Scaling and Time Change

Introduction

In Brownian Motion Foundations, we established the fundamental scaling property of Brownian motion (Theorem 1.3.8):

\[W_{ct} \overset{d}{=} \sqrt{c} W_t \quad \text{for any } c > 0\]

This self-similarity property is remarkable: Brownian motion has no intrinsic time scale. Zooming in on time by a factor \(c\) is equivalent (in distribution) to zooming out in space by a factor \(\sqrt{c}\).

This section explores three interconnected topics that build on this scaling property:

1. Consequences of scaling: How the \(\sqrt{\Delta t}\) scaling governs increment sizes and leads to Itô calculus
2. Deterministic time change: How reparameterizing time via \(\phi(t)\) produces new Gaussian processes
3. Random time change: The profound Dambis-Dubins-Schwarz theorem, which shows every continuous martingale is "Brownian motion run on a random clock"

These results form a bridge to stochastic integration (Chapter 1.3) and martingale theory (Chapter 1.2), revealing the universality of Brownian motion among continuous stochastic processes.

Brownian Scaling

1. Scaling Property

Let \(\{W_t\}_{t\ge 0}\) be a standard Brownian motion in \(\mathbb{R}^d\). For any constant \(c>0\), define

\[\widetilde{W}_t := \frac{1}{\sqrt{c}}\,W_{ct}, \qquad t\ge 0\]

Then \(\{\widetilde{W}_t\}_{t\ge 0}\) is again a standard Brownian motion (in distribution, and in fact as a process).

Proof:

Note that

\[\widetilde{W}_t-\widetilde{W}_s = \frac{1}{\sqrt{c}}\left(W_{ct}-W_{cs}\right)\sim \mathcal{N}\!\left(0,(t-s)I_d\right)\]

since \(W_{ct}-W_{cs}\sim \mathcal{N}(0,c(t-s)I_d)\).

The increments are independent (by independent increments of \(W\)), and paths are continuous. Thus \(\widetilde{W}\) satisfies all four conditions for Brownian motion. \(\square\)

Hence we write the self-similarity (scaling) property:

\[\boxed{ \{W_{ct}\}_{t\ge 0}\;\overset{d}{=}\;\{\sqrt{c}\,W_t\}_{t\ge 0} }\]

Remark: This is a restatement of Theorem 1.3.8 from Brownian Motion Foundations. We now explore its deeper implications.

2. Consequence 1: Typical Increment Size

For small \(\Delta t>0\),

\[W_{t+\Delta t}-W_t \sim \mathcal{N}(0,\Delta t)\]

Therefore:

\[\mathbb{E}\!\left[|W_{t+\Delta t}-W_t|^2\right]=\Delta t\]

The typical magnitude of an increment is of order \(\sqrt{\Delta t}\), not \(\Delta t\).

Heuristic notation: We often write

\[dW_t \sim \sqrt{dt}\]

to emphasize this scaling.

Implication for Itô calculus: When we compute \((dW_t)^2\) in stochastic integrals:

\[(dW_t)^2 = (\sqrt{dt})^2 = dt \quad \text{(not zero!)}\]

This is the source of the Itô correction terms that appear in Itô's lemma. For example, for \(f(W_t)\):

\[df(W_t) = f'(W_t) dW_t + \frac{1}{2}f''(W_t) dt\]

The \(\frac{1}{2}f''(W_t) dt\) term arises precisely because \((dW_t)^2 = dt\).

3. Consequence 2: No Preferred Time Scale

The scaling property shows Brownian motion has no preferred time scale.

Interpretation:

• Zooming in on time by factor \(c\) (looking at \(W_{ct}\))
• Is equivalent to zooming out in space by \(\sqrt{c}\) (looking at \(\sqrt{c} W_t\))

Physical analogy: If you observe Brownian motion without knowing the time units, you cannot determine the "true" time scale from the path alone. A process observed at millisecond resolution looks statistically identical (after spatial rescaling) to one observed at hour resolution.

Financial application: This is why volatility in asset returns scales with \(\sqrt{T}\):

• Daily volatility: \(\sigma\)
• \(T\)-day volatility: \(\sigma \sqrt{T}\)

This square-root-of-time scaling is fundamental to option pricing.

