Bridge to Stochastic Differential Equations¶

The previous sections established that financial returns are random, non-differentiable, heavy-tailed, and exhibit volatility clustering — none of which deterministic ODEs can reproduce. This section builds the precise conceptual bridge from discrete-time random walks to continuous-time stochastic differential equations (SDEs), which form the mathematical core of modern quantitative finance.

The bridge has three spans: (1) the discrete model that is already familiar, (2) the limit process that arises as the time step shrinks, and (3) the shorthand SDE notation that encodes that limit.

Concept Definition¶

A stochastic differential equation is an equation of the form

\[ dX_t = \mu(X_t, t)\,dt + \sigma(X_t, t)\,dW_t, \]

where:

\(X_t\) is the state variable (e.g., an asset price or interest rate),
\(\mu(X_t,t)\) is the drift function — the infinitesimal expected change,
\(\sigma(X_t,t)\) is the diffusion function — the infinitesimal volatility,
\(W_t\) is standard Brownian motion.

This equation is shorthand for the integral equation

\[ X_t = X_0 + \int_0^t \mu(X_s, s)\,ds + \int_0^t \sigma(X_s, s)\,dW_s, \]

where the first integral is an ordinary Riemann integral and the second is an Itô stochastic integral, whose rigorous definition is the subject of Section 2.1.

From Discrete Random Walk to Continuous-Time Limit¶

Step 1 — The Discrete-Time Model¶

Fix a time step \(\Delta t > 0\). Suppose log-prices evolve via

\[ \log S_{n+1} - \log S_n = \mu\,\Delta t + \sigma\sqrt{\Delta t}\cdot Z_n, \qquad Z_n \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,1). \]

Equivalently, in multiplicative form:

\[ S_{n+1} = S_n \exp\!\left(\mu\,\Delta t + \sigma\sqrt{\Delta t}\cdot Z_n\right). \]

This model has three desirable properties:

Positivity: \(S_n > 0\) for all \(n\) whenever \(S_0 > 0\).
Correct variance scaling: \(\operatorname{Var}(\log S_{n+1} - \log S_n) = \sigma^2\Delta t\), consistent with empirical returns.
Multiplicative structure: Volatility is proportional to price level.

Step 2 — Refining the Time Step¶

Let \(t = n\,\Delta t\) and write the change over one step as

\[ S(t+\Delta t) - S(t) = S(t)\!\left[\exp\!\left(\mu\,\Delta t + \sigma\sqrt{\Delta t}\cdot Z\right) - 1\right]. \]

For small \(\Delta t\) the first-order Taylor expansion \(e^x - 1 \approx x\) gives the heuristic approximation

\[ S(t+\Delta t) - S(t) \approx S(t)\!\left(\mu\,\Delta t + \sigma\sqrt{\Delta t}\cdot Z\right). \]

This expansion discards the Itô correction

The full second-order expansion gives \(e^x - 1 \approx x + \frac{1}{2}x^2\). The discarded term \(\frac{1}{2}(\sigma\sqrt{\Delta t})^2 = \frac{1}{2}\sigma^2\Delta t\) is \(O(\Delta t)\) — the same asymptotic order as the drift \(\mu\Delta t\) — and therefore cannot be neglected when identifying the drift coefficient of the limiting SDE. Its contribution is precisely the \(-\frac{\sigma^2}{2}\) drift correction derived rigorously in Section 2.2 via Itô's lemma. The first-order expansion is sufficient to motivate the structure of the SDE (drift + diffusion terms) but not to determine the exact drift coefficient.

Rearranging the first-order approximation:

\[ \frac{S(t+\Delta t) - S(t)}{\Delta t} \approx \mu\,S(t) + \sigma\,S(t)\,\frac{Z}{\sqrt{\Delta t}}. \]

As \(\Delta t \to 0\) the second term \(\sigma S(t) \cdot Z/\sqrt{\Delta t}\) does not converge pathwise — its \(L^2\) norm grows like \(1/\sqrt{\Delta t}\). This is precisely why ordinary calculus is insufficient: the natural limit of the rescaled increment is not a classical derivative but a distributional object. The correct framework takes a weak limit of the entire rescaled random walk, not a pointwise limit of the ratio above.

