I'm having trouble solving a particular integral. It is
$$ (1/\Delta t)\int_t^{t+\Delta t}I(t')dt', $$
where
$$ I(t') = \mu_c+\sigma_c \eta(t'). $$
In this second equation,
$$ \eta(t') = \lim_{dt\to0}N(0, 1/\sqrt{dt}), $$
where $N(\alpha, \beta)$ is a Gaussian random variable of mean $\alpha$ and variance $\beta^2$.
The variables I haven't defined are all constants. Can anyone help me out? It's really that limit I'm having trouble with. A numerical solution is as good as an analytical one as far as I'm concerned.
This integral comes from this book chapter. Specifically, it is part of the "diffusion approximation" described in Section 15.2.3, which begins on page 436. I'm trying to replicate the line on the right side of Figure 15.1B (page 440), the definition of which is given in the figure caption on page 441.