In the first chapter of John Cochrane's Asset Pricing textbook, in order to calculate the price in discrete time, we solve the maximization problem of $Max\space E(\Sigma\beta^j U(c_{t+j}))$ when our $c_t = e_t - \xi p_t $ and $c_{t+j} = e_{t+j} + \xi D_{t+j} $ (e as endownment, c as consumption and $\xi$ as quantity of the asset purchased.)
So, when we move to continuous time, the problem becomes to maximize $E\int e^{-\delta t} U(c_{t})\space dt$. In this form the constraint is written as $c_t = e_t - \xi p_t/d_t$.
What I don't understand is why does $dt$ show up in the constraint. Isn't it still true that the price of $\xi$ units of asset is $p_t$ so that we should write it similar to the discrete time i.e ($e_t - \xi p_t$)?
Any advice would be appreciated.