In a physics text, "Quantum Measurement Theory and it's Applications" by Kurt Jacobs, it describes the idea of a "continuous measurement" (measurement taking place over time $T$): $$dy = x_{true}dt + \beta dW$$ with a term $\beta dW$ corresponding to Wiener noise (Brownian prcoess). It then defined the continuous measurement result after time $T$ as $$y := \int_{0}^{T}xdt + \beta \int_{0}^{T}dW$$ Equivalently it shows we can write $$dy = \langle x \rangle dt + \frac{dW}{\sqrt{8k}}$$

It then states that the meaurement result can be written as $$y(t) = \langle x(t) \rangle + \frac{1}{\sqrt{8k}} \xi(t)$$ where $\xi(t) := \frac{dW}{dt}.$ Can anyone see the reasoning for this last equation and specifically why we would take $\xi(t) := \frac{dW}{dt}$?

Thanks for any assistance.


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