4. Consequence 3: Hurst Exponent

The scaling exponent \(1/2\) in \(\sqrt{c}\) is called the Hurst exponent for Brownian motion:

\[W_{ct} \overset{d}{=} c^{1/2} W_t\]

More generally, a self-similar process with exponent \(H\) satisfies:

\[X_{ct} \overset{d}{=} c^H X_t\]

• \(H = 1/2\): Brownian motion (standard diffusion)
• \(H > 1/2\): Persistent (positive autocorrelation, fractional Brownian motion)
• \(H < 1/2\): Anti-persistent (negative autocorrelation, mean-reverting)

Only \(H = 1/2\) gives processes with independent increments.

Deterministic Time Change

1. General Setup

Let \(\phi:[0,\infty)\to[0,\infty)\) be a function satisfying:

• Nondecreasing: \(\phi(s) \le \phi(t)\) for \(s < t\)
• Continuous
• \(\phi(0)=0\)

Define the time-changed process:

\[B_t := W_{\phi(t)}\]

Question: Is \(\{B_t\}\) a Brownian motion?

2. Properties of Time-Changed Process

Proposition 1.4.1

The process \(\{B_t\}\) is a continuous Gaussian process with:

1. \(\mathbb{E}[B_t]=0\)
2. \(\text{Cov}(B_s,B_t)=\min(\phi(s),\phi(t))\)

Proof:

(1) By linearity of expectation:

\[\mathbb{E}[B_t] = \mathbb{E}[W_{\phi(t)}] = 0\]

(2) Without loss of generality, assume \(s < t\). Then \(\phi(s) \le \phi(t)\), so:

\[\text{Cov}(B_s, B_t) = \text{Cov}(W_{\phi(s)}, W_{\phi(t)}) = \min(\phi(s), \phi(t)) = \phi(s)\]

The process is Gaussian because \(W\) is Gaussian.

Continuity follows from the continuity of \(W\) and \(\phi\). \(\square\)

3. When Is B_t a Brownian Motion?

Theorem 1.4.2

The time-changed process \(\{B_t\}\) has the finite-dimensional distributions of a standard Brownian motion if and only if \(\phi(t) = t\).

Proof:

(\(\Leftarrow\)) If \(\phi(t) = t\) then \(B_t = W_t\), which is Brownian motion by assumption.

(\(\Rightarrow\)) For \(B_t\) to have stationary Gaussian increments matching those of Brownian motion, we need:

\[B_t - B_s \sim \mathcal{N}(0, t-s) \quad \text{for all } s < t\]

But:

\[B_t - B_s = W_{\phi(t)} - W_{\phi(s)} \sim \mathcal{N}(0, \phi(t) - \phi(s))\]

For this to equal \(\mathcal{N}(0, t-s)\), we need:

\[\phi(t) - \phi(s) = t - s \quad \text{for all } s < t\]

This implies \(\phi(t) = t + c\) for some constant \(c\). Combined with \(\phi(0) = 0\), we get \(\phi(t) = t\). Continuity of the resulting paths is then inherited from \(W\) and \(\phi\). \(\square\)

4. Special Case: Linear Time Change

If \(\phi(t) = ct\) for some constant \(c > 0\):

\[B_t = W_{\phi(t)} = W_{ct} \;\overset{d}{=}\; \sqrt{c}\,W_t\]

This is Brownian motion up to a spatial scaling by \(\sqrt{c}\).

Interpretation: Speeding up or slowing down time by a constant factor produces Brownian motion in a rescaled space.

5. Example: Quadratic Time Change

Let \(\phi(t) = t^2\). Then:

\[B_t = W_{t^2}\]

Properties:

• \(\mathbb{E}[B_t] = 0\)
• \(\text{Var}(B_t) = t^2\) (not \(t\)!)
• \(\text{Cov}(B_s, B_t) = \min(s^2, t^2)\)

This is not a Brownian motion because:

\[\text{Var}(B_t - B_s) = t^2 - s^2 = (t-s)(t+s) \neq t-s\]

The increments are not stationary.

Random Time Change

We now consider random time changes, which lead to one of the most profound results in stochastic calculus.

1. Motivation

The quadratic variation \(\langle W \rangle_t = t\) for Brownian motion suggests that time itself can be measured by quadratic variation.

For a general continuous martingale \(M_t\) with quadratic variation \(\langle M \rangle_t\), can we "run it on a clock" that makes it look like Brownian motion?

