Mathematics.

stochastic calculus

Ito's Formula

Stochastic Processes70 minDifficulty8 out of 10

You should know: stochastic differential equations, brownian motion

Overview

Ito's formula is the chain rule of stochastic calculus. For a smooth function f applied to an Ito process X_t, it provides the stochastic differential of f(X_t). Unlike the classical chain rule, Ito's formula contains a second-order correction term involving the quadratic variation of X -- because Brownian motion has (dB_t)^2 = dt in the infinitesimal sense. Ito's formula is used to derive stochastic differential equations for functions of processes, to compute SDEs for financial instruments, and to identify martingales.

Intuition

If X_t changes by dX_t, then f(X_t) changes by f'(X_t) dX_t in classical calculus. But Brownian motion is 'rougher than usual': its quadratic variation is dt, not (dt)^2. So the Taylor expansion must include the (dX_t)^2 term, which gives f''(X_t)/2 * dt. This second-order correction is the hallmark of Ito calculus and distinguishes it from ordinary calculus.

Formal Definition

Definition

Let X_t be an Ito process: dX_t = mu_t dt + sigma_t dB_t, and let f(t, x) be C^{1,2} (once in t, twice in x). Then the stochastic differential of Y_t = f(t, X_t) is given by Ito's formula.

dXt=μt dt+σt dBt(Ito process)dX_t = \mu_t\,dt + \sigma_t\,dB_t \quad \text{(Ito process)}
Ito process
df(t,Xt)=∂f∂t dt+∂f∂x dXt+12∂2f∂x2(dXt)2df(t, X_t) = \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
Ito's formula (informal)
df(t,Xt)=(∂f∂t+μt∂f∂x+σt22∂2f∂x2)dt+σt∂f∂x dBtdf(t,X_t) = \left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{\sigma_t^2}{2}\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma_t \frac{\partial f}{\partial x}\,dB_t
Ito's formula (explicit)
(dBt)2=dt,dBt dt=0,(dt)2=0(dB_t)^2 = dt, \quad dB_t\,dt = 0, \quad (dt)^2 = 0
Ito multiplication table

Notation

NotationMeaning
dXtdX_tStochastic differential of the process X
ft,  fx,  fxxf_t,\; f_x,\; f_{xx}Partial derivatives of f with respect to t, x, and xx
[X]t[X]_tQuadratic variation of X up to time t; for X_t = B_t, [X]_t = t

Theorems

Theorem 1: Theorem 1 (Ito's Formula for Brownian Motion)
Let f∈C2(R). Then f(Bt)=f(B0)+∫0tf′(Bs) dBs+12∫0tf′′(Bs) ds.\text{Let } f \in C^2(\mathbb{R}). \text{ Then } f(B_t) = f(B_0) + \int_0^t f'(B_s)\,dB_s + \frac{1}{2}\int_0^t f''(B_s)\,ds.
Theorem 2: Theorem 2 (Multidimensional Ito Formula)
For independent Brownian motions B1,…,Bd and f∈C1,2,  df(t,Xt)=ft dt+∑ifxi dXti+12∑i,jfxixj d⟨Xi,Xj⟩t.\text{For independent Brownian motions } B^1, \ldots, B^d \text{ and } f \in C^{1,2},\; df(t,X_t) = f_t\,dt + \sum_i f_{x_i}\,dX_t^i + \frac{1}{2}\sum_{i,j} f_{x_i x_j}\,d\langle X^i, X^j\rangle_t.

Worked Examples

  1. 1

    Let f(x) = x^2. Then f'(x) = 2x and f''(x) = 2. Apply Ito's formula with dX_t = dB_t (so mu_t = 0, sigma_t = 1).

    dYt=d(Bt2)=f′(Bt) dBt+12f′′(Bt) dt=2Bt dBt+12⋅2 dt=2Bt dBt+dtdY_t = d(B_t^2) = f'(B_t)\,dB_t + \frac{1}{2}f''(B_t)\,dt = 2B_t\,dB_t + \frac{1}{2}\cdot 2\,dt = 2B_t\,dB_t + dt
  2. 2

    Integrating from 0 to T: B_T^2 - B_0^2 = 2 int_0^T B_t dB_t + T, confirming int_0^T B_t dB_t = (B_T^2 - T)/2.

    BT2=2∫0TBt dBt+T  ⟹  ∫0TBt dBt=BT2−T2B_T^2 = 2\int_0^T B_t\,dB_t + T \implies \int_0^T B_t\,dB_t = \frac{B_T^2 - T}{2}

✓ Answer

d(B_t^2) = 2B_t dB_t + dt. The +dt term is the Ito correction from (dB_t)^2 = dt.

Practice Problems

Mediumfree response

Find the SDE satisfied by Y_t = e^{B_t} using Ito's formula.

Hardfree response

Let X_t satisfy dX_t = r X_t dt + sigma X_t dB_t (GBM). Compute d(log X_t) and identify the drift.

Common Mistakes

Common Mistake

Forgetting the (1/2) f''(B_t) dt correction term

The Ito correction (1/2) sigma^2 f_{xx} dt is essential. Omitting it gives the wrong answer. This term has no classical analogue.

Common Mistake

Writing the GBM solution as S_t = S_0 exp(mu t + sigma B_t)

The correct solution is S_t = S_0 exp((mu - sigma^2/2)t + sigma B_t). The -sigma^2/2 in the exponent compensates for the Ito correction.

Quiz

In the Ito multiplication table, (dB_t)^2 equals:
For f in C^2 and B_t standard Brownian motion, Ito's formula gives:
Geometric Brownian motion S_t = S_0 exp((mu - sigma^2/2)t + sigma B_t) satisfies:

Summary

  • Ito's formula: d(f(t, X_t)) = f_t dt + f_x dX_t + (1/2) f_{xx} sigma_t^2 dt for dX_t = mu_t dt + sigma_t dB_t.
  • Key rule: (dB_t)^2 = dt; all other products involving dt are zero.
  • GBM: S_t = S_0 exp((mu - sigma^2/2)t + sigma B_t) satisfies dS_t = mu S_t dt + sigma S_t dB_t.
  • d(log S_t) = (mu - sigma^2/2) dt + sigma dB_t -- the drift of log S is mu - sigma^2/2, not mu.
  • Ito's formula is the cornerstone of stochastic calculus and the Black-Scholes derivation.

References

  1. BookOksendal, B. -- Stochastic Differential Equations, Chapter 4
  2. BookKaratzas, I. and Shreve, S. -- Brownian Motion and Stochastic Calculus, Chapter 3
  3. WebsiteWikipedia -- Ito's lemma