Local Martingales

In the unifying framework of this section, the local martingale is the raw, uncontrolled object — the starting point before any upgrade to a true martingale.

A local martingale is a process that behaves like a martingale "locally" — when stopped at appropriate times — but may fail to be a true martingale globally. This distinction is crucial in continuous-time finance, where many natural price processes are local martingales but not martingales.

Prerequisites

This section assumes familiarity with:

• Stopping Times
• Itô Integral Construction
• Quadratic Variation

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Definitions

Martingale (Recap)

A process \(\{M_t\}_{t \geq 0}\) is a martingale with respect to filtration \(\{\mathcal{F}_t\}\) if:

1. Adaptedness: \(M_t\) is \(\mathcal{F}_t\)-measurable for all \(t \geq 0\)
2. Integrability: \(\mathbb{E}[|M_t|] < \infty\) for all \(t \geq 0\)
3. Martingale property: \(\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s\) almost surely for all \(0 \leq s \leq t\)

Local Martingale

Local Martingale

An adapted process \(\{M_t\}_{t \geq 0}\) with \(M_0\) finite almost surely is a local martingale if there exists a sequence of stopping times \(\{\tau_n\}_{n=1}^{\infty}\) such that:

1. Monotonicity: \(\tau_1 \leq \tau_2 \leq \tau_3 \leq \cdots\)
2. Divergence: \(\tau_n \to \infty\) almost surely as \(n \to \infty\)
3. Stopped martingale: The stopped process \(M^{\tau_n}_t := M_{t \wedge \tau_n}\) is a martingale for each \(n\)

The sequence \(\{\tau_n\}\) is called a localizing sequence (or reducing sequence).

Remark on condition 3: For \(M_{t \wedge \tau_n}\) to be a martingale, we need \(\mathbb{E}[|M_{t \wedge \tau_n}|] < \infty\) for all \(t\). This is the sense in which localization "tames" potentially non-integrable processes.

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The Martingale Hierarchy

The following inclusions are strict:

\[ \boxed{ \text{UI Martingales} \subsetneq \text{Martingales} \subsetneq \text{Local Martingales} } \]

where UI denotes uniformly integrable. The inclusions go from strongest (UI martingales, the smallest class) to weakest (local martingales, the largest class). A local martingale that is not a true martingale is called a strict local martingale.

Connection to Convergence Theory

Uniformly integrable martingales converge in \(L^1\), not just almost surely. See Martingale Convergence for the full hierarchy of convergence results.

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What Can Go Wrong?

A local martingale fails to be a martingale when any of the following occurs:

1. Integrability Failure

The random variable \(M_t\) may satisfy \(\mathbb{E}[|M_t|] = \infty\) for some (or all) \(t > 0\).

2. Explosion to Infinity

The process may escape to \(+\infty\) (or \(-\infty\)) in finite time, i.e., \(\lim_{t \to \zeta^-} |M_t| = \infty\) where \(\zeta < \infty\) is an explosion time.

3. Mass Leakage at Infinity

Even without explosion, probability mass can "escape to infinity" in the sense that:

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

The "missing mass" corresponds to paths where \(M_t\) has grown large.

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Canonical Examples

Example 1: Itô Integrals

Consider the Itô integral:

\[ M_t = \int_0^t \sigma_s\,dW_s \]

where \(\sigma\) is an adapted process.

Integrability Hierarchy for Itô Integrals

Condition	Result
\(\int_0^t \sigma_s^2 \, ds < \infty\) a.s.	Integral exists; local martingale
\(\mathbb{E}\left[\int_0^T \sigma_s^2 \, ds\right] < \infty\)	True martingale on \([0,T]\)
Neither	Integral not defined

The a.s. condition is the existence requirement—without it, the Itô integral is not even defined. The \(L^1\) condition upgrades local martingale to true martingale.

Intuition: A driftless SDE \(dM_t = \sigma_t dW_t\) looks like a martingale—it's "pure noise" with no systematic drift. And usually it is a true martingale. But technically, Itô calculus only guarantees a local martingale; upgrading to true martingale requires verifying integrability.

