] \leq \mathcal{E}(M)_0 = e^{0} = 1 \]

Equality holds for all \(t\) if and only if \(\mathcal{E}(M)\) is a true martingale. This is exactly the gap that the Novikov and Kazamaki conditions are designed to close. \(\square\)

Exercise 7. Consider a stochastic volatility model with zero interest rate:

\[ dS_t = S_t\, \sigma_t\, dW_t^{(1)}, \qquad d\sigma_t = \alpha\, \sigma_t\, dW_t^{(2)} \]

where \(W^{(1)}, W^{(2)}\) are independent Brownian motions under \(\mathbb{P}\) and \(\alpha > 0\).

(a) Explain why the stock price \(S_t\) is a local martingale under \(\mathbb{P}\).

(b) Show that for any adapted process \(\lambda_t\) satisfying the Novikov condition \(\mathbb{E}\!\left[\exp\!\left(\frac{1}{2}\int_0^T \lambda_t^2\, dt\right)\right] < \infty\), the measure \(\mathbb{Q}^\lambda\) defined by \(d\mathbb{Q}^\lambda / d\mathbb{P}|_{\mathcal{F}_T} = \mathcal{E}\!\left(\int_0^\cdot \lambda_s\, dW_s^{(2)}\right)_T\) is an equivalent local martingale measure for \(S\). Conclude that the model admits infinitely many such measures.

(c) Using the structural pipeline from this section, identify which step fails to be unique and explain why the martingale representation theorem does not hold for the filtration generated by \(S\) alone.

Solution to Exercise 7

(a) The SDE \(dS_t = S_t \sigma_t\, dW_t^{(1)}\) has no \(dt\) term, so \(S_t\) is a stochastic integral against Brownian motion (with integrand \(S_t \sigma_t\)). Every such stochastic integral is a local martingale. Equivalently, \(S_t = S_0\, \mathcal{E}\!\left(\int_0^\cdot \sigma_s\, dW_s^{(1)}\right)_t\), which is a local martingale by Exercise 6.

(b) The Girsanov density \(\mathcal{E}\!\left(\int_0^\cdot \lambda_s\, dW_s^{(2)}\right)_T\) is a true martingale by the Novikov condition, so \(\mathbb{Q}^\lambda\) is a well-defined probability measure equivalent to \(\mathbb{P}\).

Under \(\mathbb{Q}^\lambda\), Girsanov's theorem gives:

\[ \widetilde{W}_t^{(2)} = W_t^{(2)} - \int_0^t \lambda_s\, ds \quad \text{is a } \mathbb{Q}^\lambda\text{-Brownian motion} \]

Crucially, since \(W^{(1)}\) is independent of \(\mathcal{E}(\lambda \cdot W^{(2)})\), the process \(W^{(1)}\) remains a Brownian motion under \(\mathbb{Q}^\lambda\). Therefore \(S_t\) still satisfies \(dS_t = S_t \sigma_t\, dW_t^{(1)}\) with \(W^{(1)}\) a \(\mathbb{Q}^\lambda\)-Brownian motion, so \(S_t\) remains a local martingale under \(\mathbb{Q}^\lambda\).

Since \(\lambda\) is arbitrary (subject to the Novikov condition), there are infinitely many equivalent local martingale measures.

(c) In the structural pipeline:

\[ \text{Local martingales} \to \text{Stochastic exponential} \to \text{Measure change} \to \mathbb{Q} \to \text{Pricing} \to \text{MRT} \to \text{Hedging} \]

the step "\(\text{Measure change} \to \mathbb{Q}\)" fails to be unique. Multiple valid \(\mathbb{Q}^\lambda\) exist, and each assigns different prices to claims that depend on \(\sigma_t\) (such as variance swaps or volatility options).

The martingale representation theorem fails for the filtration generated by \(S\) alone because the market is driven by two independent sources of randomness (\(W^{(1)}, W^{(2)}\)) but only one is tradeable. A \(\mathbb{Q}\)-martingale adapted to the full filtration \(\mathcal{F}_t = \sigma(W_s^{(1)}, W_s^{(2)} : s \leq t)\) generally requires integrals against both \(W^{(1)}\) and \(W^{(2)}\), but only the \(W^{(1)}\) component can be replicated by trading \(S\). This is the hallmark of an incomplete market: control of local martingales is partial, not full. \(\square\)