Step 3 — Donsker's Theorem (Functional CLT)¶

Donsker's theorem applies to the rescaled log-price walk — the cumulative sum of \(Z_i\) increments — and shows it converges to Brownian motion. The price process \(S_t\) then follows by applying the exponential map to that limit.

Donsker's Invariance Principle (1951)

Let \(Z_1, Z_2, \ldots\) be i.i.d. with mean zero and variance one (not necessarily Gaussian). Define the linearly interpolated random walk, for \(t \in [0, T)\):

\[ W^{(\Delta t)}(t) = \sqrt{\Delta t}\sum_{i=1}^{\lfloor t/\Delta t \rfloor} Z_i + \sqrt{\Delta t}\!\left(\frac{t}{\Delta t} - \left\lfloor\frac{t}{\Delta t}\right\rfloor\right) Z_{\lfloor t/\Delta t\rfloor+1}, \]

with \(W^{(\Delta t)}(T)\) defined by continuity. As \(\Delta t \to 0\), \(W^{(\Delta t)} \xrightarrow{d} W\) in the space of continuous functions on \([0,T]\), where \(W\) is standard Brownian motion.

Two remarks are essential:

Universality: The limit is Brownian motion regardless of the distribution of \(Z_i\), as long as the mean is zero and variance is one. This justifies using Brownian motion as the canonical noise process for any asset whose log-returns have finite variance.
The \(\sqrt{\Delta t}\) scaling is decisive. Scaling by \(\Delta t\) would collapse the walk to zero; scaling by \(1\) (no rescaling) would cause it to diverge. Only \(\sqrt{\Delta t}\) keeps the limit non-degenerate.

Step 4 — The Formal SDE Limit¶

With Donsker's theorem in hand, the discrete model

\[ S(t+\Delta t) - S(t) \approx \mu\,S(t)\,\Delta t + \sigma\,S(t)\,\sqrt{\Delta t}\cdot Z \]

converges in distribution, as \(\Delta t \to 0\), to the SDE

\[ \boxed{dS_t = \mu\,S_t\,dt + \sigma\,S_t\,dW_t.} \]

This is Geometric Brownian Motion (GBM), the foundation of the Black-Scholes model. Its explicit solution — derived in Section 2.2 using Itô's lemma, which recovers the \(O(\Delta t)\) term discarded above — is

\[ S_t = S_0 \exp\!\left[\left(\mu - \tfrac{\sigma^2}{2}\right)t + \sigma W_t\right]. \]

The \(-\frac{\sigma^2}{2}\) drift correction is a consequence of \((dW_t)^2 = dt \neq 0\), not of the heuristic derivation here.

Notation is heuristic, content is rigorous

Writing "\(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t\)" is a compact shorthand. The symbol \(dW_t\) does not denote an ordinary derivative — Brownian paths are nowhere differentiable, so \(W'(t)\) does not exist. The meaning is always the integral form \(S_t = S_0 + \int_0^t \mu S_s\,ds + \int_0^t \sigma S_s\,dW_s\), where the second integral is defined via Itô's construction (Section 2.1).

What is \(dW_t\)?¶

Standard Brownian motion \(W_t\) is a stochastic process satisfying:

\(W_0 = 0\) almost surely,
Independent increments: \(W_t - W_s \perp W_u - W_v\) for non-overlapping intervals \([s,t]\) and \([u,v]\),
Gaussian increments: \(W_t - W_s \sim \mathcal{N}(0,\, t-s)\) for \(s < t\),
Continuous paths: \(t \mapsto W_t\) is continuous almost surely.

The symbol \(dW_t\) is shorthand for the infinitesimal increment \(W_{t+dt} - W_t \sim \mathcal{N}(0, dt)\), satisfying

\[ \mathbb{E}[dW_t] = 0, \qquad \mathbb{E}[(dW_t)^2] = dt. \]

The second identity — \((dW_t)^2 = dt\) in the mean-square sense — has a precise pathwise version: the quadratic variation of \(W\) over \([0,T]\) is exactly \(T\):

\[ \lim_{|\mathcal{P}|\to 0} \sum_{i} \bigl(W_{t_{i+1}} - W_{t_i}\bigr)^2 = T \quad \text{almost surely.} \]

This stands in sharp contrast to ordinary calculus, where \((dx)^2 = 0\) for smooth functions. It is precisely this non-vanishing quadratic variation that forces the modification of the chain rule known as Itô's lemma:

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

The extra term \(\frac{1}{2}f''(W_t)\,dt\) has no analogue in ordinary calculus; it arises directly from \((dW_t)^2 = dt\).