Proof that it's a local martingale: Define the localizing sequence:

\[ \tau_n = \inf\left\{t \geq 0 : \int_0^t \sigma_s^2\,ds \geq n\right\} \wedge n \]

Then \(\tau_n \uparrow \infty\) a.s., and by construction:

\[ \mathbb{E}\left[\int_0^{T \wedge \tau_n} \sigma_s^2\,ds\right] \leq n < \infty \]

By the Itô isometry criterion, \(M_{t \wedge \tau_n}\) is a true martingale for each \(n\). \(\square\)

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Example 2: Stochastic Exponential (True Martingale)

The stochastic exponential of Brownian motion:

\[ Z_t = \mathcal{E}(W)_t := \exp\left(W_t - \frac{t}{2}\right) \]

satisfies the SDE \(dZ_t = Z_t\,dW_t\) with \(Z_0 = 1\).

Claim: \(Z_t\) is a true martingale with \(\mathbb{E}[Z_t] = 1\) for all \(t \geq 0\).

Proof: We verify Novikov's condition. Here \(\langle W \rangle_t = t\), so:

\[ \mathbb{E}\left[\exp\left(\frac{1}{2}\langle W \rangle_T\right)\right] = \mathbb{E}\left[\exp\left(\frac{T}{2}\right)\right] = e^{T/2} < \infty \]

By Novikov's theorem (see Novikov & Kazamaki Conditions), \(\mathcal{E}(W)\) is a true martingale. \(\square\)

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Example 3: Reciprocal of 3D Bessel Process (Strict Local Martingale)

Let \(R_t = |B_t|\) where \(B_t = (B^1_t, B^2_t, B^3_t)\) is 3-dimensional Brownian motion started from \(B_0 = x\) with \(|x| = r_0 > 0\). The process \(R_t\) is the 3-dimensional Bessel process started from \(r_0\).

Define:

\[ M_t = \frac{1}{R_t} \]

Claim: \(M_t\) is a strict local martingale (local martingale but NOT a true martingale).

Common Misconception

The failure is not because \(R_t\) hits zero. In fact, the 3D Bessel process is transient: \(R_t \to \infty\) as \(t \to \infty\) almost surely, and \(R_t > 0\) for all \(t \geq 0\) when \(r_0 > 0\).

Proof that \(M_t\) is a local martingale:

The \(d\)-dimensional Bessel process satisfies the SDE:

\[ dR_t = \frac{d-1}{2R_t}dt + dW_t \]

where \(W\) is a 1-dimensional Brownian motion. For \(d = 3\), this gives \(dR_t = \frac{1}{R_t}dt + dW_t\).

By Itô's formula applied to \(f(r) = 1/r\):

\[ d\left(\frac{1}{R_t}\right) = -\frac{1}{R_t^2}dR_t + \frac{1}{R_t^3}dt = -\frac{1}{R_t^2}\left(\frac{1}{R_t}dt + dW_t\right) + \frac{1}{R_t^3}dt = -\frac{1}{R_t^2}dW_t \]

The drift terms cancel! Thus \(M_t = 1/R_t\) satisfies:

\[ dM_t = -\frac{1}{R_t^2}dW_t \]

This is an Itô integral (no drift), hence a local martingale.

Proof that \(M_t\) is NOT a true martingale:

Using the transition density of the 3D Bessel process (see Revuz–Yor, Chapter VI, or Karatzas–Shreve §3.3.C), one can compute:

\[ \mathbb{E}\left[\frac{1}{R_t}\right] = \frac{1}{r_0} - \frac{2}{r_0}\Phi\left(-\frac{r_0}{\sqrt{t}}\right) < \frac{1}{r_0} = M_0 \]

where \(\Phi\) is the standard normal CDF. The strict inequality shows \(\mathbb{E}[M_t] < \mathbb{E}[M_0]\), violating the martingale property.

Intuition: As \(t \to \infty\), the Bessel process drifts to \(+\infty\), so \(1/R_t \to 0\). The "probability mass" that would be needed to maintain \(\mathbb{E}[M_t] = M_0\) has "leaked to infinity."

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Example 4: CEV Model with β > 1 (Strict Local Martingale)

The constant elasticity of variance (CEV) model provides a clean example of a strict local martingale arising in finance. Consider:

\[ dX_t = \sigma X_t^\beta \, dW_t, \quad X_0 = x_0 > 0 \]

where \(\sigma > 0\) and \(\beta > 1\).

Claim: For \(\beta > 1\), the process \(X_t\) is a strict local martingale.

Why this is a local martingale: The process is clearly a local martingale since it is an Itô integral with no drift term. The localizing sequence:

\[ \tau_n = \inf\{t \geq 0 : X_t \geq n \text{ or } X_t \leq 1/n\} \wedge n \]

ensures \(X_{t \wedge \tau_n}\) is bounded and hence a true martingale.