2. Setup for Random Time Change

Let \(M=\{M_t\}_{t\ge 0}\) be a continuous local martingale with:

• \(M_0=0\)
• Quadratic variation \(\langle M\rangle_t\) is strictly increasing and continuous

Define the inverse of the quadratic variation:

\[\tau(u) := \inf\{t\ge 0:\langle M\rangle_t > u\}, \qquad u\ge 0\]

This is a stopping time for each \(u\).

Interpretation: \(\tau(u)\) is the (random) time at which the quadratic variation of \(M\) reaches level \(u\).

3. Key Property of the Inverse

Proposition 1.4.3

The time-changed process \(\{M_{\tau(u)}\}_{u\ge 0}\) has quadratic variation:

\[\langle M_{\tau(\cdot)} \rangle_u = u\]

Proof sketch:

By definition of \(\tau(u)\):

\[\langle M \rangle_{\tau(u)} = u\]

The quadratic variation of the time-changed process satisfies:

\[\langle M_{\tau(\cdot)} \rangle_u = \langle M \rangle_{\tau(u)} = u \quad \square\]

This shows that "running \(M\) on the quadratic variation clock" produces a process with quadratic variation equal to \(u\).

4. Dambis-Dubins-Schwarz Theorem

The fundamental result connecting continuous martingales to Brownian motion is:

Theorem 1.4.4 (Dambis-Dubins-Schwarz)

Let \(M = \{M_t\}_{t \ge 0}\) be a continuous local martingale with \(M_0 = 0\) and \(\langle M \rangle_\infty = \infty\), and let \(\tau(u) = \inf\{t \ge 0 : \langle M\rangle_t > u\}\) be the inverse of its quadratic variation (as defined above). Then the process

\[B_u := M_{\tau(u)}, \qquad u \ge 0,\]

is a standard Brownian motion, and:

\[\boxed{M_t = B_{\langle M\rangle_t}, \quad t\ge 0.}\]

Proof: We will prove this in Chapter 1.2 after developing martingale theory. The key steps are:

1. Show \(B_u = M_{\tau(u)}\) is a continuous martingale with \(B_0 = 0\)
2. Verify \(\langle B \rangle_u = u\) (done above in Proposition 1.4.3)
3. Apply Lévy's characterization: A continuous martingale with \(\langle B \rangle_u = u\) is a Brownian motion

\(\square\)

Remark: The condition \(\langle M \rangle_\infty = \infty\) ensures that \(\tau(u) < \infty\) for all \(u\), so \(B_u\) is well-defined for all \(u \ge 0\).

5. Interpretation and Significance

The Dambis-Dubins-Schwarz theorem reveals that:

Every continuous local martingale is a Brownian motion run on a random clock.

More precisely:

• The "random clock" is the quadratic variation \(\langle M \rangle_t\)
• The process \(M_t\) evolves like Brownian motion, but time is measured by how much quadratic variation has accumulated

Why this matters:

1. Universality of Brownian motion: All continuous martingales share the same fundamental structure
2. Quadratic variation is the right "time": Not calendar time \(t\), but accumulated variance \(\langle M \rangle_t\)
3. Simplification: Many properties of general martingales reduce to properties of Brownian motion via this representation

Example: For the Itô integral \(M_t = \int_0^t H_s dW_s\):

\[\langle M \rangle_t = \int_0^t H_s^2 ds\]

So \(M_t\) is a Brownian motion under the clock \(\int_0^t H_s^2 ds\).

Link to Stochastic Integration

The time-change perspective provides deep insight into stochastic integrals.

1. Stochastic Integrals as Time-Changed Brownian Motion

For an Itô integral

\[M_t := \int_0^t H_s\,dW_s\]

where \(H_s\) is a predictable process, the quadratic variation is:

\[\langle M\rangle_t = \int_0^t H_s^2\,ds\]

By the Dambis-Dubins-Schwarz theorem, there exists a Brownian motion \(\{B_u\}\) such that:

\[\boxed{ \int_0^t H_s dW_s = B_{\int_0^t H_s^2 ds} }\]

Interpretation:

• The stochastic integral \(\int_0^t H_s dW_s\) evolves like Brownian motion
• But time is measured by \(\int_0^t H_s^2 ds\), not \(t\)
• When \(H_s\) is large, "time speeds up" (high volatility)
• When \(H_s\) is small, "time slows down" (low volatility)

2. Example: Constant Integrand

If \(H_s = \sigma\) (constant), then:

\[M_t = \int_0^t \sigma dW_s = \sigma W_t\]

\[\langle M \rangle_t = \int_0^t \sigma^2 ds = \sigma^2 t\]

By Dambis-Dubins-Schwarz:

\[\sigma W_t = B_{\sigma^2 t}\]

This is consistent with the scaling property: \(B_{\sigma^2 t} \overset{d}{=} \sigma W_t\).