Anatomy of a General SDE¶

Drift and Diffusion¶

The general Itô SDE

\[ dX_t = \mu(X_t, t)\,dt + \sigma(X_t, t)\,dW_t \]

has two components with distinct roles.

Drift term \(\mu(X_t,t)\,dt\): the deterministic trend with \(\mathbb{E}[dX_t \mid X_t] = \mu(X_t,t)\,dt\), units \([X]\) per unit time, analogue of the ODE right-hand side.

Diffusion term \(\sigma(X_t,t)\,dW_t\): random fluctuations with \(\operatorname{Var}(dX_t \mid X_t) = \sigma^2(X_t,t)\,dt\), units \([X]\) per \(\sqrt{\text{time}}\), mean zero so it adds uncertainty but not bias.

Integral Form¶

The SDE is always an abbreviation for

\[ X_t = X_0 + \underbrace{\int_0^t \mu(X_s, s)\,ds}_{\text{Riemann integral}} + \underbrace{\int_0^t \sigma(X_s, s)\,dW_s}_{\text{Itô integral}}. \]

The Itô integral cannot be defined as a Riemann–Stieltjes integral because Brownian paths have unbounded variation. Section 2.1 gives its rigorous construction as a mean-square limit of left-endpoint sums:

\[ \int_0^t \sigma(X_s,s)\,dW_s = \lim_{|\mathcal{P}|\to 0} \sum_i \sigma(X_{t_i}, t_i)\bigl(W_{t_{i+1}} - W_{t_i}\bigr). \]

Left endpoints are used because they make the integrand \(\sigma(X_{t_i}, t_i)\) measurable with respect to the information available at time \(t_i\), before the increment \(W_{t_{i+1}} - W_{t_i}\) is observed. This non-anticipativity is what gives the Itô integral its martingale property. By contrast, a right-endpoint integrand \(\sigma(X_{t_{i+1}}, t_{i+1})\) would use information about the future increment to determine its own coefficient, destroying the martingale property and introducing a systematic bias. Section 2.1 makes this precise.

The Journey in One Diagram¶

Real return data │ │ (heavy tails, clustering, non-differentiable paths) ▼ Stylized facts │ │ (ODE has zero variance, smooth paths — Failures 1–5) ▼ Deterministic models fail │ │ (multiplicative noise, Δt → 0, Donsker's theorem) ▼ Discrete random walk ──────────────────► Brownian motion W_t │ │ │ (formal limit) │ (rigorous foundation) ▼ ▼ dS_t = μ S_t dt + σ S_t dW_t ◄──── Itô calculus │ │ (solve via Itô's lemma, Section 2.2) ▼ S_t = S_0 exp[(μ − σ²/2)t + σ W_t] │ │ (extend drift/diffusion functions) ▼ General SDEs → Quantitative finance models

What Comes Next¶

The bridge is now conceptually complete. Three items remain to be made rigorous.

Section 2.1 — Itô Integration Defines \(\int_0^t f(s)\,dW_s\) as a mean-square limit, establishes the Itô isometry \(\mathbb{E}\!\left[\left(\int_0^t f\,dW\right)^2\right] = \int_0^t \mathbb{E}[f^2]\,ds\), and proves the martingale property.

Section 2.2 — Itô's Lemma Derives the stochastic chain rule \(df(X_t) = f'(X_t)\,dX_t + \frac{1}{2}f''(X_t)(dX_t)^2\), explains why the second-order term survives, and verifies the GBM solution stated above — showing explicitly how the \(-\frac{\sigma^2}{2}\) correction emerges from the non-zero quadratic variation of \(W_t\).

Section 2.3 — SDE Theory Presents canonical models (GBM, Vasicek/OU, CIR), solves them analytically where possible, and discusses numerical simulation when no closed form exists. Correlated multi-dimensional SDEs, including the Heston stochastic volatility model, are covered in Section 2.4.

Summary¶

The passage from discrete to continuous time is governed by one key observation: the \(\sqrt{\Delta t}\) scaling of random shocks is the only scaling consistent with a non-degenerate continuous-time limit. Donsker's theorem makes this precise — the rescaled log-price random walk converges to Brownian motion in distribution, universally across all finite-variance innovations, and the price process follows by the exponential map.