Why this is NOT a true martingale: For \(\beta > 1\), the process can reach infinity in finite time with positive probability. More precisely, the scale function analysis shows that infinity is an accessible boundary. Even when we define \(X_t = \infty\) for \(t \geq \zeta\) (the explosion time), we have:

\[ \mathbb{E}[X_t] < x_0 \quad \text{for all } t > 0 \]

The "missing mass" corresponds to paths that have exploded.

Financial interpretation: The CEV model with \(\beta > 1\) exhibits explosive behavior inconsistent with limited liability. This is why practitioners typically use \(\beta < 1\) (which gives absorption at zero rather than explosion at infinity).

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Mathematical Characterization

The Supermartingale Property

Non-negative Local Martingales are Supermartingales

Let \(M\) be a non-negative local martingale. Then \(M\) is a supermartingale:

\[ \mathbb{E}[M_t \mid \mathcal{F}_s] \leq M_s \quad \text{almost surely for all } 0 \leq s \leq t \]

Proof: Let \(\{\tau_n\}\) be a localizing sequence. For the stopped process:

\[ \mathbb{E}[M_{t \wedge \tau_n} \mid \mathcal{F}_s] = M_{s \wedge \tau_n} \quad \text{(martingale property)} \]

Since \(M \geq 0\), Fatou's lemma gives:

\[ \mathbb{E}[M_t \mid \mathcal{F}_s] = \mathbb{E}\left[\liminf_{n \to \infty} M_{t \wedge \tau_n} \mid \mathcal{F}_s\right] \leq \liminf_{n \to \infty} \mathbb{E}[M_{t \wedge \tau_n} \mid \mathcal{F}_s] = \liminf_{n \to \infty} M_{s \wedge \tau_n} = M_s \]

where the last equality uses \(\tau_n \to \infty\) a.s. \(\square\)

Corollary: For non-negative local martingales:

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

with equality if and only if \(M\) is a true martingale.

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Characterization via Fatou's Lemma

For a non-negative local martingale with localizing sequence \(\{\tau_n\}\):

\[ \mathbb{E}[M_{t \wedge \tau_n}] = \mathbb{E}[M_0] \quad \text{for all } n \]

Taking \(n \to \infty\) and applying Fatou's lemma:

\[ \mathbb{E}[M_t] \leq \liminf_{n \to \infty} \mathbb{E}[M_{t \wedge \tau_n}] = \mathbb{E}[M_0] \]

The inequality can be strict—this is the signature of a strict local martingale.

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Sufficient Conditions for True Martingale

A local martingale \(M\) is a true martingale if any of the following conditions holds:

1. Boundedness

\[ |M_t| \leq C \quad \text{almost surely for all } t \in [0,T] \]

for some constant \(C < \infty\).

2. Domination

\[ |M_t| \leq Y \quad \text{almost surely for all } t \in [0,T] \]

for some integrable random variable \(Y\) (i.e., \(\mathbb{E}[Y] < \infty\)).

3. L^p Boundedness (p > 1)

\[ \sup_{t \in [0,T]} \mathbb{E}[|M_t|^p] < \infty \]

This follows from the fact that \(L^p\)-bounded martingales are uniformly integrable for \(p > 1\).

4. Finite Expected Quadratic Variation

For continuous local martingales with \(M_0\) integrable:

\[ \mathbb{E}[\langle M \rangle_T] < \infty \implies M \text{ is a true martingale on } [0,T] \]

Proof sketch: By the Burkholder–Davis–Gundy inequality:

\[ \mathbb{E}\left[\sup_{t \leq T} |M_t|\right] \leq C \cdot \mathbb{E}\left[\langle M \rangle_T^{1/2}\right] \leq C \cdot \mathbb{E}[\langle M \rangle_T]^{1/2} < \infty \]

Hence \(M\) is dominated by an integrable random variable. \(\square\)

5. Novikov's Condition (for Stochastic Exponentials)

For a continuous local martingale \(M\) with \(M_0 = 0\):

\[ \mathbb{E}\left[\exp\left(\frac{1}{2}\langle M \rangle_T\right)\right] < \infty \implies \mathcal{E}(M)_t \text{ is a true martingale on } [0,T] \]

where \(\mathcal{E}(M)_t = \exp(M_t - \frac{1}{2}\langle M \rangle_t)\) is the stochastic exponential.

6. Kazamaki's Condition (Weaker than Novikov's Condition)

If \(\mathcal{E}(M/2)\) is a submartingale, then \(\mathcal{E}(M)\) is a true martingale on \([0,T]\).