3. Example: Time-Dependent Integrand

If \(H_s = \sqrt{s}\), then:

\[M_t = \int_0^t \sqrt{s} dW_s\]

\[\langle M \rangle_t = \int_0^t s ds = \frac{t^2}{2}\]

By Dambis-Dubins-Schwarz:

\[\int_0^t \sqrt{s} dW_s = B_{t^2/2}\]

The integral accumulates quadratic variation at rate \(s\) at time \(s\), so the clock runs faster as time progresses.

Applications in Finance

1. Stochastic Volatility Models

In a stochastic volatility model, the asset price satisfies:

\[dS_t = \mu S_t dt + \sigma_t S_t dW_t\]

The log-return has quadratic variation:

\[\langle \log S \rangle_t = \int_0^t \sigma_s^2 ds\]

By Dambis-Dubins-Schwarz, the log-return is a Brownian motion run on the "integrated variance clock" \(\int_0^t \sigma_s^2 ds\).

Implication: Even though volatility \(\sigma_t\) is random, the fundamental structure remains Brownian.

2. Option Pricing and Realized Variance

The Black-Scholes formula uses:

\[\text{Variance} = \sigma^2 T\]

But if volatility varies, the relevant quantity is:

\[\text{Realized Variance} = \int_0^T \sigma_t^2 dt\]

This is the "true time" as measured by quadratic variation.

3. Time-Changed Lévy Processes

More generally, one can time-change Lévy processes (not just Brownian motion) to model jumps in asset prices. The "business time" is again measured by some activity clock (e.g., trading volume, information arrival).

Simulation: Verifying Scaling and Time Change

This simulation verifies the scaling property and demonstrates deterministic time change.

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

np.random.seed(42) # Fixed seed for reproducibility

Parameters

T = 1.0 num_paths = 10000 num_steps = 1000 dt = T / num_steps

Generate Brownian motion paths

dW = np.random.normal(0, np.sqrt(dt), size=(num_paths, num_steps)) W = np.cumsum(np.hstack([np.zeros((num_paths, 1)), dW]), axis=1) time_grid = np.linspace(0, T, num_steps + 1)

=============================================================================

Part 1: Verify Scaling Property W_{ct} =^d sqrt(c) * W_t

=============================================================================

c_values = [0.25, 1.0, 4.0]

fig, axes = plt.subplots(1, 3, figsize=(15, 4)) for idx, c in enumerate(c_values): ax = axes[idx]

# For W_{cT}, we need paths up to time cT
# Generate extended paths if needed
if c > 1:
    extended_steps = int(c * num_steps)
    dW_ext = np.random.normal(0, np.sqrt(dt), size=(num_paths, extended_steps))
    W_ext = np.cumsum(np.hstack([np.zeros((num_paths, 1)), dW_ext]), axis=1)
    W_cT = W_ext[:, int(c * num_steps)]
else:
    W_cT = W[:, int(c * num_steps)]

sqrt_c_W_T = np.sqrt(c) * W[:, -1]

# Compare distributions
bins = np.linspace(-4, 4, 50)
ax.hist(W_cT, bins=bins, density=True, alpha=0.5, label=f'$W_{{{c}T}}$')
ax.hist(sqrt_c_W_T, bins=bins, density=True, alpha=0.5, label=f'$\\sqrt{{{c}}}W_T$')
ax.set_title(f'$c = {c}$: Var$(W_{{cT}})$ = {np.var(W_cT):.3f}', fontsize=11)
ax.legend()
ax.grid(alpha=0.3)

plt.suptitle('Scaling Property: \(W_{ct} \\overset{d}{=} \\sqrt{c} W_t\)', fontsize=14) plt.tight_layout() plt.savefig('figures/fig07_scaling.png', dpi=150, bbox_inches='tight') plt.show()

=============================================================================

Part 2: Deterministic Time Change - Compare W_t vs W_{t^2}

=============================================================================

fig, axes = plt.subplots(1, 2, figsize=(12, 4))

