Why Itô’s Formula Includes a Second-Derivative Term

Summary

The note contrasts ordinary Newtonian calculus with Itô stochastic calculus through Brownian motion, a continuous-time process with independent Gaussian increments. It presents Itô’s formula for a function of Brownian motion: the change in the function includes an integral involving its first derivative and a time integral involving its second derivative.

The central lesson is that stochastic integration changes the familiar chain rule: a second-order term remains instead of disappearing. This observation underlies many results in stochastic modeling, including tools used in quantitative finance. The note is only a brief conceptual explanation; it does not derive the formula, define the stochastic integral in detail, or explore its assumptions and extensions. Its statement about bounded functions is also not a complete account of the conditions needed for the formula.

Key ideas

• Itô calculus commonly models randomness using Brownian motion with Gaussian increments.
• Itô’s formula describes how a function of Brownian motion changes over time.
• The stochastic chain rule includes a term involving the second derivative.
• The note introduces the idea but does not provide a derivation or full assumptions.

Tags

# Difference between stochastic calculus and newton calculus


# Difference between stochastic calculus and newton calculus












As I am not a student of hard core mathematics,I just want to know how stochastic calculus is different from newton calculus. What make stochastic calculus different from simple newton calculus ?

## Answer by Richi Wa (score 2, accepted)

https://quant.stackexchange.com/a/21170

Talking about stochastic calculus in the sense of Ito the basic buidling block is a process with iid Gaussian increments called Brownian motion $(B_t)_{t \ge 0}$.

Then a basic observation that can be generalized in numerous ways is that for a bounded function $f$ it holds that $$ f(B_T) = f(B_0) + \int_{0}^T f'(B_t) dB_t + 1/2 \int_{0}^T f''(B_t) dt, $$ where the definition of the integral with respect to Brownian motion is fundamental. Furthermore in usual calculus the $f''$ would not be present in the above equation. In stochastic calculus the second order derivative does not vanish. This is what pops up everywhere.

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