Kazamaki's condition is strictly weaker than Novikov's. See Novikov & Kazamaki Conditions for details and proofs.

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Logical Relationships Between Conditions

How the Conditions Relate

The conditions above are not independent. For continuous local martingales:

Implication	Mechanism
(1) ⟹ (2)	Boundedness is domination with \(Y = C\)
(3) ⟹ (2)	Doob's maximal inequality: \(\mathbb{E}[\sup_t \|M_t\|^p] \leq \left(\frac{p}{p-1}\right)^p \mathbb{E}[\|M_T\|^p]\), so \(Y = \sup_t \|M_t\|\) works
(4) ⟹ (2)	BDG inequality: \(\mathbb{E}[\sup_t \|M_t\|] \leq C \cdot \mathbb{E}[\langle M \rangle_T^{1/2}] < \infty\)
(5) ⟹ (6)	Novikov implies Kazamaki (see proof)

The common thread: all conditions ultimately ensure uniform integrability, which prevents mass from escaping to infinity.

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Connection to Infinitesimal Generators

Let \(X_t\) be a diffusion with infinitesimal generator:

\[ \mathcal{L} = \mu(x)\frac{\partial}{\partial x} + \frac{1}{2}\sigma^2(x)\frac{\partial^2}{\partial x^2} \]

For \(f \in C^2\), define the process \(Y_t = f(X_t)\).

Generator Criterion

If \(\mathcal{L}f(x) = 0\) for all \(x\) in the state space, then \(f(X_t)\) is a local martingale.

To upgrade to a true martingale, verify any of the six sufficient conditions above—for example:

• 1. Boundedness: \(|f(X_t)| \leq C\)
• 1. Domination: \(|f(X_t)| \leq Y\) with \(\mathbb{E}[Y] < \infty\)
• 1. \(L^p\) Boundedness (\(p > 1\)): \(\sup_{t \in [0,T]} \mathbb{E}[|f(X_t)|^p] < \infty\)
• 1. Finite Expected Quadratic Variation: \(\mathbb{E}[\langle f(X) \rangle_T] < \infty\)
• 1. Novikov's Condition (for Stochastic Exponentials)
• 1. Kazamaki's Condition (Weaker than Novikov's Condition)

Connection to Dynkin's formula: By Itô's formula:

\[ f(X_t) - f(X_0) = \int_0^t \mathcal{L}f(X_s)\,ds + \int_0^t f'(X_s)\sigma(X_s)\,dW_s \]

When \(\mathcal{L}f = 0\), the drift integral vanishes, leaving only the stochastic integral (which is a local martingale).

See Generator and Martingales for the full treatment.

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Financial Implications

Discounted Asset Prices

Under the risk-neutral measure \(\mathbb{Q}\), the discounted asset price:

\[ \tilde{S}_t = e^{-rt}S_t \]

should be a martingale for the market to be free of arbitrage (First Fundamental Theorem of Asset Pricing).

In practice, \(\tilde{S}_t\) is often only a local martingale. The distinction matters.

Strict Local Martingales and Financial Bubbles

Bubble Characterization

If the discounted price process is a strict local martingale under \(\mathbb{Q}\):

\[ \mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] < S_0 \]

This implies the current price \(S_0\) exceeds its "fundamental value" (the discounted expected future price). This is the mathematical signature of a financial bubble.

Reference: Jarrow, Protter, and Shimbo (2010), "Asset Price Bubbles in Incomplete Markets," Mathematical Finance.

Put-Call Parity Failure

The standard put-call parity:

\[ C(K,T) - P(K,T) = S_0 - Ke^{-rT} \]

relies on \(\mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] = S_0\). When the stock price is a strict local martingale:

\[ C(K,T) - P(K,T) = \mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] - Ke^{-rT} < S_0 - Ke^{-rT} \]

Put-call parity fails, and the put price includes a "bubble premium."

Connection to Girsanov's Theorem

When performing measure changes via Girsanov's theorem, the Radon–Nikodym derivative:

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_t} = Z_t = \mathcal{E}\left(-\int_0^\cdot \theta_s\,dW_s\right)_t \]

must be a true martingale (not just a local martingale) for the measure change to be valid. This is precisely where Novikov and Kazamaki conditions enter.

See Girsanov's Theorem for the full treatment.

