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I am working with Hamilton-Jacobi-Bellman Equations and the following result appeared.

Suppose $V_t= \frac{\partial V}{\partial t}$

and that $ E_{t}[Y]=E\left[Y \mid F_{t}\right] $ , where $F_t$ represents the information at time t

I do not understand why the following holds:

$E[V_t dt]=V_t$

You could write the expectation as an integral but this didn't clearify anything for me

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    $\begingroup$ Why do you believe that $E[V_t dt] = V_t$? $\endgroup$
    – Michael
    Jun 13, 2022 at 17:27
  • $\begingroup$ It is written in the book $\endgroup$
    – WHN
    Jun 13, 2022 at 18:06
  • $\begingroup$ I am guessing the book intended to write $E[V_t|F_t]=V_t$, assuming $V_t$ is $F_t$-measurable. $\endgroup$
    – Michael
    Jun 13, 2022 at 21:12

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