The resulting continuous-time object, the SDE

\[ dX_t = \mu(X_t,t)\,dt + \sigma(X_t,t)\,dW_t, \]

captures what the deterministic ODE cannot: randomness, non-differentiable paths, and a volatility that may depend on state and time. Making this construction rigorous, and learning to compute with it, is the goal of Chapter 2.

Next: Section 2.1 — Itô Integration. \(\square\)

Exercises¶

Exercise 1. Starting from the discrete model \(\log S_{n+1} - \log S_n = \mu\,\Delta t + \sigma\sqrt{\Delta t}\cdot Z_n\) with \(Z_n \sim \mathcal{N}(0,1)\) i.i.d., derive the mean and variance of the log-price \(\log S_N\) after \(N\) steps. Show that \(\operatorname{Var}(\log S_N) = \sigma^2 N\Delta t = \sigma^2 T\) where \(T = N\Delta t\), confirming that variance scales linearly with total time regardless of the step size \(\Delta t\).

Solution to Exercise 1

After \(N\) steps, the log-price is:

\[ \log S_N = \log S_0 + \sum_{n=0}^{N-1}(\log S_{n+1} - \log S_n) = \log S_0 + \sum_{n=0}^{N-1}(\mu\,\Delta t + \sigma\sqrt{\Delta t}\cdot Z_n) \]

Mean: Since \(\mathbb{E}[Z_n] = 0\) for all \(n\):

\[ \mathbb{E}[\log S_N] = \log S_0 + \sum_{n=0}^{N-1}\mu\,\Delta t = \log S_0 + N\mu\,\Delta t = \log S_0 + \mu T \]

where \(T = N\Delta t\).

Variance: Since the \(Z_n\) are i.i.d. with \(\operatorname{Var}(Z_n) = 1\), and \(\log S_0\) is a constant:

\[ \operatorname{Var}(\log S_N) = \operatorname{Var}\!\left(\sum_{n=0}^{N-1}\sigma\sqrt{\Delta t}\cdot Z_n\right) = \sum_{n=0}^{N-1}\sigma^2\Delta t\cdot\operatorname{Var}(Z_n) = N\sigma^2\Delta t \]

Since \(T = N\Delta t\):

\[ \operatorname{Var}(\log S_N) = \sigma^2 N\Delta t = \sigma^2 T \]

The variance depends only on the total time \(T = N\Delta t\), not on the individual values of \(N\) and \(\Delta t\). Whether we use \(N = 252\) steps of \(\Delta t = 1/252\) or \(N = 25200\) steps of \(\Delta t = 1/25200\), the variance is \(\sigma^2 T\) in both cases. This confirms that the variance of the log-price scales linearly with total time, independent of the discretisation — a necessary consistency condition for the continuous-time limit to be well-defined.

Exercise 2. Donsker's theorem states that the rescaled random walk \(W^{(\Delta t)}(t) = \sqrt{\Delta t}\sum_{i=1}^{\lfloor t/\Delta t\rfloor} Z_i\) converges in distribution to Brownian motion. Suppose the \(Z_i\) are i.i.d. with \(\mathbb{E}[Z_i] = 0\), \(\operatorname{Var}(Z_i) = 1\), but \(Z_i\) follows a Rademacher distribution (\(P(Z_i = +1) = P(Z_i = -1) = 1/2\)) rather than a Gaussian. Does Donsker's theorem still apply? What is the limiting process? Explain why this universality result is important for the use of Brownian motion in finance.

Solution to Exercise 2

Yes, Donsker's theorem still applies. The theorem requires only that the \(Z_i\) are i.i.d. with \(\mathbb{E}[Z_i] = 0\) and \(\operatorname{Var}(Z_i) = 1\). The Rademacher distribution satisfies both conditions:

\[ \mathbb{E}[Z_i] = \frac{1}{2}(+1) + \frac{1}{2}(-1) = 0 \]

\[ \operatorname{Var}(Z_i) = \mathbb{E}[Z_i^2] = \frac{1}{2}(1)^2 + \frac{1}{2}(-1)^2 = 1 \]

The limiting process is standard Brownian motion \(W_t\), the same limit as for Gaussian increments.