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Summary Table

Property	Martingale	Local Martingale	Strict Local Martingale
Definition	\(\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s\)	\(M_{t\wedge\tau_n}\) is martingale	Local mart., not true mart.
Integrability	Required: $\mathbb{E}[	M_t	] < \infty$
Mean preservation	\(\mathbb{E}[M_t] = \mathbb{E}[M_0]\)	\(\mathbb{E}[M_t] \leq \mathbb{E}[M_0]\)	\(\mathbb{E}[M_t] < \mathbb{E}[M_0]\)
If \(M \geq 0\)	Supermartingale	Supermartingale	Strict supermartingale
Explosion	Cannot explode	Can explode	May or may not explode
Financial interpretation	Fair game	Locally fair	Bubble possible

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Key Takeaways

\[ \boxed{ \mathcal{L}f = 0 \implies f(X_t) \text{ is a local martingale} } \]

\[ \boxed{ \mathcal{L}f = 0 \text{ + sufficient condition (1–6)} \implies f(X_t) \text{ is a true martingale} } \]

\[ \boxed{ \text{Non-negative local martingale} \implies \text{Supermartingale} } \]

\[ \boxed{ \mathbb{E}[M_t] < \mathbb{E}[M_0] \text{ for non-negative } M \iff M \text{ is a strict local martingale} } \]

The Bottom Line

The distinction between local martingales and true martingales is essential for:

1. Rigorous Itô calculus: Ensuring stochastic integrals have the expected properties
2. Measure changes: Validating Girsanov transformations via Novikov/Kazamaki
3. Financial modeling: Detecting and modeling asset price bubbles
4. PDE connections: Understanding when Feynman–Kac representations hold

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Python Simulation: Mass Leakage in Strict Local Martingales