Left: Variance comparison

t_points = np.linspace(0.1, 1.0, 10) var_W_t = [] var_W_t2 = []

for t in t_points: idx_t = int(t * num_steps) idx_t2 = int(t**2 * num_steps) var_W_t.append(np.var(W[:, idx_t])) var_W_t2.append(np.var(W[:, idx_t2]))

axes[0].plot(t_points, var_W_t, 'o-', label='Var\((W_t)\) (empirical)', markersize=8) axes[0].plot(t_points, t_points, 'k--', label='\(t\) (theoretical)', linewidth=2) axes[0].plot(t_points, var_W_t2, 's-', label='Var\((W_{t^2})\) (empirical)', markersize=8) axes[0].plot(t_points, t_points**2, 'r--', label='\(t^2\) (theoretical)', linewidth=2) axes[0].set_xlabel('\(t\)', fontsize=12) axes[0].set_ylabel('Variance', fontsize=12) axes[0].set_title('Variance: \(W_t\) vs \(W_{t^2}\)', fontsize=13) axes[0].legend() axes[0].grid(alpha=0.3)

Right: Sample paths

axes[1].plot(time_grid, W[0, :], label='\(W_t\)', alpha=0.8) t2_grid = time_grid**2 axes[1].plot(time_grid, W[0, (t2_grid * num_steps).astype(int).clip(0, num_steps)], label='\(W_{t^2}\)', alpha=0.8) axes[1].set_xlabel('\(t\)', fontsize=12) axes[1].set_ylabel('Value', fontsize=12) axes[1].set_title('Sample Paths: \(W_t\) vs \(W_{t^2}\)', fontsize=13) axes[1].legend() axes[1].grid(alpha=0.3)

plt.tight_layout() plt.savefig('figures/fig08_time_change.png', dpi=150, bbox_inches='tight') plt.show()

Print verification

print("Scaling Property Verification:") print("-" * 50) for c in c_values: if c > 1: extended_steps = int(c * num_steps) dW_ext = np.random.normal(0, np.sqrt(dt), size=(num_paths, extended_steps)) W_ext = np.cumsum(np.hstack([np.zeros((num_paths, 1)), dW_ext]), axis=1) var_Wct = np.var(W_ext[:, int(c * num_steps)]) else: var_Wct = np.var(W[:, int(c * num_steps)]) print(f"c = {c}: Var(W_{{cT}}) = {var_Wct:.4f}, theoretical = {c*T:.4f}")

print("\nTime Change Verification:") print("-" * 50) print(f"Var(W_{{0.5}}) = {np.var(W[:, int(0.5num_steps)]):.4f}, theoretical = 0.5") print(f"Var(W_{{0.25}}) = {np.var(W[:, int(0.25num_steps)]):.4f}, theoretical = 0.25") print(f"Var(W_{{(0.5)^2}}) = Var(W_{{0.25}}) (time-changed process at t=0.5)") ```

Output: ``` Scaling Property Verification:

---

c = 0.25: Var(W_{cT}) = 0.2500, theoretical = 0.2500 c = 1.0: Var(W_{cT}) = 0.9837, theoretical = 1.0000 c = 4.0: Var(W_{cT}) = 3.9144, theoretical = 4.0000

Time Change Verification:

Var(W_{0.5}) = 0.5030, theoretical = 0.5 Var(W_{0.25}) = 0.2500, theoretical = 0.25 Var(W_{(0.5)^2}) = Var(W_{0.25}) (time-changed process at t=0.5) ```

Interpretation:

• Scaling property (top): The distributions of \(W_{cT}\) and \(\sqrt{c}W_T\) match for all values of \(c\), confirming \(W_{ct} \overset{d}{=} \sqrt{c}W_t\)
• Time change (bottom left): Var\((W_t) = t\) while Var\((W_{t^2}) = t^2\), showing the time-changed process has non-stationary increments
• Sample paths (bottom right): \(W_{t^2}\) appears "compressed" near \(t=0\) and "stretched" near \(t=1\) compared to \(W_t\)

Summary

This section explored three levels of time transformation:

1. Scaling (\(W_{ct} \overset{d}{=} \sqrt{c} W_t\)): No intrinsic time scale, \(\sqrt{\Delta t}\) increments
2. Deterministic time change (\(W_{\phi(t)}\)): Produces Gaussian processes, but only \(\phi(t) = t\) gives Brownian motion
3. Random time change (Dambis-Dubins-Schwarz): Every continuous martingale is Brownian motion on a random clock

