Itô Processes¶

1. Concept Definition¶

An Itô process is a continuous adapted process \(X_t\) that combines an ordinary (Lebesgue) integral and a stochastic (Itô) integral:

where:

• \(X_0\) is an \(\mathcal{F}_0\)-measurable initial value
• \(\mu_t\) is an adapted process with \(\int_0^T |\mu_s|\, ds < \infty\) a.s. — the drift coefficient
• \(\sigma_t\) is an adapted process with \(\int_0^T \sigma_s^2\, ds < \infty\) a.s. — the diffusion coefficient
• \(B_t\) is a standard Brownian motion

Differential notation. We write this compactly as:

with the understanding that this is shorthand for the integral form above. The symbol \(dX_t\) is not a derivative—it is an infinitesimal increment. The equality \(dX_t = \mu_t\,dt + \sigma_t\,dB_t\) means \(X_t - X_s = \int_s^t \mu_u\,du + \int_s^t \sigma_u\,dB_u\) for all \(s < t\).

Structure. An Itô process is a semimartingale: the sum of a finite-variation process (the drift integral) and a local martingale (the stochastic integral).

• The drift term \(\int_0^t \mu_s\, ds\) represents deterministic, predictable evolution.
• The diffusion term \(\int_0^t \sigma_s\, dB_s\) represents random fluctuations.

2. Explanation¶

Heuristic multiplication rules¶

The compact notation \(dX_t = \mu_t\,dt + \sigma_t\,dB_t\) comes with algebraic rules that encode the quadratic variation structure of Brownian motion:

These rules capture the essence of stochastic calculus: deterministic infinitesimals are negligible compared to stochastic ones, and the quadratic variation of Brownian motion is of order \(dt\). They are shorthand for rigorous statements about quadratic covariation.

Martingale characterization¶

An Itô process is a martingale if and only if its drift coefficient is zero.

Theorem. An Itô process \(X_t = X_0 + \int_0^t \mu_s\,ds + \int_0^t \sigma_s\,dB_s\) is:

• a martingale if and only if \(\mu_t = 0\) a.e.
• a submartingale if \(\mu_t \ge 0\) a.e.
• a supermartingale if \(\mu_t \le 0\) a.e.

Proof. For \(s < t\):

The martingale property \(\mathbb{E}[X_t \mid \mathcal{F}_s] = X_s\) holds if and only if \(\mathbb{E}[\int_s^t \mu_u\,du \mid \mathcal{F}_s] = 0\), which holds if and only if \(\mu_t = 0\) a.e. \(\square\)

Quadratic variation¶

The quadratic variation of an Itô process is determined entirely by its diffusion coefficient.

Theorem. \([X,X]_t = \int_0^t \sigma_s^2\, ds\).

Proof. The drift integral \(\int_0^t \mu_s\,ds\) has finite variation, so its quadratic variation is zero. The quadratic variation is then:

where the last equality is the quadratic variation property of the Itô integral. \(\square\)

Corollary. The instantaneous variance (spot volatility squared) of \(X_t\) at time \(t\) is \(\sigma_t^2\,dt\). The diffusion coefficient \(|\sigma_t|\) is the instantaneous volatility.

Doob-Meyer decomposition¶

For a general Itô process, the Doob-Meyer decomposition is explicit:

where \(M_t\) is a continuous local martingale and \(A_t\) is a continuous process of finite variation. This decomposition is unique up to indistinguishability.

3. Diagram¶

flowchart LR
    A["Itô process: dX_t = μ_t dt + σ_t dB_t"]

    A --> B["Drift term ∫μ_s ds — finite variation, differentiable"]
    A --> C["Diffusion term ∫σ_s dB_s — martingale, QV = ∫σ_s² ds"]

    B --> D["μ_t = 0 ⟺ martingale; μ_t ≥ 0 ⟺ submartingale; μ_t ≤ 0 ⟺ supermartingale"]
    C --> E["Itô isometry: Var = E[∫σ_s² ds]"]

    B --> F["[X,X]_t = ∫σ_s² ds (drift contributes zero)"]
    C --> F

4. Examples¶

Example 1: Brownian motion with drift¶

Here \(\mu\) and \(\sigma\) are constants. The solution is:

Properties: \(\mathbb{E}[X_t] = x + \mu t\), \(\operatorname{Var}(X_t) = \sigma^2 t\), and \(X_t \sim \mathcal{N}(x + \mu t,\, \sigma^2 t)\).

Quadratic variation: \([X,X]_t = \sigma^2 t\).

Used in physics (particle with constant force), and as the local approximation for stock returns in the Black-Scholes model.

Example 2: Ornstein-Uhlenbeck process¶

where \(\theta > 0\) (mean-reversion rate) and \(\sigma > 0\) (volatility).

Drift: \(\mu_t = -\theta X_t\) — mean-reverting toward zero. The drift is negative when \(X_t > 0\) and positive when \(X_t < 0\).