This universality is important for finance because it means the choice of Brownian motion as the canonical noise process does not depend on the specific distribution of individual price shocks. Real daily returns are non-Gaussian (heavy-tailed, skewed), yet the rescaled cumulative return process still converges to Brownian motion as the time step shrinks, provided the increments have finite variance. This justifies using the SDE framework \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t\) as a continuous-time model regardless of the precise distributional form of discrete-time returns. The only requirement is finite variance — when this fails (e.g., for stable distributions with \(\alpha < 2\)), the limit is no longer Brownian motion but a Levy process, requiring a different framework.

Exercise 3. For standard Brownian motion \(W_t\), verify from the defining properties that:

(a) \(\mathbb{E}[W_t] = 0\) for all \(t \geq 0\),

(b) \(\operatorname{Var}(W_t) = t\),

(c) \(\operatorname{Cov}(W_s, W_t) = \min(s, t)\) for \(s, t \geq 0\).

For part (c), use the decomposition \(W_t = (W_t - W_s) + W_s\) and the independent-increments property.

Solution to Exercise 3

(a) By property 3 (Gaussian increments), \(W_t = W_t - W_0 \sim \mathcal{N}(0, t - 0) = \mathcal{N}(0, t)\). Therefore:

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

(b) From the same property:

\[ \operatorname{Var}(W_t) = \mathbb{E}[W_t^2] - (\mathbb{E}[W_t])^2 = \mathbb{E}[W_t^2] - 0 = t \]

(c) Assume without loss of generality that \(s \leq t\). Decompose \(W_t = (W_t - W_s) + W_s\). Then:

\[ \operatorname{Cov}(W_s, W_t) = \operatorname{Cov}(W_s, (W_t - W_s) + W_s) \]

\[ = \operatorname{Cov}(W_s, W_t - W_s) + \operatorname{Cov}(W_s, W_s) \]

By property 2 (independent increments), \(W_t - W_s\) is independent of \(W_s = W_s - W_0\) since the intervals \([0, s]\) and \([s, t]\) are non-overlapping. For independent random variables, the covariance is zero:

\[ \operatorname{Cov}(W_s, W_t - W_s) = 0 \]

Therefore:

\[ \operatorname{Cov}(W_s, W_t) = 0 + \operatorname{Var}(W_s) = s = \min(s, t) \]

By symmetry (\(\operatorname{Cov}(W_s, W_t) = \operatorname{Cov}(W_t, W_s)\)), the result holds for \(s > t\) as well, giving \(\operatorname{Cov}(W_s, W_t) = \min(s, t)\) for all \(s, t \geq 0\).

Exercise 4. Consider the general SDE \(dX_t = \mu(X_t,t)\,dt + \sigma(X_t,t)\,dW_t\). For each of the following models, identify the drift function \(\mu(x,t)\) and diffusion function \(\sigma(x,t)\), and state whether the diffusion is additive or multiplicative:

(a) Geometric Brownian Motion: \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t\)

(b) Ornstein–Uhlenbeck: \(dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t\)

(c) Cox–Ingersoll–Ross: \(dr_t = \kappa(\theta - r_t)\,dt + \xi\sqrt{r_t}\,dW_t\)

Solution to Exercise 4

(a) Geometric Brownian Motion: \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t\)

Drift: \(\mu(x, t) = \mu x\)
Diffusion: \(\sigma(x, t) = \sigma x\)
The diffusion is multiplicative because \(\sigma(x,t)\) depends on the state \(x\) (proportional to the current price).

(b) Ornstein--Uhlenbeck: \(dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t\)

Drift: \(\mu(x, t) = \kappa(\theta - x)\) (mean-reverting toward \(\theta\))
Diffusion: \(\sigma(x, t) = \sigma\) (a constant)
The diffusion is additive because \(\sigma(x,t) = \sigma\) does not depend on the state \(x\).

(c) Cox--Ingersoll--Ross: \(dr_t = \kappa(\theta - r_t)\,dt + \xi\sqrt{r_t}\,dW_t\)

Drift: \(\mu(x, t) = \kappa(\theta - x)\) (mean-reverting toward \(\theta\), same structure as OU)
Diffusion: \(\sigma(x, t) = \xi\sqrt{x}\)
The diffusion is multiplicative because \(\sigma(x,t) = \xi\sqrt{x}\) depends on the state \(x\). This square-root diffusion ensures that the volatility of the interest rate decreases as \(r_t\) approaches zero, which helps keep \(r_t\) non-negative (under the Feller condition \(2\kappa\theta \geq \xi^2\)).

