Itô’s Lemma as a Second-Order Taylor Expansion

Summary

The document explains how Itô’s lemma resembles a second-order Taylor expansion for a function of time and an Itô process. It gives the terms involving the time derivative, the first derivative with respect to the process, and the second spatial derivative multiplied by the squared increment. It also notes that the derivatives of a deterministic function remain deterministic functions when evaluated along the process.

The explanation is a compact mathematical clarification rather than a worked example. It does not spell out the stochastic differential rule that interprets the squared increment, so the expression should not be read as an ordinary Taylor series in which all higher-order terms can be discarded. The result applies under the stated smoothness condition on the function and the assumption that the underlying process is an Itô process.

Key ideas

• Itô’s lemma can be written in a form resembling a second-order Taylor expansion.
• The formula includes an explicit time-derivative term as well as derivatives with respect to the process.
• The second spatial derivative is paired with the squared process increment.
• The stated result assumes an Itô process and a function with the required time and spatial differentiability.

Tags

# Can I write Ito's Lemma as a taylor expension?


# Can I write Ito's Lemma as a taylor expension?












instead of using Wikipedia's definition: $$ {d}(f(X_t,t)) = \frac{\partial f}{\partial t}(X_t,t)\,\mathrm{d}t + \frac{\partial f}{\partial x}(X_t,t) \, \mathrm{d}X_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2}(X_t,t)\sigma_t^2 \, \mathrm{d}t.$$

Can I write it like that:

$$ d(f(X_t,t)) = \frac{\partial f}{\partial Xt} \, \mathrm{d}Xt + \frac{1}{2} \frac{\partial^2 f}{\partial Xt^2} \, \mathrm{d}Xt^2$$

Thanks

## Answer by ir7 (score 7)

https://quant.stackexchange.com/a/57727

Ito Lemma (as 'Taylor expansion'): For $X$ an Ito process and $f = f(t, x) ∈ C^{1,2}(\mathbb{R}^2)$ a deterministic function, the stochastic process $$Y_t = f(t,X_t)$$ is an Ito process and we have $$df (t,X_t) = \partial_tf(t,X_t)\,dt + \partial_xf(t,X_t)\,dX_t + \frac{1}{2} \partial_{xx}^2f(t,X_t)(dX_t)^2. $$

Note: Functions

$$g(t,x)= \partial_tf(t,x), $$

$$h(t,x) = \partial_xf(t,x), $$

$$k(t,x) = \partial_{xx}^2f(t,x) $$ are also deterministic. So:

$$df (t,X_t) = g(t,X_t)\,dt + h(t,X_t)\,dX_t + \frac{1}{2} k(t,X_t)(dX_t)^2. $$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.