Stochastic Calculus: Itô Integration and the Itô Formula

Tier: R (Research). Builds on probability spaces, measure theory, real analysis and functional analysis. The subject is the mathematics of randomness that changes in time, and it is the language of mathematical finance, stochastic PDEs, and the modern theory of regularity for rough functions.

Attempt these from memory before reading.

• What is a filtration? What does it mean for a process to be adapted?
• State the three properties that define Brownian motion.
• What is the quadratic variation of a function, and what is it for a differentiable function?
• State the ordinary chain rule.
• What does the Lebesgue integral of a function against a measure of unbounded variation require?

You want to model a stock price, a diffusing particle, or a noisy signal. The natural object is an integral

where $W$ is Brownian motion; the integral of a strategy against a random noise. The problem is that $dW(s)$ is not a measure. Brownian paths have unbounded variation on every interval, so the Riemann–Stieltjes integral does not exist: the sums do not converge no matter how you partition.

That is not a technicality. It is the mathematical statement that the noise is rough, and roughness is the whole point. The Itô integral is the repair, and the repair has a price: the chain rule acquires an extra term, and that term is what makes stochastic calculus a different subject from ordinary calculus.

Definition (F, in this subject). A standard Brownian motion $W$ is a process with:

1. $W_0 = 0$ almost surely.
2. Independent increments: $W_{t} - W_{s}$ is independent of the past for $s < t$.
3. Gaussian increments: $W_t - W_s \sim \mathcal{N}(0, t-s)$.
4. Continuous paths.

Three properties worth knowing by heart.

The quadratic variation is the central object. For a differentiable function $f$, the sums $\sum (f(t_{i+1}) - f(t_i))^2$ over a partition tend to $0$ like the mesh. For Brownian motion they tend to $t$; a positive, finite, non-random limit. Brownian motion accumulates exactly $t$ worth of roughness, and that is a deterministic fact about almost every path.

For a deterministic integrand $f$ and an integrator $g$ of bounded variation, the Riemann–Stieltjes sums $\sum f(\xi_i)(g(t_{i+1}) - g(t_i))$ converge regardless of where the evaluation point $\xi_i$ sits in $[t_i, t_{i+1}]$.

For Brownian motion, the choice of $\xi_i$ changes the answer. Taking $\xi_i$ at the left endpoint gives one limit; taking it at the midpoint gives a different limit; the difference is exactly half the quadratic variation of the cross term.

So there is no integral free of that choice. Itô’s choice is the left endpoint, which has a probabilistic meaning: $\xi_i$ is then known given the past, so the integrand is non-anticipating. That is what makes the integral adapted, and adaptedness is what makes it a martingale; the property that does all the work.

The price and the payoff. The Itô integral is not a pathwise object; it is defined as an $L^2$ limit. It has no meaning for a fixed realisation of $W$. In exchange it is a martingale with

which is the Itô isometry, and it is what makes the integral usable.

Theorem (R, Itô 1944). Let $X_t$ satisfy $dX_t = \mu_t\,dt + \sigma_t\,dW_t$ and let $f$ be twice continuously differentiable. Then

and for Brownian motion $d\langle W \rangle_t = dt$, so

Where the extra term comes from. Taylor-expand $f(X_t)$ to second order:

In ordinary calculus $(dX)^2$ is $o(dX)$ and vanishes. For Brownian motion, $(dW)^2$ is exactly $dt$; that is the statement $[W]_t = t$. The second-order term is of the same order as the first, so it cannot be dropped.

Worked example (R, in full). Find the SDE for $Y_t = W_t^2$.

Take $f(x) = x^2$. Then $f'(x) = 2x$ and $f''(x) = 2$, so

Rearranging, $W_t^2 - t = \int_0^t 2W_s\,dW_s$, which shows directly that the Itô integral is a martingale: $W_t^2 - t$ has zero drift. In ordinary calculus you would get $d(W^2) = 2W\,dW$ and no $dt$ term; the $dt$ is the Itô correction, and it is the quadratic variation appearing.

Worked example (R, geometric Brownian motion). Financial models use

Find $d(\log S_t)$. With $f(x) = \log x$, $f'(x) = 1/x$, $f''(x) = -1/x^2$:

So $\log S_t = \log S_0 + (\mu - \sigma^2/2)t + \sigma W_t$, and $S_t$ is log-normal with drift $\mu - \sigma^2/2$, not $\mu$. Missing the $-\sigma^2/2$ is the most common error in the subject, and it comes precisely from dropping the Itô term.

Three proofs. The first establishes the isometry, which is the integral’s definition; the second establishes quadratic variation, which the Itô formula assumes; the third is the formula itself. The order matters, because the third uses the second and would be circular without it.

Theorem 1 (Itô isometry). For simple $\sigma$, $\left\|\int_0^t \sigma\,dW\right\|_2^2 = \mathbb{E}\int_0^t \sigma^2\,ds$, and the simple adapted maps are dense in $L^2(\Omega \times [0,t])$, so the isometry extends the integral to all of $L^2$.

Proof. Let $\sigma_t = \sigma_{t_i}$ on $(t_i, t_{i+1}]$ and $\Delta W_i = W_{t_{i+1}} - W_{t_i}$. Then

For a cross term, condition on $\mathcal{F}_{t_j}$: $\sigma_{t_j}$ and $\Delta W_i$ for $i<j$ are $\mathcal{F}_{t_j}$-measurable, and $\mathbb{E}[\Delta W_j \mid \mathcal{F}_{t_j}] = 0$ by independent increments, so the term is zero. For the diagonal, $\mathbb{E}[\sigma_{t_i}^2\Delta W_i^2] = \mathbb{E}[\sigma_{t_i}^2](t_{i+1}-t_i)$ by the same conditioning. Summing gives the isometry.

