I would like to understand how to go from a deterministic process, say multiplicative growth $$ d X_t = \rho X_t d t\ , $$ to the corresponding stochastic process, say GBM, $$ d X_t = \mu X_t d t + \sigma d W_t\ , $$ by explicitly considering a mapping from deterministic variables to stochastic ones, i.e. in the example above, $\rho \to \rho_t$. The reason I wish to do this it that I wish to understand how to "perturb" the deterministic processes that I am currently working with in a systematic way.
To stick with the above example, from what I understand about GBM, it arises by saying, "instead of $\rho$ being constant, let us assume that it is drawn from a normal distribution with a certain mean and variance." So let $\rho \mapsto \rho_t$, where $\rho_t$ follows a random walk $$ d \rho_t = \sigma d W_t $$ with initial condition $\rho_0 = \mu$. We therefore have $$ \rho_t = \mu + \sigma W_t\ , $$ which makes sense to me (even though I am not sure how rigorous this is). If I insert this into the very first equation, I end up with $$ d X_t = \mu X_t dt + \sigma X_t W_t dt\ , $$ which is almost correct. Is there any rule or heuristic that allows me to write $W_t dt = dW_t$? If so, I would be done. But all of this leaves me feeling a little queasy. Any improvements or suggestions on how to improve this "derivation" are appreciated!
Indeed as mentioned in the comments a natural approach to approximating SDEs by ODEs are the Wong-Zakai type theorems.
A good book reference is the Stroock-Varadhan book "Multidimensional Diffusion Processes" chapter on limit theorems 11.1.4 Theorem.
See "A Wong-Zakai theorem for SDEs with singular drift" for many references. As mentioned there, the main idea is to first mollify the noise i.e.
$B_t=\int 1_{[0,t]}\xi(s)$ can be mollified as $B_t^{\epsilon}=\int 1_{[0,t]}\xi_{\epsilon}(s)$ for $\xi_{\epsilon}(s)=\int \phi_{\epsilon}(r-s)\xi(r)dr$
In the original Wong-Zakai article, it was proved that in dimension one, under the assumptions that $b \in\mathcal{C}_b^1([0,T]\times\mathbb{R})$ and $\sigma \in\mathcal{C}_b^2([0,T]\times\mathbb{R})$, if $(w^n_t)_{n\geq1}$ is a continuous and piecewise smooth sequence of approximations to the Brownian motion $(W_t)_{t\geq0}$, then the solutions $X_t^{w^n}$ to the following equation $$d X_t^{w^n} = b(t,X_t^{w^n}) \, d t + \sigma(t,X_t^{w^n}) \, d w^n_t,\quad X_0=x_0\in\mathbb{R},\quad t\geq0,$$ converge to $X_t$ almost surely, where $X_t$ solves \begin{align}\label{sde00stro} d X_t=b(t,X_t)d t + \sigma(t,X_t)\circ d W_t,\quad X_0=x_0\in\mathbb{R},\quad t\geq0. \end{align} Here $\sigma(t,X_t)\circ d W_t:=\sigma(t,X_t) \, d W_t + \frac{1}{2}\sigma(t,X_t)\frac{\partial \sigma(t,X_t)}{\partial x} \, d t$, i.e. the $\circ$ integral denotes the usual Stratonovich integral, and $\frac{1}{2}\sigma(t,X_t)\frac{\partial \sigma(t,X_t)}{\partial x} \, d t$ is the Itô correction term. The fact that the limiting SDE has to be understood in Stratonovich sense is not surprising: if the approximating equations satisfy the usual chain rule, the same should be true for the limiting equation.