Exercise 5. The Itô integral uses left-endpoint evaluation: \(\sum_i \sigma(X_{t_i}, t_i)(W_{t_{i+1}} - W_{t_i})\). Explain why using right-endpoint evaluation \(\sigma(X_{t_{i+1}}, t_{i+1})\) instead would violate the non-anticipativity requirement. What practical consequence does this have for the martingale property of the Itô integral? Specifically, show that \(\mathbb{E}\!\left[\sigma(X_{t_i}, t_i)(W_{t_{i+1}} - W_{t_i}) \mid \mathcal{F}_{t_i}\right] = 0\) when \(\sigma(X_{t_i}, t_i)\) is \(\mathcal{F}_{t_i}\)-measurable, and explain why this fails if \(\sigma(X_{t_{i+1}}, t_{i+1})\) is used.

Solution to Exercise 5

Non-anticipativity with left-endpoint evaluation. When the integrand is \(\sigma(X_{t_i}, t_i)\), it depends only on the state at time \(t_i\). The increment \(W_{t_{i+1}} - W_{t_i}\) is the Brownian motion increment over the interval \((t_i, t_{i+1}]\). By the independent-increments property, \(W_{t_{i+1}} - W_{t_i}\) is independent of \(\mathcal{F}_{t_i}\) (the information available up to time \(t_i\)). Since \(\sigma(X_{t_i}, t_i)\) is \(\mathcal{F}_{t_i}\)-measurable, the integrand is determined before the increment is realised — this is non-anticipativity.

Why right-endpoint evaluation violates non-anticipativity. If we use \(\sigma(X_{t_{i+1}}, t_{i+1})\), the integrand depends on \(X_{t_{i+1}}\), which itself depends on the increment \(W_{t_{i+1}} - W_{t_i}\) (through the SDE dynamics). The coefficient of the Brownian increment would therefore depend on the increment itself — this is "looking into the future" and violates non-anticipativity.

Martingale property with left-endpoint evaluation. For the left-endpoint sum, consider a single term. Since \(\sigma(X_{t_i}, t_i)\) is \(\mathcal{F}_{t_i}\)-measurable and \(W_{t_{i+1}} - W_{t_i}\) is independent of \(\mathcal{F}_{t_i}\):

\[ \mathbb{E}[\sigma(X_{t_i}, t_i)(W_{t_{i+1}} - W_{t_i}) \mid \mathcal{F}_{t_i}] = \sigma(X_{t_i}, t_i)\,\mathbb{E}[W_{t_{i+1}} - W_{t_i} \mid \mathcal{F}_{t_i}] \]

\[ = \sigma(X_{t_i}, t_i) \cdot 0 = 0 \]

The key step is pulling \(\sigma(X_{t_i}, t_i)\) outside the conditional expectation (valid because it is \(\mathcal{F}_{t_i}\)-measurable), and then using \(\mathbb{E}[W_{t_{i+1}} - W_{t_i} \mid \mathcal{F}_{t_i}] = 0\) (independent increments with zero mean).

Why this fails with right-endpoint evaluation. If we use \(\sigma(X_{t_{i+1}}, t_{i+1})\), it is not \(\mathcal{F}_{t_i}\)-measurable, so we cannot pull it outside the conditional expectation. Moreover, \(\sigma(X_{t_{i+1}}, t_{i+1})\) and \(W_{t_{i+1}} - W_{t_i}\) are generally correlated (since \(X_{t_{i+1}}\) depends on \(W_{t_{i+1}} - W_{t_i}\)), so:

\[ \mathbb{E}[\sigma(X_{t_{i+1}}, t_{i+1})(W_{t_{i+1}} - W_{t_i}) \mid \mathcal{F}_{t_i}] \neq 0 \]

in general. The non-zero conditional expectation introduces a systematic drift, destroying the martingale property of the stochastic integral. This is the essential difference between Ito and Stratonovich integration.

