Why Itô’s Lemma Does Not Apply Directly to Path-Dependent Integrals
Summary
The document addresses an attempt to derive stochastic differentials for integrals involving Brownian motion by treating the integral itself as a function of time and the current Brownian value. The response clarifies that the differential notation for a stochastic integral is shorthand for its integral definition. For example, writing the differential of an integral with deterministic integrand does not mean that its integrand can be treated as a partial derivative with respect to time.
The key condition is that the function used in Itô’s lemma must be expressible in terms of the current time and state, such as a function of time and the current Brownian motion. An accumulated integral generally depends on the Brownian path over the full interval, so it is not automatically such a function and the proposed partial derivatives are invalid. The response rejects both attempted applications on that basis. It gives a conceptual correction rather than a worked alternative derivation, and assumes the reader already recognizes the distinction between a state function and a path-dependent quantity.
Key ideas
- The differential form of a stochastic integral is notation for its integral definition.
- Itô’s lemma applies to suitable functions of the current time and state.
- An accumulated stochastic integral can depend on the entire path, not only the current Brownian value.
- Treating such an integral as a function of time and the current state leads to invalid partial derivatives.
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Full text
# Stochastic differential equation of a Brownian Motion
# Stochastic differential equation of a Brownian Motion
I have two questions about Ito's Lemma with respect to calculating SDEs. The examples are simple enough, but I haven't found an answer yet.
Take $W_t$ as a standard Brownian motion and $g(s)$ as some function of $s$. Assume that all regularities etc. are fulfilled and take $F$ as some function. I know that if $F = \int_0^tg(s)dW_s$, then the corresponding SDE is $dF = g(t)dW_t$. However, applying Ito's Lemma, I'm not sure how this SDE is derived. I am unsure about the next part:
- $dF = \underbrace{\frac{\partial F}{\partial t}}_{=g(t)dW_t}dt + \underbrace{\frac{\partial F}{\partial W_t}}_{=g(t)}dW_t + \underbrace{\frac{1}{2}\frac{\partial^2 F}{\partial W_t^2}}_{=0}dt = g(t)dW_tdt + g(t)dW_t = g(t)dW_t$
Question 1: is $\frac{\partial F}{\partial t} = g(t)dW_t$ correct? Or should this be zero?
Now take $F=\int_0^tW_s^2dW_s$. My approach would be:
- $dF = \underbrace{\frac{\partial F}{\partial t}}_{=W_t^2dW_t}dt + \underbrace{\frac{\partial F}{\partial W_t}}_{=W_t^2}dW_t + \underbrace{\frac{1}{2}\frac{\partial^2 F}{\partial W_t^2}}_{=W_t}dt = W_t^2dW_t+W_tdt$
Question 2: Are the partial derivatives in the above example correct?
## Answer by Gordon (score 6, accepted)
https://quant.stackexchange.com/a/31110
In stochastic calculus, only stochastic integrals are defined. The differential form is just a notation. That is, $$dF=g(t)dW_t$$ is just another expression for the integral $$F=\int_0^t g(s) dW_s.$$ See, for example, in this book or this book, all Ito's lemmas are expressed in integral forms.
For your question, note that $F$ is not a function of $t$ and $W_t$, that is, it is not of the form $F(t, W_t)$. In fact, it depends on the whole path of $W_s$ from $0$ to $t$. Then Ito's lemma can not be applied to $F$. The application for both of your questions are incorrect.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.