Itô's Lemma and Stochastic Calculus › Derivation of Itô's Lemma

Now we derive Itô's Lemma, which tells us how to compute the differential of a function $f(X_t, t)$ when $X_t$ follows an Itô process. This derivation follows the logic of Taylor expansion but carefully accounts for the non-vanishing quadratic variation of Brownian motion that we just established.

Let $X_t$ satisfy the SDE:

$$dX_t = \mu(X_t, t) \, dt + \sigma(X_t, t) \, dW_t$$

where:

• $X_t$: stochastic process value

• $\mu(X_t, t)$: drift coefficient

• $\sigma(X_t, t)$: diffusion coefficient

• $dt$: infinitesimal time increment

• $dW_t$: Brownian increment

and let $f(x, t)$ be a function that is twice continuously differentiable in $x$ and once continuously differentiable in $t$. We want to find an expression for $df(X_t, t)$, the infinitesimal change in the function value as both time and the stochastic process evolve. The smoothness requirements on $f$ ensure that the Taylor expansion is valid and the derivatives we need exist.

Since $f$ depends on both the process $X_t$ and time $t$, we need the two-variable Taylor expansion. The two-variable Taylor expansion of $f$ around $(X_t, t)$ is:

$$\begin{aligned} f(X_t + \Delta X_t, t + \Delta t) &= f(X_t, t) + \frac{\partial f}{\partial t} \Delta t + \frac{\partial f}{\partial x} \Delta X_t \\ &\quad + \frac{1}{2} \frac{\partial^2 f}{\partial t^2} (\Delta t)^2 + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} (\Delta X_t)^2 + \frac{\partial^2 f}{\partial x \partial t} \Delta X_t \Delta t + \cdots \end{aligned}$$

where:

• $\frac{\partial f}{\partial t}$: partial derivative with respect to time

• $\frac{\partial f}{\partial x}$: partial derivative with respect to the state variable

• $\frac{\partial^2 f}{\partial x^2}$: second partial derivative with respect to the state variable

• $\Delta X_t$: change in the process $X_t$

• $\Delta t$: time increment

All partial derivatives are evaluated at $(X_t, t)$. In ordinary calculus, we would keep only the first-order terms and discard everything else. However, as we established in the previous section, the term $(\Delta X_t)^2$ contains a component that behaves like $\Delta t$, not like $(\Delta t)^2$. We must therefore carefully analyze which terms survive in the limit.

We have $\Delta X_t = \mu \Delta t + \sigma \Delta W_t$. To apply the Taylor expansion correctly, we need to compute $(\Delta X_t)^2$. Expanding the square:

$$\begin{aligned} (\Delta X_t)^2 &= (\mu \Delta t + \sigma \Delta W_t)^2 && \text{(substitute } \Delta X_t \text{)} \\ &= \mu^2 (\Delta t)^2 + 2\mu\sigma \Delta t \Delta W_t + \sigma^2 (\Delta W_t)^2 && \text{(expand square)} \end{aligned}$$

where:

• $\Delta X_t$: increment of the Itô process

• $\mu$: drift coefficient

• $\sigma$: diffusion coefficient

• $\Delta W_t$: Brownian increment

Now we apply the heuristic rules for infinitesimal products, which encode the key insight about Brownian motion's quadratic variation:

• $(\Delta t)^2 \to 0$ (second-order in $\Delta t$, negligible in the limit)

• $\Delta t \cdot \Delta W_t \to 0$ (since $\Delta W_t \sim \sqrt{\Delta t}$, this product is of order $(\Delta t)^{3/2}$, which vanishes)

• $(\Delta W_t)^2 \to \Delta t$ (the key stochastic calculus result we established earlier)

Therefore, in the limit as $\Delta t \to 0$:

$$(\Delta X_t)^2 \to \sigma^2 \, dt$$

where:

• $\to$: denotes convergence in the limit as $\Delta t \to 0$

• $\sigma^2 \, dt$: the deterministic limit of the squared stochastic increment

This step distinguishes stochastic calculus from ordinary calculus. The squared increment does not vanish; instead, it contributes a term proportional to $dt$.

Having established how each term behaves in the limit, we can now assemble the final expression. The change in $f$ is:

$$\Delta f = \frac{\partial f}{\partial t} \Delta t + \frac{\partial f}{\partial x} \Delta X_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \sigma^2 \Delta t + \text{(higher order terms)}$$

Substituting $\Delta X_t = \mu \Delta t + \sigma \Delta W_t$ into this expression:

$$\Delta f = \frac{\partial f}{\partial t} \Delta t + \frac{\partial f}{\partial x} (\mu \Delta t + \sigma \Delta W_t) + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \sigma^2 \Delta t$$

Rearranging by grouping the deterministic terms (those multiplying $\Delta t$) and the stochastic terms (those multiplying $\Delta W_t$):

$$\Delta f = \left( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial x^2} \right) \Delta t + \sigma \frac{\partial f}{\partial x} \Delta W_t$$

Taking the limit as $\Delta t \to 0$, we obtain the fundamental result of stochastic calculus, known as Itô's Lemma:

The key difference from the ordinary chain rule is the term $\frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial x^2}$, which arises from the quadratic variation of Brownian motion. This "correction term" is sometimes called the Itô correction or second-order term. Itô represents the accumulated effect of the infinitely many small but collectively significant jumps in Brownian motion. When a function is convex (positive second derivative), this correction adds to the drift; when the function is concave (negative second derivative), it subtracts from the drift.

To appreciate what stochastic calculus adds to the familiar framework, consider what Itô's Lemma reduces to in the absence of randomness. If there were no randomness ($\sigma = 0$), we would have:

$$\begin{aligned} df &= \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} \mu \, dt \\ &= \left( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} \right) dt \end{aligned}$$

where:

• $df$: total differential of $f$

• $\mu$: drift rate (velocity)

• $dt$: time increment

This is exactly the total derivative from ordinary calculus, expressed in differential form. The presence of randomness adds two new elements to this familiar picture:

1. The second-order correction in the drift, which appears because the process's random fluctuations create a systematic effect on the function value
2. A diffusion term in $dW_t$, which means the function of the process inherits randomness from the underlying stochastic process