# Integral of Wiener process over time

This should hopefully be an easy question to answer, but I am new to Stochastic Calculus and am gapping as to why the following is true, for a brownian motion $$W_t$$:

$$d(\int W_t dt ) = W_t dt$$

I have seen many of the relevant linked posts about the integral, but all of them use this as a basic fact.

I have tried just applying Ito's lemma, which yields the first term when taking the partial derivative w.r.t. $$t$$, but I am wondering why there is no $$dW_t$$ term. In particular, why don't we have:

$$\frac{\partial}{\partial W_t} \int W_t dt = \int \frac{\partial}{\partial W_t} W_t dt = \int dt = t$$.

• Sorry but $\frac{\partial}{\partial W_t}$ is completely meaningless, there is no such thing. The intuition for the other statement is that "d" and $\int dt$ operators are inverses of each other. Differentiation wrt time is the opposite of integration wrt time. Commented Dec 21, 2018 at 18:36
• ah... that would explain why I felt something was totally off. thanks for setting me straight! Commented Dec 22, 2018 at 4:11

Compare $$\int_o^t W_t dt$$ and $$\int_o^{t+dt} W_t dt$$

The increment between the first integral and the second is equal to $$W_t dt$$ (i.e. the value of the integrand at the upper limit of integration ($$W_t$$) multiplied by the length of time by which the integral has been extended to the right ($$dt$$).

That is what we mean when we write $$d(\int W_t dt ) = W_t dt$$

(The limits of integration have been left out, but it is a definite integral that we are talking about here).

• This was actually very helpful, thanks for clearing it up Commented Dec 22, 2018 at 4:55

The ito lemma applies to a function $$f(t,W_t)$$. To help you understand why you are confused i would ask: since you believe that you can apply the ito lemma, can you explictly provide the function $$f$$ you believe is applicable in this case ?

• My original thought was that $\int W_t dt$ was a function of $W_t$, but that seems wrong. It seems like I am generally missing a good chunk of the intuition here. Commented Dec 22, 2018 at 4:40
• If you think you can write such function $(x,t)\rightarrow f(t,x)$ of 2 variables then what would be the value of $f(1,2)$ for example ? Probably it will help you understand what is incorrect with your mental picture.
– Ezy
Commented Dec 22, 2018 at 9:24
• I'm not sure its possible, which I guess is partly because $W_t$ has time-dependence, whereas if it was replaced with some variable $x$, then the integral would be a function of $x$. Commented Dec 22, 2018 at 16:12