I have a stochastic differential equation of the type: $$ dX(t) = \mu(t) X(t)dt + \sigma(t) X(t) dW(t) \tag{1} $$ However, my $\mu(t)$ is a complicated function of $t$ as well as $W(t)$, somewhat like:
$$ \mu(t) = t \cdot a(t) + b(t) + \sigma W(t) \tag{2} $$
I feel like $W(t)dt$ that I get by substituting $(2)$ into $(1)$ doesn't look "right". And I'm not sure how to get the $X(t)$ from that. Edit What I mean, is that my equation $(1)$ becomes: $$ dX(t) = (t \cdot a(t) + b(t))X(t)dt + \sigma W(t)X(t)dt + \sigma(t) X(t) dW(t) \tag{3} $$ My questions are:
Assuming that I obtain a $X(t)$, does that single-handedly prove that $(1)$ is uniquely solvable? I am not sure what conditions will demonstrate that something is uniquely solvable.
Is the $ W(t)dt $ term that I get in equation $(3)$ sensible?
The equation
$$ dX(t) = \left[(t \cdot a(t) + b(t)+ \sigma W(t)\right] X(t)dt + \sigma(t) X(t) dW(t)=\mu(t)X(t)dt + \sigma(t) X(t) dW(t) $$ is a linear equation and it has a general solution give here Solution to General Linear SDE, and to be clear all you need from these various functions is continuity.