Itô's lemma
Itô's lemma is the stochastic version of the chain rule. It tells us how a function of an Itô process changes when the underlying process contains Brownian noise.
Start with the ordinary chain rule
Suppose \(x=x(t)\) is differentiable and
\[y=f(x,t).\]Ordinary calculus gives
\[dy=f_t\,dt+f_x\,dx.\]If \(x\) satisfies
\[dx=a\,dt,\]then
\[dy=(f_t+af_x)dt.\]No second-derivative term appears because ordinary increments are small enough that terms such as \((dx)^2\) vanish faster than \(dt\).
Now replace the ordinary process by an Itô process
Let
\[\boxed{dX_t=a(X_t,t)dt+b(X_t,t)dW_t}.\]and define
\[Y_t=f(X_t,t).\]We want to find \(dY_t\).
Why the ordinary chain rule fails
A Brownian increment has characteristic size
\[dW_t\sim\sqrt{dt}.\]Therefore
\[(dW_t)^2\sim dt.\]This is completely different from an ordinary smooth increment, whose square is negligible compared with \(dt\).
Use a second-order Taylor expansion
For a small change in both time and state,
\[df\approx f_tdt+f_xdX+\frac12f_{xx}(dX)^2.\]Terms of higher order will vanish in the stochastic limit, but the quadratic term cannot automatically be discarded.
Substitute the SDE
Since
\[dX=a\,dt+b\,dW,\]we have
\[(dX)^2=(a\,dt+b\,dW)^2.\]Expanding,
\[(dX)^2=a^2(dt)^2+2ab\,dt\,dW+b^2(dW)^2.\]Apply the Itô multiplication rules
In Itô calculus,
\[\boxed{(dt)^2=0,\qquad dt\,dW=0,\qquad(dW)^2=dt}.\]Therefore
\[(dX)^2=b^2dt.\]Substituting back into the Taylor expansion gives
\[df=f_tdt+f_x(a\,dt+b\,dW)+\frac12f_{xx}b^2dt.\]Collecting the \(dt\) and \(dW\) terms gives Itô's lemma:
\[\boxed{df=\left(f_t+af_x+\frac12b^2f_{xx}\right)dt+b f_x\,dW_t}.\]What each term means
| Term | Meaning |
|---|---|
| \(f_tdt\) | change caused explicitly by time |
| \(af_xdt\) | ordinary chain-rule contribution from drift |
| \(bf_xdW_t\) | transformed random fluctuation |
| \(\frac12b^2f_{xx}dt\) | Itô correction caused by Brownian quadratic variation |
The extra term depends on curvature
The Itô correction contains
\[f_{xx}.\]If \(f\) is linear, then \(f_{xx}=0\), so no correction appears. If \(f\) is curved, the correction is generally non-zero.
Example 1: \(f(X)=X\)
Take
\[f(X)=X.\]Then
\[f_x=1,\qquad f_{xx}=0.\]Itô's lemma gives
\[df=a\,dt+b\,dW,\]which is just the original SDE. A linear transformation creates no Itô correction.
Example 2: \(f(X)=X^2\)
Now take
\[f(X)=X^2.\]Then
\[f_x=2X,\qquad f_{xx}=2.\]For
\[dX=a\,dt+b\,dW,\]Itô's lemma gives
\[d(X^2)=\left(2aX+b^2\right)dt+2bX\,dW.\]The simplest demonstration: Brownian motion squared
Let
\[X_t=W_t.\]Then \(a=0\) and \(b=1\). For
\[f(W)=W^2,\]Itô's lemma gives
\[\boxed{d(W_t^2)=2W_t\,dW_t+dt}.\]The extra \(dt\) is the Itô correction.
Why that extra \(dt\) matters
Taking expectations,
\[E[d(W_t^2)]=E[2W_t\,dW_t]+dt.\]The Itô integral has mean zero under standard conditions, so
\[dE[W_t^2]=dt.\]Integrating from 0 to \(t\),
\[\boxed{E[W_t^2]=t}.\]This matches the known Brownian variance \(\operatorname{Var}(W_t)=t\). Without the Itô correction, we would incorrectly obtain zero.
Visualising the effect of curvature
Why symmetric noise can change the mean after transformation
Brownian noise has mean zero, but a nonlinear transformation can convert symmetric fluctuations into a systematic effect.
For example, if a random perturbation is \(+\varepsilon\) or \(-\varepsilon\), both give
\[(\pm\varepsilon)^2=\varepsilon^2.\]So after squaring, both directions contribute positively.
Example 3: logarithm
Suppose
\[dX_t=\mu X_tdt+\sigma X_tdW_t\]and define
\[Y_t=\ln X_t.\]Then
\[f_x=\frac1X,\qquad f_{xx}=-\frac1{X^2}.\]Substituting into Itô's lemma gives
\[d\ln X_t=\left(\mu-\frac12\sigma^2\right)dt+\sigma dW_t.\]The drift of \(\ln X_t\) is therefore not simply \(\mu\); it receives the Itô correction
\[-\frac12\sigma^2.\]Time-dependent functions
If
\[f=f(X_t,t),\]the full formula is
\[\boxed{df=\left(f_t+af_x+\frac12b^2f_{xx}\right)dt+b f_xdW_t}.\]If \(f\) has no explicit time dependence, then \(f_t=0\).
Why the factor \(1/2\) appears
It comes directly from the second-order Taylor expansion:
\[f(x+dx)\approx f(x)+f_xdx+\frac12f_{xx}(dx)^2.\]It is not a special stochastic convention added afterwards. What is special is that \((dW)^2\) survives instead of vanishing.
Itô's lemma versus the ordinary chain rule
| Ordinary calculus | Itô calculus |
|---|---|
| \(df=f_tdt+f_xdx\) | \(df=(f_t+af_x+\frac12b^2f_{xx})dt+bf_xdW\) |
| second-order increment terms vanish | Brownian square contributes at order \(dt\) |
| smooth paths | Brownian-driven rough paths |
A practical procedure
Worked procedure in one line
If
\[dX=a\,dt+b\,dW\]and \(Y=f(X,t)\), then calculate
\[f_t,\qquad f_x,\qquad f_{xx},\]and substitute them into
\[\boxed{dY=\left(f_t+af_x+\frac12b^2f_{xx}\right)dt+bf_xdW}.\]Why Itô's lemma matters in mathematical biology
Itô's lemma is used when we transform stochastic biological variables, derive equations for functions of them, study moments, or obtain alternative forms of stochastic population and epidemic models.
For example, it can be used to derive equations for \(X_t^2\), \(\ln X_t\), or other biologically meaningful functions of a stochastic population.
What comes next?
The next useful step is to connect this formula with quadratic variation more explicitly and then use Itô's lemma in stochastic modelling and moment calculations.