Key insights:

• Quadratic variation \(\langle M \rangle_t\) is the "right" measure of time for martingales
• Brownian motion is universal among continuous martingales
• Stochastic integrals \(\int H_s dW_s\) are Brownian motions run on the clock \(\int H_s^2 ds\)

These ideas are fundamental to:

• Itô calculus (Chapter 1.4): Understanding why \((dW_t)^2 = dt\)
• Martingale theory (Chapter 1.2): Characterizing continuous martingales
• Stochastic integration (Chapter 1.3): Defining \(\int H_s dW_s\) rigorously
• Volatility modeling: Measuring time by integrated variance

The next sections will make these connections rigorous and develop the full machinery of stochastic calculus.

Exercises

Scaling Property

Let \(a > 0\) and define \(X_t := \frac{1}{\sqrt{a}} W_{at}\).

1. Show that \((X_t)_{t \ge 0}\) is a standard Brownian motion by verifying all four defining properties.

Solution to Exercise 1

Define \(X_t = \frac{1}{\sqrt{a}} W_{at}\). We verify the four defining properties of Brownian motion.

(i) Initial condition: \(X_0 = \frac{1}{\sqrt{a}} W_0 = 0\) a.s.

(ii) Independent increments: For \(0 \leq t_0 < t_1 < \cdots < t_n\):

\[ X_{t_k} - X_{t_{k-1}} = \frac{1}{\sqrt{a}}(W_{at_k} - W_{at_{k-1}}) \]

Since \(at_0 < at_1 < \cdots < at_n\), the increments \(W_{at_k} - W_{at_{k-1}}\) are independent by the independent increments property of \(W\). Scaling by \(1/\sqrt{a}\) preserves independence.

(iii) Gaussian stationary increments: For \(s < t\):

\[ X_t - X_s = \frac{1}{\sqrt{a}}(W_{at} - W_{as}) \sim \frac{1}{\sqrt{a}} \mathcal{N}(0, a(t-s)) = \mathcal{N}(0, t-s) \]

(iv) Continuity of paths: Since \(t \mapsto at\) is continuous and \(t \mapsto W_t\) is continuous a.s., the composition \(t \mapsto W_{at}\) is continuous a.s. Dividing by the constant \(\sqrt{a}\) preserves continuity.

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1. Explain why this property is called self-similarity and what it implies about the visual appearance of Brownian paths at different time scales.

Solution to Exercise 2

Self-similarity means that the statistical properties of the process are preserved under simultaneous rescaling of time and space. Specifically, \(W_{ct} \overset{d}{=} \sqrt{c}\,W_t\) means the process at time scale \(c\) is identical in distribution to the process at the original time scale after spatial rescaling by \(\sqrt{c}\).

Visual implication: If you zoom into a Brownian path by magnifying the time axis by a factor \(c\) and the space axis by \(\sqrt{c}\), the resulting picture is statistically indistinguishable from the original. At any resolution — whether you view the path over milliseconds or years — the roughness and structure look the same. There is no characteristic scale at which the path becomes "smooth" or "different."

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1. How does the scaling property affect: (a) The variance \(\text{Var}(X_t)\) compared to \(\text{Var}(W_t)\)? (b) The relationship between "time units" and "space units"?

Solution to Exercise 3

(a) Since \(X_t = \frac{1}{\sqrt{a}} W_{at}\) is a standard Brownian motion (by Exercise 1):

\[ \text{Var}(X_t) = t \]

And indeed \(\text{Var}(X_t) = \frac{1}{a}\text{Var}(W_{at}) = \frac{1}{a} \cdot at = t = \text{Var}(W_t)\). The variance is the same.

(b) The scaling \(W_{ct} \overset{d}{=} \sqrt{c}\,W_t\) relates time and space units: multiplying time by \(c\) is equivalent to multiplying space by \(\sqrt{c}\). This gives the fundamental relationship "space scales as the square root of time." In finance, this is why daily volatility \(\sigma_{\text{daily}}\) annualizes as \(\sigma_{\text{annual}} = \sigma_{\text{daily}}\sqrt{252}\).