The following simulation demonstrates how \(\mathbb{E}[M_t]\) can decrease over time for a strict local martingale.

```python import numpy as np import matplotlib.pyplot as plt

def simulate_inverse_bessel_3d(r0, T, dt, n_paths): """ Simulate 1/R_t where R_t is a 3D Bessel process. This is a strict local martingale. """ n_steps = int(T / dt) t = np.linspace(0, T, n_steps + 1)

# Simulate 3D Brownian motion
dW = np.sqrt(dt) * np.random.randn(n_paths, n_steps, 3)
B = np.zeros((n_paths, n_steps + 1, 3))
B[:, 0, :] = r0 / np.sqrt(3)  # Start at distance r0 from origin

for i in range(n_steps):
    B[:, i+1, :] = B[:, i, :] + dW[:, i, :]

# Compute R_t = |B_t|
R = np.sqrt(np.sum(B**2, axis=2))
R = np.maximum(R, 1e-10)  # Avoid division by zero

# M_t = 1/R_t
M = 1.0 / R

return t, M, R

Parameters

r0 = 1.0 T = 5.0 dt = 0.001 n_paths = 50000

np.random.seed(42) t, M, R = simulate_inverse_bessel_3d(r0, T, dt, n_paths)

Compute E[M_t] over time

E_M = np.mean(M, axis=0)

Theoretical initial value

M0 = 1.0 / r0

Plot

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

Left: E[M_t] over time

ax1 = axes[0] ax1.plot(t, E_M, 'b-', linewidth=2, label=r'\(\mathbb{E}[M_t]\) (Monte Carlo)') ax1.axhline(y=M0, color='r', linestyle='--', linewidth=2, label=r'\(M_0 = 1/r_0\)') ax1.set_xlabel('Time \(t\)', fontsize=12) ax1.set_ylabel(r'\(\mathbb{E}[M_t]\)', fontsize=12) ax1.set_title('Mass Leakage in Strict Local Martingale\n(Inverse 3D Bessel Process)', fontsize=12) ax1.legend(fontsize=11) ax1.grid(True, alpha=0.3) ax1.set_ylim([0, M0 * 1.1])

Right: Sample paths

ax2 = axes[1] n_show = 20 for i in range(n_show): ax2.plot(t, M[i, :], alpha=0.5, linewidth=0.5) ax2.set_xlabel('Time \(t\)', fontsize=12) ax2.set_ylabel(r'\(M_t = 1/R_t\)', fontsize=12) ax2.set_title(f'Sample Paths ({n_show} shown)', fontsize=12) ax2.grid(True, alpha=0.3) ax2.set_ylim([0, 5])

plt.tight_layout() plt.savefig('strict_local_martingale_simulation.png', dpi=150, bbox_inches='tight') plt.show()

Print summary statistics

print(f"Initial value M_0 = 1/r_0 = {M0:.4f}") print(f"E[M_T] at T={T}: {E_M[-1]:.4f}") print(f"Mass leakage: {(M0 - E_M[-1])/M0 * 100:.2f}%") ```

Output:

Initial value M_0 = 1/r_0 = 1.0000 E[M_T] at T=5.0: 0.3471 Mass leakage: 65.29%

Interpretation: The plot shows \(\mathbb{E}[M_t]\) decreasing below \(M_0 = 1\), demonstrating the strict local martingale property. The "leaked mass" corresponds to paths where \(R_t\) has drifted far from the origin—as the 3D Bessel process is transient and escapes to infinity, \(1/R_t \to 0\), but the expectation cannot be preserved because the probability mass needed to compensate has "escaped to infinity."

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Exercises

Exercise 1. Let \(M_t = \int_0^t \sigma_s\,dW_s\) where \(\sigma_s = 1/(1 - s)\) for \(s \in [0, 1)\). Show that \(\int_0^1 \sigma_s^2\,ds = +\infty\) but \(\int_0^t \sigma_s^2\,ds < \infty\) for every \(t < 1\). Construct a localizing sequence \(\{\tau_n\}\) that makes \(M_{t \wedge \tau_n}\) a true martingale for each \(n\).

Solution to Exercise 1

We have \(\sigma_s = 1/(1-s)\) for \(s \in [0,1)\). For \(t < 1\):

\[ \int_0^t \sigma_s^2\,ds = \int_0^t \frac{1}{(1-s)^2}\,ds = \left[\frac{1}{1-s}\right]_0^t = \frac{1}{1-t} - 1 = \frac{t}{1-t} < \infty \]

As \(t \to 1^-\), this diverges: \(\int_0^1 \sigma_s^2\,ds = \lim_{t \to 1^-} \frac{t}{1-t} = +\infty\).

For the localizing sequence, define:

\[ \tau_n = \inf\left\{t \geq 0 : \int_0^t \sigma_s^2\,ds \geq n\right\} \wedge \left(1 - \frac{1}{n}\right) \]

Then \(\tau_n \uparrow 1\) a.s. as \(n \to \infty\), and by construction:

\[ \int_0^{T \wedge \tau_n} \sigma_s^2\,ds \leq n < \infty \]

Since \(\mathbb{E}\left[\int_0^{T \wedge \tau_n} \sigma_s^2\,ds\right] \leq n < \infty\), the Itô isometry criterion guarantees that \(M_{t \wedge \tau_n} = \int_0^{t \wedge \tau_n} \sigma_s\,dW_s\) is a true (square-integrable) martingale for each \(n\).

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Exercise 2. Prove that every true martingale is a local martingale. Then explain why the converse fails by giving the key property that a strict local martingale violates. (Hint: consider the integrability condition.)

Solution to Exercise 2

