Itô Calculus — Mathematics Knowledge System

Overview

Itô calculus is the extension of classical calculus to functions of Brownian motion and, more generally, semimartingales. Because Brownian paths have infinite first-order variation but finite quadratic variation [B]_t = t, the classical Riemann–Stieltjes integral ∫ H dB fails. Itô (1944) defined a stochastic integral ∫ H_s dB_s as a limit of non-anticipating Riemann sums, and derived the cornerstone Itô formula — the stochastic analogue of the chain rule — which contains an additional second-order correction term arising from the non-zero quadratic variation.

Intuition

In ordinary calculus, df = f'(x)dx. For a function f(B_t) of Brownian motion, the chain rule must be corrected because (dB_t)² = dt (non-zero quadratic variation). Expanding f(B_{t+dt}) - f(B_t) via Taylor series: df ≈ f'(B_t)dB_t + (1/2)f''(B_t)(dB_t)² = f'(B_t)dB_t + (1/2)f''(B_t)dt. The extra second-derivative term is the Itô correction.

Formal Definition

Definition

Let {B_t} be a standard Brownian motion on (Ω, F, P) with filtration {F_t}. An elementary process is H_t = Σ_k h_k 1_{(t_k, t_{k+1}]}(t) where h_k is F_{t_k}-measurable and bounded.

\int_0^T H_s \, dB_s = \sum_k h_k (B_{t_{k+1}} - B_{t_k})

Itô integral for elementary processes

ITO-ELEM

E\!\left[\left(\int_0^T H_s \, dB_s\right)^2\right] = E\!\left[\int_0^T H_s^2 \, ds\right]

Itô isometry

ITO-ISO

f(B_t) = f(B_0) + \int_0^t f'(B_s)\,dB_s + \frac{1}{2}\int_0^t f''(B_s)\,ds

Itô's formula for f(B_t), f ∈ C²

ITO-FORMULA

f(t, X_t) = f(0, X_0) + \int_0^t \partial_t f \, ds + \int_0^t \partial_x f \, dX_s + \frac{1}{2}\int_0^t \partial_{xx} f \, d\langle X \rangle_s

General Itô formula for semimartingale X_t

ITO-GENERAL

Notation

Notation	Meaning
$\int_0^t H_s \, dB_s$	Itô stochastic integral of adapted process H against Brownian motion
$[X, Y]_t$	Quadratic covariation of semimartingales X, Y
$dX_t = \mu_t \, dt + \sigma_t \, dB_t$	Informal SDE notation for an Itô process

Proofs

Itô's formula (for f(B_t), f ∈ C²)

$f(B_t) - f(B_0) = \sum_k [f(B_{t_{k+1}}) - f(B_{t_k})]$ (Telescoping sum over partition 0=t₀<t₁<...<tₙ=t)
$\approx \sum_k f'(B_{t_k})\Delta B_k + \frac{1}{2}\sum_k f''(B_{t_k})(\Delta B_k)^2$ (Taylor expansion to second order; third order terms vanish as mesh → 0)
$\sum_k (\Delta B_k)^2 \to [B]_t = t \text{ in probability}$ (Definition of quadratic variation of Brownian motion)
$f(B_t) = f(B_0) + \int_0^t f'(B_s)\,dB_s + \frac{1}{2}\int_0^t f''(B_s)\,ds$ (Taking limits: Riemann sums → Itô integral; quadratic variation → Lebesgue integral)

Properties

Itô isometry

E\!\left[\left(\int_0^T H_s\,dB_s\right)^2\right] = E\!\left[\int_0^T H_s^2\,ds\right]

Condition: H ∈ L²([0,T]×Ω) adapted

Martingale property

\left\{\int_0^t H_s\,dB_s\right\}_{t\geq 0} \text{ is a local martingale}

Condition: H adapted and locally square-integrable

Integration by parts (Itô product rule)

d(X_t Y_t) = X_t \, dY_t + Y_t \, dX_t + d[X,Y]_t

Condition: X, Y semimartingales

Kunita–Watanabe inequality

\left|\int_0^T H_s K_s \, d[M,N]_s\right|^2 \leq \int_0^T H_s^2 \, d[M]_s \cdot \int_0^T K_s^2 \, d[N]_s

Condition: M, N local martingales

Theorems

Theorem 1: Itô's Formula (Multi-dimensional)

\text{For } f \in C^{1,2}([0,T]\times\mathbb{R}^n) \text{ and } X_t = X_0 + \int_0^t b_s\,ds + \int_0^t \sigma_s\,dB_s: \\ df(t,X_t) = \partial_t f\,dt + \nabla_x f\cdot dX_t + \frac{1}{2}\text{tr}(\sigma_t\sigma_t^\top D^2_x f)\,dt

