# Solving for the limit of a Gaussian random variable within an integral

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.

• So your $\eta$ is white noise? This limit doesn't exist in the conventional sense, so you have to be careful with how you use it. – Ian Aug 26 '15 at 22:23
• @Ian exactly. actually, according to the paper i'm working from, the definition for it i gave in the question is only a "heuristic" definition. i'm going to link to the specific section of the paper that's relevant in my question. – dbliss Aug 26 '15 at 22:24
• I think that what is intended is that $\eta$ should have the property that $W(T)=\int_0^T \eta(t) dt$ is the Wiener process, also called (in mathematical circles) Brownian motion. – Ian Aug 26 '15 at 22:26
• @Ian OK, so it sounds like i should try to implement a white noise process computationally, and then solve the integral numerically. is there any other work we can do on this with pure math? or, more generally, do you have any recommendations? – dbliss Aug 26 '15 at 22:32
• @Ian i guess i'm still not sure how to define the white noise process such that it's amenable to computational analysis. for example, i have a function that returns a Gaussian random process provided a mean and SD. what would i pass it as an SD in this case? infinity? that doesn't seem like it would get me what i need, but i'm not sure how else to interpret the limit in my question. – dbliss Aug 26 '15 at 22:45