Every true martingale is a local martingale: Let \(M_t\) be a true martingale. Define \(\tau_n = n\) for all \(n \geq 1\). Then \(\tau_n \to \infty\), and \(M_{t \wedge \tau_n} = M_{t \wedge n}\) is a martingale (a stopped martingale is still a martingale). Hence \(M\) is a local martingale with localizing sequence \(\{\tau_n = n\}\).

The converse fails: A strict local martingale \(M_t\) violates the integrability condition. Specifically, for a true martingale we need \(\mathbb{E}[|M_t|] < \infty\) for all \(t\) and \(\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s\). A strict local martingale may have \(\mathbb{E}[|M_t|] < \infty\) but satisfy only the inequality \(\mathbb{E}[M_t \mid \mathcal{F}_s] \leq M_s\) (supermartingale property for non-negative case) rather than equality. The key property violated is mean preservation: for a non-negative strict local martingale, \(\mathbb{E}[M_t] < \mathbb{E}[M_0]\), meaning the expectation strictly decreases over time due to "mass leaking to infinity."

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Exercise 3. Let \(M_t\) be a non-negative local martingale with \(M_0 = 1\). Using Fatou's lemma, prove the supermartingale inequality \(\mathbb{E}[M_t] \leq 1\) for all \(t \geq 0\). Explain the financial interpretation of \(1 - \mathbb{E}[M_t]\) when \(M_t\) is the discounted price of an asset under the risk-neutral measure.

Solution to Exercise 3

Let \(M_t\) be a non-negative local martingale with \(M_0 = 1\) and localizing sequence \(\{\tau_n\}\). For each \(n\), \(M_{t \wedge \tau_n}\) is a true martingale, so:

\[ \mathbb{E}[M_{t \wedge \tau_n}] = \mathbb{E}[M_0] = 1 \]

As \(n \to \infty\), \(\tau_n \to \infty\) a.s., so \(M_{t \wedge \tau_n} \to M_t\) a.s. Since \(M_t \geq 0\), Fatou's lemma gives:

\[ \mathbb{E}[M_t] = \mathbb{E}\left[\liminf_{n \to \infty} M_{t \wedge \tau_n}\right] \leq \liminf_{n \to \infty} \mathbb{E}[M_{t \wedge \tau_n}] = 1 \]

Financial interpretation: When \(M_t\) is the discounted price of an asset under \(\mathbb{Q}\), the quantity \(1 - \mathbb{E}[M_t]\) represents the bubble component. If \(M_t\) is a strict local martingale, \(\mathbb{E}[M_t] < 1 = M_0\), meaning the current asset price exceeds its "fundamental value" \(\mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T]\) by the amount \(S_0(1 - \mathbb{E}[M_t])\). This excess is the mathematical signature of a financial bubble.

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Exercise 4. Consider the CEV model \(dX_t = \sigma X_t^{\beta}\,dW_t\) with \(X_0 = 1\) and \(\sigma = 0.5\). For \(\beta = 0.5\), verify that \(X_t\) is a true martingale by checking that \(\mathbb{E}[\langle X \rangle_T] < \infty\) is plausible. For \(\beta = 1.5\), explain qualitatively why \(X_t\) is only a strict local martingale.

Solution to Exercise 4

For \(\beta = 0.5\): The SDE is \(dX_t = 0.5 X_t^{0.5}\,dW_t\), so the quadratic variation is:

\[ \langle X \rangle_T = \int_0^T (0.5)^2 X_s\,ds = 0.25 \int_0^T X_s\,ds \]

Since \(X_t\) is a non-negative local martingale with \(X_0 = 1\), we have \(\mathbb{E}[X_t] \leq 1\) for all \(t\). Thus:

\[ \mathbb{E}[\langle X \rangle_T] = 0.25 \int_0^T \mathbb{E}[X_s]\,ds \leq 0.25T < \infty \]

By the sufficient condition (finite expected quadratic variation), \(X_t\) is a true martingale on \([0, T]\).

For \(\beta = 1.5\): The diffusion coefficient is \(\sigma(x) = 0.5 x^{1.5}\), which grows superlinearly. For \(\beta > 1\), the process can "explode" — reach infinity in finite time with positive probability. This happens because the volatility grows so rapidly as \(X_t\) increases that the process is pushed to infinity. The explosion causes \(\mathbb{E}[X_t] < X_0 = 1\) since the "mass" associated with exploded paths is lost. The scale function analysis for the boundary at infinity shows it is accessible (reached in finite time), confirming that \(X_t\) is only a strict local martingale.

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Exercise 5. In the Black-Scholes model under \(\mathbb{Q}\), the discounted stock price \(\tilde{S}_t = e^{-rt}S_t\) satisfies \(d\tilde{S}_t = \sigma \tilde{S}_t\,dW_t^{\mathbb{Q}}\). Show that this is a true martingale by verifying that

\[ \mathbb{E}\left[\int_0^T \sigma^2 \tilde{S}_s^2\,ds\right] < \infty \]

