By Gabriel Dunin-Borkowski – Last Updated:
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We start by defining an Itô process Xt with drift μt and diffusion σt. We also recall that Wt is a Wiener (Brownian) process, with \( dW_t \sim O\left(\sqrt{\Delta t}\right) \).
Let $$dX_t = \mu_t\,dt + \sigma_t\,dW_t,$$ where \(W_t\) is a standard Brownian motion.
We consider a function \(f = f_t(X_t)\). We want to determine how \(f_t(X_t)\) evolves over an infinitesimal time step, i.e., we want \(df_t\) - How does \(f_t\) grow? What are its dynamics? That is - what is its total differential?
We use Taylor's theorem in multiple dimensions to expand the increment (difference) Δf. We keep terms up to second order in Δx, since higher-order terms become negligible in the limit. We'll show how it is that they become negligible and which ones do become negligible and which ones don't.
$$\Delta f = f_{t+\Delta t}\bigl(x+\Delta x\bigr) \;-\; f_t(x).$$ $$\Delta f = f\bigl(t+\Delta t,\,x+\Delta x\bigr) \;-\; f(t,x).$$
Hence, by Taylor's theorem: $$\Delta f \;=\; \frac{\partial f}{\partial t}\,\Delta t \;+\; \frac{1}{2}\,\frac{\partial^2 f}{\partial t^2}\,(\Delta t)^2 \;+\; \frac{\partial f}{\partial x}\,\Delta x \;+\; \frac{1}{2}\,\frac{\partial^2 f}{\partial x^2}\,(\Delta x)^2 \;+\;\cdots.$$
Because \((\Delta t)^2 \ll \Delta t\) when \(\Delta t \to 0\), we can discard terms like \(\mathcal{O}((\Delta t)^2)\). That is, higher exponents dominate when the exponent base goes to infinity, but smaller exponents dominate when the exponent base goes to zero. Thus, in limit-land, we have:
$$\Delta f \;\approx\; \frac{\partial f}{\partial t}\,\Delta t \;+\; \frac{\partial f}{\partial x}\,\Delta x \;+\; \frac{1}{2}\,\frac{\partial^2 f}{\partial x^2}\,(\Delta x)^2.$$
Recall \(dX_t = \mu_t\,dt + \sigma_t\,dW_t\). In the infinitesimal limit, $$\Delta X_t \;\to\; \sigma_t\,dW_t \quad\text{as}\quad \Delta t \to 0.$$ Squaring this, $$(\Delta X_t)^2 \;\to\; \sigma_t^2\, (dW_t)^2.$$ And in Itô calculus, \((dW_t)^2 = dt\). To motivate this, recall that, \( dW_t \sim O\left(\sqrt{\Delta t}\right) \), and therefore \( \left(dW_t\right)^2 \sim O\left(\Delta t\right) \). So, if we take the time difference to approach zero:
$$(\Delta X_t)^2 \;\to\; \sigma_t^2\,dt.$$
Putting it all together, in differential form:
$$df_t(X_t) \;=\; \frac{\partial f}{\partial t}\,dt \;+\; \frac{\partial f}{\partial x}\,dX_t \;+\; \frac{1}{2}\,\frac{\partial^2 f}{\partial x^2}\,\sigma_t^2\,dt.$$
This expression is the renowned Itô's lemma. The second-order term in x does not vanish because (dWt)2 is of order dt, yielding a non-zero contribution. The distribution of test statistics have non-degenerate asymptotic behavior.
Due to the non-vanishing quadratic variation of \(\{W_t\}\), the second derivative term remains in the final differential of \(f\). This is the essence of Itô's lemma.
Because quadratic variation (of Brownian motion) never disappears, the second-order term in the Taylor expansion remains.