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Deterministic Time Change

1. Let \(\phi(t) = t^2\) and define \(B_t = W_{\phi(t)} = W_{t^2}\). (a) Compute \(\text{Var}(B_t)\) and \(\text{Cov}(B_s, B_t)\) for \(s < t\). (b) Is \(B_t\) a Brownian motion? Why or why not? (c) Are the increments \(B_t - B_s\) stationary?

Solution to Exercise 4

(a) \(B_t = W_{t^2}\), so:

\[ \text{Var}(B_t) = \text{Var}(W_{t^2}) = t^2 \]

For \(s < t\): \(\text{Cov}(B_s, B_t) = \text{Cov}(W_{s^2}, W_{t^2}) = \min(s^2, t^2) = s^2\).

(b) \(B_t\) is not a Brownian motion. For Brownian motion we need \(\text{Var}(B_t) = t\), but \(\text{Var}(B_t) = t^2 \neq t\) (for \(t \neq 0, 1\)).

(c) The increments are not stationary:

\[ \text{Var}(B_t - B_s) = \text{Var}(W_{t^2} - W_{s^2}) = t^2 - s^2 = (t-s)(t+s) \]

This depends on \(s\) and \(t\) separately, not just on \(t - s\). For instance, \(\text{Var}(B_2 - B_1) = 4 - 1 = 3\) while \(\text{Var}(B_3 - B_2) = 9 - 4 = 5\), even though both intervals have length 1.

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1. For what functions \(\phi(t)\) is \(W_{\phi(t)}\) a Brownian motion? Prove your answer.

Solution to Exercise 5

\(W_{\phi(t)}\) is a standard Brownian motion if and only if \(\phi(t) = t\).

Proof. For \(W_{\phi(t)}\) to be a Brownian motion, we need \(W_{\phi(t)} - W_{\phi(s)} \sim \mathcal{N}(0, t-s)\) for all \(s < t\). But:

\[ W_{\phi(t)} - W_{\phi(s)} \sim \mathcal{N}(0, \phi(t) - \phi(s)) \]

So we need \(\phi(t) - \phi(s) = t - s\) for all \(0 \leq s < t\). Setting \(s = 0\): \(\phi(t) - \phi(0) = t - 0\), and since \(\phi(0) = 0\), we get \(\phi(t) = t\) for all \(t\).

Conversely, if \(\phi(t) = t\) then \(W_{\phi(t)} = W_t\), which is a Brownian motion by assumption.

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1. Let \(\phi(t) = e^t - 1\). Compute the covariance \(\text{Cov}(W_{\phi(s)}, W_{\phi(t)})\) and describe how this time change affects the process.

Solution to Exercise 6

With \(\phi(t) = e^t - 1\):

\[ \text{Cov}(W_{\phi(s)}, W_{\phi(t)}) = \min(\phi(s), \phi(t)) = \min(e^s - 1, e^t - 1) \]

For \(s < t\), this equals \(e^s - 1\).

Effect of the time change: Since \(\phi'(t) = e^t\), the clock runs exponentially faster as \(t\) increases. Near \(t = 0\), \(\phi(t) \approx t\) so the process behaves like standard Brownian motion. For large \(t\), \(\phi(t) \approx e^t\), so the variance \(\text{Var}(W_{\phi(t)}) = e^t - 1\) grows exponentially rather than linearly. The increments become increasingly volatile over time.

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Hurst Exponent

1. A self-similar process with Hurst exponent \(H\) satisfies \(X_{ct} \overset{d}{=} c^H X_t\). (a) Verify that standard Brownian motion has \(H = 1/2\). (b) If \(H > 1/2\), what does this imply about the correlation of increments? (c) Why is \(H = 1/2\) the only value compatible with independent increments?

Solution to Exercise 7

(a) By the scaling property, \(W_{ct} \overset{d}{=} \sqrt{c}\,W_t\). This means \(X_{ct} \overset{d}{=} c^{1/2} X_t\) where \(X = W\), confirming \(H = 1/2\).

(b) If \(H > 1/2\), the process exhibits persistent behavior: the increments are positively correlated. If the process went up in the past, it is more likely to continue going up. This is sometimes called "long-range dependence" or "trending" behavior.