Theorem 2: Girsanov's Theorem

\text{If } \theta_t \text{ satisfies Novikov's condition } E[e^{\frac{1}{2}\int_0^T \theta_t^2\,dt}] < \infty, \text{ then under } d\tilde{P} = \mathcal{E}(-\theta \cdot B)_T \, dP: \; \tilde{B}_t = B_t + \int_0^t \theta_s\,ds \text{ is a Brownian motion}

Theorem 3: Feynman–Kac Formula

u(t,x) = E_x\!\left[\int_t^T e^{-\int_t^s r(X_u)\,du} f(X_s)\,ds + e^{-\int_t^T r(X_u)\,du} g(X_T)\right] \text{ solves } \partial_t u + \mathcal{L}u - ru + f = 0

Applications

Black–Scholes: Itô's formula applied to f(t,S_t) = e^{-rt}C(t,S_t) yields the Black–Scholes PDE, whose solution gives the fair option price.

Worked Examples

Apply Itô's formula with f(x) = x²:
$d(B_t^2) = 2B_t \, dB_t + \frac{1}{2}\cdot 2 \, d[B]_t = 2B_t \, dB_t + dt$
Integrate from 0 to t:
$B_t^2 = B_0^2 + \int_0^t 2B_s \, dB_s + t = \int_0^t 2B_s \, dB_s + t$
Therefore B_t² - t = ∫₀ᵗ 2B_s dB_s, which is an Itô integral — hence a local martingale. Since E[B_t²] = t < ∞, it is a martingale.
$B_t^2 - t = \int_0^t 2B_s \, dB_s$

Answer: B_t² - t is a martingale, being the Itô integral ∫₀ᵗ 2B_s dB_s.

Practice Problems

Difficulty 9/10

Prove the Itô isometry: E[(∫₀ᵀ H_s dB_s)²] = E[∫₀ᵀ H_s² ds] for elementary processes H.

Difficulty 9/10

Compute ∫₀ᵗ B_s dB_s using Itô's formula.

Difficulty 10/10

State Girsanov's theorem and use it to change from physical measure P to risk-neutral measure Q in the Black–Scholes model.

Common Mistakes

Common Mistake

The Itô integral behaves like a Riemann–Stieltjes integral

The Itô integral is a limit of left-endpoint Riemann sums (non-anticipating), not midpoint (Stratonovich). The choice of endpoint matters: ∫ B_t dB_t = (B_T² - T)/2 (Itô) vs. B_T²/2 (Stratonovich).

Common Mistake

Itô's formula is just the chain rule

Itô's formula has an extra term (1/2)f''(B_t)dt absent in ordinary calculus, arising from the non-zero quadratic variation [B]_t = t. This term is essential and cannot be neglected.

Historical Background

Kiyosi Itô (1944, 1951) constructed the stochastic integral and derived his formula in a series of landmark papers. The connection to PDEs (Feynman–Kac formula) and finance (Black–Scholes, 1973) followed. Stratonovich (1966) introduced an alternative symmetric integral better suited to physical applications.

1944
Itô defines the stochastic integral and proves the Itô formula
Kiyosi Itô
1951
Itô's formula for multi-dimensional SDEs
Kiyosi Itô
1966
Stratonovich integral introduced as symmetric alternative
Ruslan Stratonovich
1973
Black–Scholes formula derived via Itô's lemma
Fischer Black, Myron Scholes
1976
Kunita–Watanabe inequality and general semimartingale theory
Hiroshi Kunita, Shinzo Watanabe

Summary

Itô's stochastic integral ∫ H dB is defined for adapted processes via L² completion of elementary integrals.
Itô isometry: E[(∫H dB)²] = E[∫H² dt] — extends the integral to all adapted L² processes.
Itô's formula: df(B_t) = f'(B_t)dB_t + (1/2)f''(B_t)dt — the stochastic chain rule with quadratic variation correction.
Girsanov's theorem: change of measure converts one drift to another, enabling risk-neutral pricing.
Feynman–Kac links SDEs to PDEs: solutions of parabolic PDEs are expectations of functionals of diffusions.

References

BookOksendal, B. (2010). Stochastic Differential Equations (6th ed.). Springer.
BookKaratzas, I. & Shreve, S. (1991). Brownian Motion and Stochastic Calculus. Springer.
BookRevuz, D. & Yor, M. (1999). Continuous Martingales and Brownian Motion (3rd ed.). Springer.