With the
Itô integral and its properties
in hand we can name the central object of mathematical finance. An Itô process
is anything built from a drift integral plus a stochastic integral:
X_t = X_0 + \int_0^t a_s\, ds + \int_0^t b_s\, dW_s,
with adapted integrands a and b. The first
integral is an ordinary (finite-variation) Riemann integral; the second is the Itô integral, a
martingale. The whole thing is written compactly in differential form:
dX_t = a_t\, dt + b_t\, dW_t.
Here a_t is the drift — the predictable trend, the
smooth part — and b_t is the diffusion (or
volatility) — the size of the random kick. Almost every model you will meet,
from a stock price to an interest rate, is an Itô process; the job of stochastic calculus is to
compute with them.
The box algebra, line by line
To do calculus with dX_t = a_t\, dt + b_t\, dW_t we need to know how
the differentials multiply. The rules are forced by a single scaling fact: over a short step of
length dt, a Brownian increment has standard deviation
\sqrt{dt}, so it is of size
dW \sim \sqrt{dt}.
We work out each product by comparing its order of magnitude to dt,
keeping anything of order dt and discarding anything smaller.
Step 1 — (dt)^2 = 0. The product of two time-steps is
of order (dt)^2, which is smaller than
dt as dt \to 0
((dt)^2 / dt = dt \to 0). Anything of higher order than
dt is negligible, so we set it to zero:
(dt)\cdot(dt) = (dt)^2 = 0.
Step 2 — dt\cdot dW = 0. Using
dW \sim \sqrt{dt}, the cross product is of order
dt \cdot \sqrt{dt} = (dt)^{3/2} — again higher order than
dt ((dt)^{3/2}/dt = \sqrt{dt} \to 0), hence
negligible:
(dt)\cdot(dW) = (dt)^{3/2} = 0.
Step 3 — (dW)^2 = dt. This is the one that does
not vanish. The squared Brownian increment is of order
(\sqrt{dt})^2 = dt — exactly the order we keep — and we know its precise
value from the
quadratic variation:
summing squared increments converges to elapsed time, the differential shorthand of which is
(dW)\cdot(dW) = (dW)^2 = dt.
These three lines are the whole multiplication table — tabulated in the theorem below. Note the
asymmetry that makes stochastic calculus its own subject: (dt)^2 and
dt\, dW die, but (dW)^2 survives as a
genuine dt.
The quadratic variation of X
Now apply the table to an Itô process. The quadratic variation tracks the squared infinitesimal
moves, so we compute (dX_t)^2 directly.
Step 4 — square the differential. With
dX_t = a_t\, dt + b_t\, dW_t, expand the square
(p + q)^2 = p^2 + 2pq + q^2:
(dX_t)^2 = (a_t\, dt + b_t\, dW_t)^2 = a_t^2\,(dt)^2 + 2\,a_t b_t\,(dt)(dW_t) + b_t^2\,(dW_t)^2.
Step 5 — apply the box rules term by term. The first term has
(dt)^2 = 0 (Step 1); the cross term has
(dt)(dW_t) = 0 (Step 2); only the last term survives, with
(dW_t)^2 = dt (Step 3):
(dX_t)^2 = a_t^2\cdot 0 + 2\,a_t b_t\cdot 0 + b_t^2\cdot dt = b_t^2\, dt.
Step 6 — read off the quadratic variation. Accumulating the squared moves gives
d[X]_t = b_t^2\, dt, \qquad\text{equivalently}\qquad [X]_t = \int_0^t b_s^2\, ds.
The drift a has disappeared from the quadratic variation
entirely — the smooth part contributes no roughness. Only the diffusion
b drives the quadratic variation, which is exactly the
[M]_t = \int_0^t H^2\, ds we met for the Itô integral, with
H = b. The drift moves the process; the diffusion makes it
rough.
For an Itô process dX_t = a_t\, dt + b_t\, dW_t, the differentials
multiply by the rules
- dt \cdot dt = (dt)^2 = 0;
- dt \cdot dW = dW \cdot dt = 0;
- dW \cdot dW = (dW)^2 = dt.
Consequently the quadratic variation of X sees only the diffusion:
(dX_t)^2 = b_t^2\, dt, \qquad d[X]_t = b_t^2\, dt, \qquad [X]_t = \int_0^t b_s^2\, ds.
Every line of the box table is just bookkeeping of orders of magnitude, and they all
flow from one fact: \operatorname{Var}(dW) = dt, so the typical size
of a Brownian increment is \sqrt{dt}, not dt.
Line them up:
- a time step is order dt;
- a Brownian step is order \sqrt{dt} — larger than
dt when dt is small (since
\sqrt{dt} \gg dt);
- its square is order dt — back to the size we keep;
- the cross term is order \sqrt{dt}\cdot dt = (dt)^{3/2} —
smaller than dt, so it dies.
In ordinary calculus a function's increment is order dt and its square
is order (dt)^2 — negligible, which is why second-order terms drop out
of the chain rule. Brownian motion sits one rung lower: its increment is the larger
\sqrt{dt}, so the square is order dt
and refuses to vanish. That surviving (dW)^2 = dt is the extra term
that ordinary calculus lacks — and it is precisely what produces the
\tfrac12 f''\, dt correction in
Itô's lemma, the chain rule of this calculus.