Why Itô’s Formula Differentiates the Function, Not Brownian Motion
Summary
The document clears up an apparent conflict between the nondifferentiability of a Wiener process and the derivative term in Itô’s formula. For a function such as F(X) = X², the derivative is taken with respect to the function’s argument, X; it does not require differentiating the Brownian path with respect to time. The example gives the Itô differential dF = 2X dX + dt, where the extra dt term accounts for Brownian quadratic variation.
The answers also distinguish the differential notation from the underlying integral statement: the formula represents an equality of stochastic integrals. This is a conceptual explanation, not a derivation of Itô’s lemma or a discussion of its assumptions. One response loosely suggests using Itô’s lemma to differentiate Brownian processes; the accepted explanation is more precise that the function must be twice differentiable, while the process itself need not be.
Key ideas
- Itô’s formula differentiates the function of a stochastic process, not the Brownian path over time.
- For F(X) = X², the formula includes a dt term in addition to 2X dX.
- The differential form of Itô’s formula is shorthand for an equality expressed through stochastic integrals.
- A Brownian motion has no conventional time derivative, but Itô’s formula can still be applied to smooth functions of it.
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# How to differentiate a brownian motion?
# How to differentiate a brownian motion?
By definition a wiener process cannot be differentiated.
But when we use Ito's lemma on $F = X^2$, where X is wiener process
we have total change in
$$dF = 2XdX + dt$$
How can we calculate $\frac{dF}{dX}$ when by definition it cannot be differentiated? Isin't this contradiction by definition?
## Answer by ocstl (score 6, accepted)
https://quant.stackexchange.com/a/17546
In order to apply Ito's lemma, your function needs to be a twice-differentiable function. There is no issue with the non-differentiability of the Wiener process. $\frac{dF}{dX}$ involves differentiating F, not the Wiener process X.
Using a simple analogy: instantaneous velocity ($\frac{dD}{dt}$) is the derivative of position (D) over time; what is differentiated is not time, but distance. I believe this is where your confusion stems from.
## Answer by statmlben (score 2)
https://quant.stackexchange.com/a/17534
We write the differential form of Ito formula for simplification. Actually, the differential form for Ito formula $$ dF(W(t)) = 2W(t)dW(t) + dt $$ means the integral form for Ito formula, $$ \int{dF} = \int{2W(t)dW(t)} + \int{dt} $$ which make sense in mathemaitcs.
## Answer by Barnaby (score 1)
https://quant.stackexchange.com/a/17532
You are right that a Wiener process can not be differenciated in the conventional way since the derivative in respect to time does not exist. For this reason Ito lemma should be used to integrate and differenciate Brownian or Wiener processes as these are considered ito processes.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.