Exercise 6. The heuristic first-order expansion of the discrete model gives

\[ S(t+\Delta t) - S(t) \approx \mu\,S(t)\,\Delta t + \sigma\,S(t)\,\sqrt{\Delta t}\cdot Z \]

The second-order expansion includes the additional term \(\frac{1}{2}\sigma^2 S(t)\,\Delta t\). Show that this extra term is \(O(\Delta t)\) and therefore the same order as the drift \(\mu\,S(t)\,\Delta t\). Explain why ignoring it changes the drift coefficient of the limiting SDE, and identify the resulting \(-\sigma^2/2\) correction in the GBM solution \(S_t = S_0\exp[(\mu - \sigma^2/2)t + \sigma W_t]\).

Solution to Exercise 6

Starting from the discrete multiplicative model:

\[ S(t+\Delta t) = S(t)\exp(\mu\Delta t + \sigma\sqrt{\Delta t}\cdot Z) \]

The exact increment is:

\[ S(t+\Delta t) - S(t) = S(t)\left[\exp(\mu\Delta t + \sigma\sqrt{\Delta t}\cdot Z) - 1\right] \]

Let \(x = \mu\Delta t + \sigma\sqrt{\Delta t}\cdot Z\). The second-order Taylor expansion of \(e^x - 1\) is:

\[ e^x - 1 \approx x + \frac{1}{2}x^2 \]

Expanding \(x^2\):

\[ x^2 = (\mu\Delta t + \sigma\sqrt{\Delta t}\cdot Z)^2 = \mu^2(\Delta t)^2 + 2\mu\sigma(\Delta t)^{3/2}Z + \sigma^2\Delta t\cdot Z^2 \]

The dominant term is \(\sigma^2\Delta t \cdot Z^2\), which is \(O(\Delta t)\) (since \(\mathbb{E}[Z^2] = 1\)). The other terms are \(O((\Delta t)^{3/2})\) or higher and vanish in the limit. Therefore the second-order correction is:

\[ \frac{1}{2}x^2 \approx \frac{1}{2}\sigma^2\Delta t\cdot Z^2 \]

This is \(O(\Delta t)\) — the same order as the drift term \(\mu\Delta t\). Since \(\mathbb{E}[Z^2] = 1\), the expected value of this correction is \(\frac{1}{2}\sigma^2\Delta t\), contributing an additional drift of \(+\frac{1}{2}\sigma^2\) per unit time.

Why ignoring it changes the drift. The full expansion of the increment is:

\[ S(t+\Delta t) - S(t) \approx S(t)\!\left[\mu\Delta t + \sigma\sqrt{\Delta t}\cdot Z + \tfrac{1}{2}\sigma^2\Delta t \cdot Z^2\right] \]

Taking expectations: \(\mathbb{E}[Z] = 0\) and \(\mathbb{E}[Z^2] = 1\), so the expected increment per unit time is \(\mu + \frac{1}{2}\sigma^2\), not \(\mu\). The first-order expansion misses the \(\frac{1}{2}\sigma^2\) term, leading to an incorrect drift.

Connection to the GBM solution. The log-price SDE is \(d(\log S_t) = \mu\,dt + \sigma\,dW_t\) (from the discrete model). Applying Ito's lemma to \(f(S) = \log S\) with \(dS_t = \tilde{\mu}S_t\,dt + \sigma S_t\,dW_t\) yields:

\[ d(\log S_t) = \left(\tilde{\mu} - \frac{\sigma^2}{2}\right)dt + \sigma\,dW_t \]

The \(-\sigma^2/2\) arises from the \((dW_t)^2 = dt\) quadratic variation term in Ito's lemma. Integrating:

\[ \log S_t = \log S_0 + \left(\tilde{\mu} - \frac{\sigma^2}{2}\right)t + \sigma W_t \]

\[ S_t = S_0\exp\!\left[\left(\tilde{\mu} - \frac{\sigma^2}{2}\right)t + \sigma W_t\right] \]

Here \(\tilde{\mu}\) is the drift of the price-level SDE, and \(\tilde{\mu} - \sigma^2/2\) is the drift of the log-price. The \(-\sigma^2/2\) correction is the continuous-time manifestation of the second-order term \(\frac{1}{2}\sigma^2\Delta t\) that the first-order heuristic expansion discarded. This term is the hallmark of Ito calculus and has no analogue in ordinary calculus.