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:

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

  • $\begingroup$ 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. $\endgroup$ – noob2 Dec 21 '18 at 18:36
  • $\begingroup$ ah... that would explain why I felt something was totally off. thanks for setting me straight! $\endgroup$ – rarnor Dec 22 '18 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).

  • $\begingroup$ This was actually very helpful, thanks for clearing it up $\endgroup$ – rarnor Dec 22 '18 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 ?

  • $\begingroup$ 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. $\endgroup$ – rarnor Dec 22 '18 at 4:40
  • $\begingroup$ 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. $\endgroup$ – Ezy Dec 22 '18 at 9:24
  • $\begingroup$ 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 $x$. $\endgroup$ – rarnor Dec 22 '18 at 16:12

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