Itô's lemma
is only as useful as the examples you can drive it through. Here we crank the handle three
times — on W_t^2, on W_t^3, and on the
single most important process in the whole subject, the
exponential martingale — each derivation line by line. Throughout we use the
pure-Brownian form
df(W_t) = f'(W_t)\,dW_t + \tfrac{1}{2}f''(W_t)\,dt
and, for the time-dependent example, the full form
df(t, W_t) = (f_t + \tfrac12 f_{xx})\,dt + f_x\,dW_t.
(a) d(W_t^2) = 2W_t\,dW_t + dt
Step 1 — pick f(x) = x^2 and differentiate:
f'(x) = 2x, f''(x) = 2.
Step 2 — substitute into the lemma:
d(W_t^2) = f'(W_t)\,dW_t + \tfrac{1}{2}f''(W_t)\,dt = 2W_t\,dW_t + \tfrac{1}{2}\cdot 2\,dt.
Step 3 — simplify:
d(W_t^2) = 2W_t\,dW_t + dt.
The +\,dt is the Itô correction; ordinary calculus would have
stopped at 2W_t\,dW_t.
(b) d(W_t^3) = 3W_t^2\,dW_t + 3W_t\,dt
Step 1 — pick f(x) = x^3:
f'(x) = 3x^2, f''(x) = 6x.
Step 2 — substitute:
d(W_t^3) = 3W_t^2\,dW_t + \tfrac{1}{2}\cdot 6W_t\,dt.
Step 3 — simplify:
d(W_t^3) = 3W_t^2\,dW_t + 3W_t\,dt.
Again the second term, 3W_t\,dt, is pure correction — naive calculus
would have given only 3W_t^2\,dW_t.
(c) The exponential martingale — the centrepiece
Define, for a fixed constant \sigma,
M_t = \exp\!\Big(\sigma W_t - \tfrac{1}{2}\sigma^2 t\Big).
The promise: M_t is a martingale, and the
-\tfrac12\sigma^2 t in the exponent is engineered to make it so. We
prove it by applying Itô to the time-dependent function
f(t, x) = \exp(\sigma x - \tfrac12\sigma^2 t), with
X_t = W_t (so the process has a = 0,
b = 1).
Step 1 — compute the partial derivatives. Note each derivative reproduces a
copy of f itself, which we write as M_t:
f_t = -\tfrac{1}{2}\sigma^2\,M_t, \qquad f_x = \sigma\,M_t, \qquad f_{xx} = \sigma^2\,M_t.
Step 2 — plug into the time-dependent lemma
dM_t = (f_t + \tfrac12 f_{xx})\,dt + f_x\,dW_t:
dM_t = \Big(-\tfrac{1}{2}\sigma^2 M_t + \tfrac{1}{2}\sigma^2 M_t\Big)dt + \sigma M_t\,dW_t.
Step 3 — watch the drift cancel. The two
dt-coefficients are exact negatives: the
-\tfrac12\sigma^2 M_t from the time-derivative
f_t kills the +\tfrac12\sigma^2 M_t from
the Itô correction \tfrac12 f_{xx}:
-\tfrac{1}{2}\sigma^2 M_t + \tfrac{1}{2}\sigma^2 M_t = 0.
Step 4 — collect. With the drift gone, only the Brownian term remains:
\boxed{\,dM_t = \sigma M_t\,dW_t.\,}
A process with no drift — a pure dW term with an
adapted integrand — is a (local)
martingale.
And a martingale keeps its mean fixed at its starting value
M_0 = e^{0} = 1:
\mathbb{E}[M_t] = \mathbb{E}[M_0] = 1 \qquad \text{for every } t.
Fix \sigma \in \mathbb{R} and let
M_t = \exp\!\big(\sigma W_t - \tfrac12\sigma^2 t\big). Then:
-
M_t satisfies the driftless SDE
dM_t = \sigma M_t\,dW_t, hence is a martingale.
-
Its mean is pinned for all time:
\mathbb{E}[M_t] = 1.
-
The compensator -\tfrac12\sigma^2 t in the exponent is exactly
what cancels the \tfrac12\sigma^2 Itô correction — remove it and
e^{\sigma W_t} is not a martingale.
The cancellation is not luck; it is reverse-engineered. The exponent of a bare
e^{\sigma W_t} would, by Itô, generate a correction drift
\tfrac12\sigma^2 e^{\sigma W_t}\,dt. To neutralise it you subtract
a deterministic ramp whose own time-derivative is the equal-and-opposite
-\tfrac12\sigma^2 — and that ramp is precisely
-\tfrac12\sigma^2 t.
You can see the same constant through the
moment generating function
of the normal. Since W_t \sim N(0, t), the normal MGF gives
\mathbb{E}\big[e^{\sigma W_t}\big] = e^{\tfrac{1}{2}\sigma^2 t}.
So e^{\sigma W_t} has mean e^{\sigma^2 t/2},
which grows with t — not a martingale. Dividing by exactly that
growing factor renormalises the mean back to 1:
M_t = \frac{e^{\sigma W_t}}{\mathbb{E}[e^{\sigma W_t}]} = \frac{e^{\sigma W_t}}{e^{\sigma^2 t/2}} = e^{\sigma W_t - \frac12\sigma^2 t}.
The drift-cancellation and the MGF-normalisation are the same fact wearing two hats.
This little process is a giant. As a likelihood-ratio it is the engine of the
Girsanov change of measure that builds the
risk-neutral world; and with \sigma read as a volatility it is the
martingale part of
geometric Brownian motion,
the Black–Scholes model of a stock price.