(c) For a process with independent increments, the variance of the increment over \([0, ct]\) must equal the sum of variances over \([0, t]\) and \([t, ct]\). If \(X_{ct} \overset{d}{=} c^H X_t\), then \(\text{Var}(X_{ct}) = c^{2H}\text{Var}(X_t)\). By independent stationary increments, \(\text{Var}(X_{ct}) = ct \cdot \sigma^2\). Setting equal: \(c^{2H} t\sigma^2 = ct\sigma^2\), so \(c^{2H} = c\) for all \(c > 0\), which requires \(2H = 1\), i.e., \(H = 1/2\).

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Random Time Change (Dambis-Dubins-Schwarz)

1. Let \(M_t = \int_0^t \sigma dW_s = \sigma W_t\) for constant \(\sigma > 0\). (a) Compute the quadratic variation \(\langle M \rangle_t\). (b) Verify that \(M_t = B_{\langle M \rangle_t}\) for some Brownian motion \(B\).

Solution to Exercise 8

(a) For \(M_t = \sigma W_t\), the quadratic variation is:

\[ \langle M \rangle_t = \sigma^2 \langle W \rangle_t = \sigma^2 t \]

(b) By the Dambis-Dubins-Schwarz theorem, there exists a Brownian motion \(B\) such that \(M_t = B_{\langle M \rangle_t} = B_{\sigma^2 t}\). Indeed:

\[ B_{\sigma^2 t} \overset{d}{=} \sqrt{\sigma^2 t}\,Z = \sigma\sqrt{t}\,Z \quad \text{where } Z \sim \mathcal{N}(0,1) \]

And \(M_t = \sigma W_t \sim \mathcal{N}(0, \sigma^2 t)\), which matches. This is also consistent with the scaling property: \(B_{\sigma^2 t} \overset{d}{=} \sigma B_t \overset{d}{=} \sigma W_t = M_t\).

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1. Let \(M_t = \int_0^t s \, dW_s\). (a) Compute \(\langle M \rangle_t\). (b) Express \(M_t\) as a time-changed Brownian motion.

Solution to Exercise 9

(a) For \(M_t = \int_0^t s\,dW_s\), the quadratic variation is computed using the Itô isometry formula:

\[ \langle M \rangle_t = \int_0^t s^2\,ds = \frac{t^3}{3} \]

(b) By the Dambis-Dubins-Schwarz theorem, there exists a Brownian motion \(B\) such that:

\[ M_t = B_{\langle M \rangle_t} = B_{t^3/3} \]

The stochastic integral \(\int_0^t s\,dW_s\) is a Brownian motion run on the clock \(t^3/3\). Since the integrand \(H_s = s\) grows with time, the clock accelerates: the process accumulates quadratic variation at rate \(s^2\) at time \(s\), so early times contribute less randomness than later times.

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1. (Stochastic Volatility Interpretation) In a model where \(dS_t = \sigma_t S_t dW_t\), explain why the "realized variance" \(\int_0^T \sigma_t^2 dt\) represents the "true time" experienced by the asset price.

Solution to Exercise 10

In the stochastic volatility model \(dS_t = \sigma_t S_t\,dW_t\) (ignoring drift for simplicity), the log-price satisfies \(d\log S_t = -\frac{1}{2}\sigma_t^2\,dt + \sigma_t\,dW_t\). The martingale part is \(M_t = \int_0^t \sigma_s\,dW_s\), with quadratic variation:

\[ \langle M \rangle_t = \int_0^t \sigma_s^2\,ds \]

By Dambis-Dubins-Schwarz, \(M_t = B_{\int_0^t \sigma_s^2 ds}\) for some Brownian motion \(B\).

The realized variance \(\int_0^T \sigma_t^2\,dt\) represents the "true time" because it measures the total amount of randomness experienced by the asset price. When volatility is high, the asset accumulates more randomness per unit of calendar time, so its "internal clock" runs faster. When volatility is low, the clock runs slower. The realized variance is the total elapsed time on this internal clock, which is why option prices in stochastic volatility models depend on the distribution of \(\int_0^T \sigma_t^2\,dt\) rather than just \(\sigma^2 T\).

References

• Dambis, K. E. (1965). On the decomposition of continuous submartingales. Theory of Probability & Its Applications, 10(3), 401-410.
• Dubins, L. E., & Schwarz, G. (1965). On continuous martingales. Proceedings of the National Academy of Sciences, 53(5), 913-916.
• Karatzas, I., & Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer. (Chapter 1, Section 6)
• Revuz, D., & Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer. (Chapter V)