Diffusion: \(\sigma_t = \sigma\) — constant.

Not a martingale: the drift \(\mu_t = -\theta X_t\) is generally nonzero.

Properties: \(X_t\) is Gaussian, and as \(t \to \infty\), \(X_t\) converges in distribution to \(\mathcal{N}(0,\, \sigma^2/(2\theta))\). Used in physics (Langevin equation) and finance (interest rate and volatility models).

Example 3: Geometric Brownian motion (informal preview)¶

The coefficients \(\mu_t = \mu S_t\) and \(\sigma_t = \sigma S_t\) depend on the unknown process \(S_t\) itself—this is a stochastic differential equation rather than an Itô process with given coefficients. Applying Itô's formula to \(\log S_t\) shows the solution is:

(The \(-\sigma^2/2\) correction arises from the quadratic variation of \(\sigma B_t\)—a hallmark of Itô's formula.)

This process is the foundation of the Black-Scholes option pricing model.

Example 4: Representation via integration by parts¶

The process \(X_t = \int_0^t B_s\, ds\) (integrated Brownian motion) is not directly an Itô process in the form above. However, using Itô's formula applied to the product \(tB_t\):

Integrating: \(tB_t = \int_0^t B_s\, ds + \int_0^t s\, dB_s\). Solving for the time integral:

So \(X_t\) is an Itô process with \(dX_t = B_t\, dt + (-t)\, dB_t\), that is, \(\mu_t = B_t\) and \(\sigma_t = -t\).

Quadratic variation: \([X,X]_t = \int_0^t s^2\, ds = t^3/3\).

This example shows that the class of Itô processes is closed under integration by parts (which requires Itô's formula, studied in the next section).

5. Summary¶

Itô processes are the fundamental building blocks of stochastic calculus.

General form: \(X_t = X_0 + \int_0^t \mu_s\,ds + \int_0^t \sigma_s\,dB_s\)

Martingale condition: \(\mu_t = 0\) a.e.

Quadratic variation: determined entirely by \(\sigma_t\); the drift contributes nothing.

The next section develops Itô's lemma—the chain rule of stochastic calculus—which computes \(df(t, X_t)\) for \(f \in C^{1,2}\). This is where the heuristic rule \(dB_t \cdot dB_t = dt\) becomes manifest in explicit calculations, and is the principal tool for solving SDEs and deriving option pricing formulas.

Exercises¶

Exercise 1. Let \(X_t = 3 + 2t + 5B_t\). Identify the drift coefficient \(\mu_t\), the diffusion coefficient \(\sigma_t\), and the initial value \(X_0\). Compute \(\mathbb{E}[X_t]\), \(\operatorname{Var}(X_t)\), and the quadratic variation \([X,X]_t\).

Exercise 2. Let \(dX_t = \alpha X_t\, dt + \sigma X_t\, dB_t\) with \(X_0 = 1\). Is \(X_t\) a martingale? A supermartingale? Justify your answer using the martingale characterization theorem for Ito processes.

Exercise 3. Let \(X_t\) be an Ito process with \(dX_t = \mu_t\, dt + \sigma_t\, dB_t\). Using the heuristic multiplication rules \((dt)^2 = 0\), \(dB_t \cdot dt = 0\), and \((dB_t)^2 = dt\), compute \((dX_t)^2\) and verify that it equals \(\sigma_t^2\, dt\), consistent with \(d[X,X]_t = \sigma_t^2\, dt\).

Exercise 4. Consider the Ornstein-Uhlenbeck process \(dX_t = -\theta X_t\, dt + \sigma\, dB_t\) with \(X_0 = x_0\). Write down the explicit Doob-Meyer decomposition \(X_t = M_t + A_t\), identifying the local martingale part \(M_t\) and the finite-variation part \(A_t\). Compute the quadratic variation \([X,X]_t\).

Exercise 5. Let \(X_t = t B_t\). Use the integration by parts identity \(d(tB_t) = B_t\, dt + t\, dB_t\) to write \(X_t\) in integral form. Identify \(\mu_t\) and \(\sigma_t\), and determine whether \(X_t\) is a martingale.

Exercise 6. Let \(dX_t = \mu_t\, dt + \sigma_t\, dB_t\) and \(dY_t = \nu_t\, dt + \rho_t\, dB_t\), where both processes are driven by the same Brownian motion. Using the multiplication rules, show that the quadratic covariation satisfies \(d[X,Y]_t = \sigma_t \rho_t\, dt\).

Exercise 7. For the geometric Brownian motion \(S_t = S_0 \exp\!\left((\mu - \frac{\sigma^2}{2})t + \sigma B_t\right)\), verify that \(\mathbb{E}[S_t] = S_0 e^{\mu t}\) and \(\operatorname{Var}(S_t) = S_0^2 e^{2\mu t}(e^{\sigma^2 t} - 1)\). Why does the drift of \(\log S_t\) differ from \(\mu\)?