Simple adapted processes are dense in $L^2(\Omega\times[0,t])$, and because the map is an isometry on them it is $L^2$-continuous, so it extends uniquely to the closure. That extension is the Itô integral: an $L^2$ object, defined by continuity rather than pathwise. The cross terms did not merely come out small — they vanished because of adaptedness, the same property that fixed the choice of left endpoint above. $\blacksquare$

Theorem 2 (quadratic variation of Brownian motion). For partitions with mesh $\to 0$,

Proof. Write $Q_n$ for the sum and $\Delta t_i = t_{i+1}-t_i$. Then $\mathbb{E}Q_n = \sum_i \Delta t_i = t$, so it is enough to show the variance vanishes. The increments are independent, and $\operatorname{Var}(\Delta W_i^2) = 2(\Delta t_i)^2$ for a Gaussian, so

$\blacksquare$

The single line $\sum 2(\Delta t_i)^2 \le 2t\max_i\Delta t_i$ is the entire proof, and it is worth pausing on: quadratic variation is a limit, not an identity, and for any finite partition $Q_n$ has mean $t$ but fluctuates. The shorthand $(dW)^2 = dt$ is a statement about this $L^2$ limit, exactly as true and exactly as informal as that.

Theorem 3 (Itô’s formula). For $f \in C^2(\mathbb{R})$,

Proof. Partition $[0,t]$ and Taylor-expand about $W_{t_i}$:

with $R_i = \tfrac12 (f''(\xi_i) - f''(W_{t_i}))(\Delta W_i)^2$. Summing,

Term by term. The first converges in $L^2$ to the Itô integral, by Theorem 1. The second is an ordinary Riemann sum, converging to $\tfrac12\int_0^t f''\,ds$. The third is the term Riemann-Stieltjes theory does not have, and it vanishes because of Theorem 2: by Cauchy-Schwarz in $L^2$,

For the remainder, $f''$ is uniformly continuous, so $\max_i|f''(\xi_i) - f''(W_{t_i})| \to 0$ in probability while $\sum_i(\Delta W_i)^2$ is $O_P(1)$ by Theorem 2; hence $\sum_i R_i \to 0$ in probability. $\blacksquare$

The proof explains the formula rather than merely verifying it. The extra term is not a correction that happens to be needed: it is the second-order Taylor term failing to be negligible, and the reason it fails is that $\sum(\Delta W_i)^2$ has a nonzero limit. For a bounded-variation integrator that sum tends to zero, which is why ordinary calculus has no such term. Itô’s formula is Taylor’s theorem with a different answer to the question “how large is $(\Delta X)^2$” — and the answer is a property of the integrator, not of $f$.

The last row is the research-level content. The Stratonovich integral uses the midpoint evaluation point instead of the left endpoint. It agrees with ordinary calculus ($d(W^2) = 2W \circ dW$, no correction term), but it is not a martingale. Itô is the natural choice for predicting (left endpoint = no foresight); Stratonovich is the natural choice for modelling physics (invariant under coordinate changes). The conversion between them is explicit, and choosing the wrong one silently changes the drift.

• Rough paths. Lyons’ theory constructs the integral for any rough signal: not just Brownian motion – by treating the iterated integrals as extra data. It removes the probabilistic structure entirely and gives pathwise meaning to what Itô defines in $L^2$. It also explains why Itô’s correction exists: it is the Lévy area.
• Malliavin calculus. A differential calculus on Wiener space, giving probabilistic proofs of Hörmander’s hypoellipticity theorem and a way to compute sensitivities of expectations with respect to the noise.
• Stochastic PDEs. The stochastic heat, wave and Navier–Stokes equations. Regularity structures (Hairer, Fields Medal 2014) extended rough paths to give a meaning to SPDEs whose nonlinearity is more singular than the noise.
• Regularisation by noise. An ODE with irregular drift that has no solution classically can acquire a unique one when noise is added; noise creating regularity rather than destroying it.

• State the Itô isometry, and say what it is for.
• Why does the Itô formula have an extra term, and where does it come from?
• Compute the SDE for $W_t^3$.
• What is the difference between the Itô and Stratonovich integrals, and when is each the right choice?
• State one way in which rough paths generalise Itô’s integral.

Mix these with material from Probability Spaces, Measure Theory, Sequences and Limits and Functional Analysis.

1. (Probability spaces.) Show that the Itô integral is a martingale with respect to the filtration generated by $W$, using the independent-increments property.
2. (Measure theory.) Show $[W]_t = t$ almost surely for the dyadic partition, using the law of large numbers.
3. (Real analysis.) Show that a function of bounded variation has zero quadratic variation, so the Itô correction vanishes.
4. (Functional analysis.) Show the Itô isometry says the map $\sigma \mapsto \int \sigma\,dW$ is an isometry from $L^2(\Omega \times [0,t])$ into $L^2(\Omega)$, and deduce it extends to all square-integrable adapted integrands.
5. (Research.) Show that for $f \in C^2$, the Stratonovich and Itô integrals of $f'(W)$ against $dW$ differ by $\frac{1}{2}[f'(W)]_t$.

• Probability Spaces - the axioms, filtrations and adaptedness
• Measure Theory; the integral the Itô isometry lives in
• Functional Analysis: the $L^2$ structure
• Posterior Asymptotics: the statistical counterpart
• How to Use These Notes; the method