Explain why this condition guarantees the validity of risk-neutral pricing in the Black-Scholes model.

Solution to Exercise 5

Under \(\mathbb{Q}\), \(\tilde{S}_t = e^{-rt}S_t\) with \(S_t = S_0 \exp((r - \sigma^2/2)t + \sigma W_t^{\mathbb{Q}})\), so:

\[ \tilde{S}_t = S_0 \exp\left(-\frac{\sigma^2}{2}t + \sigma W_t^{\mathbb{Q}}\right) \]

Therefore \(\tilde{S}_t^2 = S_0^2 \exp(-\sigma^2 t + 2\sigma W_t^{\mathbb{Q}})\). Taking expectations:

\[ \mathbb{E}[\tilde{S}_t^2] = S_0^2 \exp(-\sigma^2 t) \cdot \mathbb{E}[\exp(2\sigma W_t^{\mathbb{Q}})] = S_0^2 \exp(-\sigma^2 t) \cdot \exp(2\sigma^2 t) = S_0^2 \exp(\sigma^2 t) \]

Now:

\[ \mathbb{E}\left[\int_0^T \sigma^2 \tilde{S}_s^2\,ds\right] = \sigma^2 \int_0^T \mathbb{E}[\tilde{S}_s^2]\,ds = \sigma^2 S_0^2 \int_0^T e^{\sigma^2 s}\,ds = S_0^2(e^{\sigma^2 T} - 1) < \infty \]

Since \(\mathbb{E}\left[\int_0^T \sigma^2 \tilde{S}_s^2\,ds\right] < \infty\), the Itô integral \(\int_0^t \sigma \tilde{S}_s\,dW_s^{\mathbb{Q}}\) is a true martingale (not just a local martingale). This ensures \(\tilde{S}_t\) is a true \(\mathbb{Q}\)-martingale, which validates the risk-neutral pricing formula \(V_0 = \mathbb{E}^{\mathbb{Q}}[e^{-rT}\Phi(S_T)]\) in the Black-Scholes model.

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Exercise 6. Suppose the discounted price process \(\tilde{S}_t\) is a strict local martingale under \(\mathbb{Q}\) with \(\mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] = 0.95\,S_0\). Compute the bubble component \(\beta_0\). Then, using the modified put-call parity \(C - P = \mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] - Ke^{-rT}\), show that the classical put-call parity fails and determine the sign of the error.

Solution to Exercise 6

The bubble component is:

\[ \beta_0 = S_0 - \mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] = S_0 - 0.95\,S_0 = 0.05\,S_0 \]

So 5% of the current price is due to the bubble.

For put-call parity, in the standard (true martingale) case:

\[ C - P = S_0 - Ke^{-rT} \]

Under the strict local martingale setting, the modified put-call parity is:

\[ C - P = \mathbb{E}^{\mathbb{Q}}[e^{-rT}S_T] - Ke^{-rT} = 0.95\,S_0 - Ke^{-rT} \]

Comparing with the classical formula:

\[ (C - P)_{\text{classical}} - (C - P)_{\text{actual}} = S_0 - 0.95\,S_0 = 0.05\,S_0 = \beta_0 > 0 \]

The classical put-call parity overestimates \(C - P\) by the bubble component \(\beta_0 = 0.05\,S_0\). Equivalently, the put price is higher than what classical parity would predict (it includes a "bubble premium"), while the call price is lower. The error is positive: \(S_0 - Ke^{-rT} > C - P\).

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Exercise 7. For the 3D Bessel process reciprocal \(M_t = 1/R_t\) starting from \(R_0 = r_0 > 0\), verify the Ito computation: apply Ito's formula to \(f(r) = 1/r\) and the SDE \(dR_t = (1/R_t)\,dt + dW_t\) to obtain \(dM_t = -M_t^2\,dW_t\). Explain why the absence of a \(dt\) term confirms \(M_t\) is a local martingale, and why the drift terms from \(f'\) and \(f''\) cancel exactly.

Solution to Exercise 7

Let \(f(r) = 1/r\), so \(f'(r) = -1/r^2\) and \(f''(r) = 2/r^3\). By Itô's formula applied to \(M_t = f(R_t)\):

\[ dM_t = f'(R_t)\,dR_t + \frac{1}{2}f''(R_t)\,(dR_t)^2 \]

With \(dR_t = \frac{1}{R_t}\,dt + dW_t\), we have \((dR_t)^2 = dt\) (since \((dW_t)^2 = dt\) and all other terms vanish). Substituting:

\[ dM_t = -\frac{1}{R_t^2}\left(\frac{1}{R_t}\,dt + dW_t\right) + \frac{1}{2}\cdot\frac{2}{R_t^3}\,dt \]

\[ = -\frac{1}{R_t^3}\,dt - \frac{1}{R_t^2}\,dW_t + \frac{1}{R_t^3}\,dt \]

\[ = -\frac{1}{R_t^2}\,dW_t = -M_t^2\,dW_t \]

The drift terms cancel exactly: \(f'(R_t) \cdot \mu(R_t) + \frac{1}{2}f''(R_t) \cdot 1 = -R_t^{-3} + R_t^{-3} = 0\). This happens because \(f(r) = 1/r\) is a harmonic function for the 3D Bessel generator \(\mathcal{L} = \frac{1}{r}\frac{\partial}{\partial r} + \frac{1}{2}\frac{\partial^2}{\partial r^2}\), i.e., \(\mathcal{L}f = 0\).

The absence of a \(dt\) term means \(M_t\) is a pure stochastic integral \(M_t = M_0 + \int_0^t (-M_s^2)\,dW_s\), which is by definition a local martingale. The drift cancellation is not coincidental — it reflects the deep connection between harmonic functions and martingales via the generator criterion.