1. Stochastic Box Calculus & Quadratic Variation Rules

In stochastic calculus, infinitesimal differentials of Brownian motion satisfy non-zero second-order terms. By the Law of Large Numbers on $(\Delta W_k)^2 \sim \Delta t \cdot \chi_1^2$, the fundamental multiplication table is:

$$(dW_t)^2 = dt, \quad dW_t \, dt = 0, \quad (dt)^2 = 0$$

2. Itô’s Lemma for Scalar Itô Processes

Let $X_t$ be an Itô process satisfying stochastic differential equation:

$$dX_t = \mu_t \, dt + \sigma_t \, dW_t$$

Let $f(t, x) \in C^{1,2}([0, \infty) \times \mathbb{R})$ be twice continuously differentiable in $x$ and once in $t$.

Theorem (Itô’s Formula)

The transformed process $Y_t = f(t, X_t)$ is an Itô process with stochastic differential:

$$df(t, X_t) = \left( \frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2} \sigma_t^2 \frac{\partial^2 f}{\partial x^2} \right) dt + \sigma_t \frac{\partial f}{\partial x} dW_t$$

Derivation via Second-Order Taylor Expansion

Expanding $f(t+dt, X_t+dX_t)$ up to second-order differentials:

$$df = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial x}dX_t + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}(dX_t)^2 + \mathcal{O}(dt^{3/2})$$

Substituting $dX_t = \mu_t dt + \sigma_t dW_t$ and applying box calculus $(dX_t)^2 = \sigma_t^2 (dW_t)^2 = \sigma_t^2 dt$:

$$df = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial x}(\mu_t dt + \sigma_t dW_t) + \frac{1}{2}\sigma_t^2 \frac{\partial^2 f}{\partial x^2}dt \quad \blacksquare$$

3. Multidimensional Itô Lemma

For $\mathbf{X}_t \in \mathbb{R}^n$ driven by $d$-dimensional Brownian motion $\mathbf{W}_t$ with covariance $d\langle X_i, X_j \rangle_t = (\boldsymbol{\sigma}_t \boldsymbol{\sigma}_t^T)_{ij} dt$:

$$df(t, \mathbf{X}_t) = \left( \frac{\partial f}{\partial t} + \sum_{i=1}^n \mu_i \frac{\partial f}{\partial x_i} + \frac{1}{2}\sum_{i=1}^n \sum_{j=1}^n (\boldsymbol{\sigma}\boldsymbol{\sigma}^T)_{ij} \frac{\partial^2 f}{\partial x_i \partial x_j} \right) dt + \sum_{i=1}^n \sum_{k=1}^d \frac{\partial f}{\partial x_i} \sigma_{ik} \, dW_t^k$$

4. Python Verification: Itô Lemma Application to $f(W_t) = W_t^2$

import numpy as np

def verify_ito_lemma_quadratic(num_paths: int = 100_000, steps: int = 1000, T: float = 1.0):
    """Verifies Itô lemma for Y_t = W_t^2: d(W_t^2) = dt + 2 W_t dW_t => E[W_T^2] = T."""
    dt = T / steps
    dW = np.random.normal(0.0, np.sqrt(dt), size=(num_paths, steps))
    W = np.hstack([np.zeros((num_paths, 1)), np.cumsum(dW, axis=1)])
    
    # Pathwise integrated differential: ∫_0^T (dt + 2 W_s dW_s)
    ito_integral_part = np.sum(2.0 * W[:, :-1] * dW, axis=1)
    drift_part = T
    
    reconstructed_W2 = drift_part + ito_integral_part
    actual_W2 = W[:, -1] ** 2
    
    print(f"Mean Actual Terminal W_T²:         {np.mean(actual_W2):.6f}")
    print(f"Mean Itô Reconstructed Terminal:   {np.mean(reconstructed_W2):.6f}")
    print(f"Theoretical Expectation T:         {T:.6f}")

if __name__ == "__main__":
    verify_ito_lemma_quadratic()