Stochastic Integrals

Motivation: why this matters in quant finance

A self-financing trading gain has the form $\int_0^T H_t\,dS_t$ : holdings times price changes. In a Brownian model, the price path is continuous but too rough for ordinary Riemann-Stieltjes integration, so this expression needs a new definition before it can support hedging or pricing.

Lawler introduces stochastic integration by analogy with betting. The integrand is the amount invested at time $s$ , and the Brownian increment is the next fair-game movement. The crucial finance constraint is non-anticipation: a hedge at time $s$ may use $\mathcal{F}_s$ , but not the next increment $W_{s+\Delta t}-W_s$ .

This lesson builds the Itô integral used throughout the stochastic-calculus track. It is the object behind stochastic differential equations, geometric Brownian motion, and the martingale calculations in the Black-Scholes framework.

The informal idea

Start with strategies that change only at finitely many times. On each interval, freeze the stake before the next Brownian move. Then define the gain by summing

\text{stake before the move}\times\text{Brownian increment}.

For more general adapted processes, approximate them by simple strategies and take an $L^2$ limit. This is why Itô integration is not "area under a random curve." The integrator is Brownian noise, and the construction is built around conditional information.

The left-endpoint convention is not a numerical preference. It is the no-look-ahead rule in mathematical form. If the integrand could depend on the future Brownian increment, the fair-game and martingale properties would fail.

Formal definitions

Let $W_t$ be a standard Brownian motion with respect to a filtration $\{\mathcal{F}_t\}$ . A simple adapted process is a process $A_t$ for which there are times

0=t_0<t_1<\cdots<t_n<\infty

and square-integrable random variables $Y_j$ such that $Y_j$ is $\mathcal{F}_{t_j}$ -measurable and

A_t=Y_j,\qquad t_j\leq t<t_{j+1}.

For such a process, define

\int_0^t A_s\,dW_s =\sum_{i=0}^{j-1}Y_i\left(W_{t_{i+1}}-W_{t_i}\right) +Y_j\left(W_t-W_{t_j}\right), \qquad t_j\leq t<t_{j+1}.

For bounded adapted processes with continuous paths, Lawler defines the integral by choosing simple adapted approximations $A^{(n)}$ satisfying

\lim_{n\to\infty}\mathbb{E}\left[\int_0^t\left(A_s^{(n)}-A_s\right)^2\,ds\right]=0

and setting

\int_0^t A_s\,dW_s =\lim_{n\to\infty}\int_0^t A_s^{(n)}\,dW_s

in $L^2$ . The construction extends further by localization to continuous or piecewise continuous adapted integrands.

Differential notation is shorthand. Writing

dX_t=A_t\,dW_t

means

X_t=X_0+\int_0^t A_s\,dW_s.

Key properties

Linearity

For constants $a,b$ and adapted integrands $A,C$ ,

\int_0^t (aA_s+bC_s)\,dW_s =a\int_0^t A_s\,dW_s+b\int_0^t C_s\,dW_s.

The same additivity holds across time intervals. In finance, this is why gains from combined trading strategies decompose linearly.

Martingale property

If the integrability conditions are strong enough, the process

Z_t=\int_0^t A_s\,dW_s

is a martingale. The proof for simple processes uses the fact that each stake $Y_i$ is known before the Brownian increment and that future Brownian increments have conditional mean zero.

This is the mathematical form of a fair self-financing gain under a martingale price model.

Variance rule

For square-integrable integrands,

\text{Var}(Z_t)=\mathbb{E}[Z_t^2] =\int_0^t \mathbb{E}[A_s^2]\,ds.

This is the core identity later isolated as the Itô isometry. It says the risk of the stochastic gain is controlled by the squared exposure process.

Continuity

For the classes Lawler constructs in Chapter 3, the stochastic integral has continuous paths. Brownian noise is rough, but the Itô integral against Brownian motion does not introduce jumps.

Quadratic variation of the integral

Z_t=\int_0^t A_s\,dW_s,

then

\langle Z\rangle_t=\int_0^t A_s^2\,ds.

Lawler describes this as the total amount of randomness or total amount of betting accumulated up to time $t$ .

Worked examples

Example 1: a constant volatility exposure

Let $A_s=\sigma$ . Then

\int_0^t \sigma\,dW_s=\sigma W_t.

Its mean is zero and its variance is

\mathbb{E}\left[\left(\int_0^t\sigma\,dW_s\right)^2\right] =\sigma^2t.

This is the simplest model for accumulated Brownian return noise.

Example 2: why $\int_0^t W_s\,dW_s$ is not ordinary calculus

Ordinary calculus might suggest

\int_0^t W_s\,dW_s=\frac{1}{2}W_t^2.

Lawler points out that this cannot be right: the stochastic integral is a martingale starting at zero, so it has expectation zero, while

\mathbb{E}\left[\frac{1}{2}W_t^2\right]=\frac{t}{2}.

Itô's formula gives the correct identity:

\int_0^t W_s\,dW_s=\frac{1}{2}\left(W_t^2-t\right).

The missing $-t/2$ is the first concrete sign that Brownian quadratic variation changes calculus.

Example 3: stochastic Euler interpretation

The differential equation

dX_t=\phi(X_t)\,dW_t

means

X_t=X_0+\int_0^t \phi(X_s)\,dW_s.

For simulation, Lawler writes the stochastic Euler step as

X_{t+\Delta t}=X_t+\phi(X_t)\sqrt{\Delta t}\,N,

where $N\sim\mathcal{N}(0,1)$ . The integrand is evaluated at the current state, not after seeing the next shock. That is the same non-anticipative convention as the definition of the integral.

Common confusions and pitfalls

"A stochastic integral is an area under a curve." The $dt$ integral accumulates area. The $dW_t$ integral accumulates Brownian increments weighted by adapted stakes.

"Right endpoints should be more accurate." In Itô integration, right endpoints generally use information from after the Brownian move. That violates the no-look-ahead condition.

"Mean zero means riskless." A stochastic integral can be a martingale and still have large variance. The variance rule measures that risk.

"The differential notation is the definition." The integral equation is the definition. Symbols like $dX_t=A_t\,dW_t$ are shorthand for a precise integral statement.

"Every continuous adapted integrand automatically gives a true martingale." Lawler later shows that localization matters. Without sufficient square integrability, the integral can fail to be a true martingale even though it is locally well behaved.

Where this goes next

Itô Isometry: Extracts the variance rule into the main $L^2$ identity for Brownian integrals.
Itô's Lemma: Computes stochastic integrals by transforming functions of Brownian motion.
Quadratic Variation: Explains why $\int W_s\,dW_s$ contains the correction term $-t/2$ .
Stochastic Differential Equations: Uses Itô integrals to define equations such as $dX_t=m(t,X_t)\,dt+\sigma(t,X_t)\,dW_t$ .
Local Martingales and Semimartingales: Clarifies what remains true when square-integrability is weakened.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 3 §3.1 (What is stochastic calculus?), §3.2 (Stochastic integral), §3.3 (Itô's formula).