Why do various SDE books consider only equations with the following form:
$$\displaystyle \mathrm {d} X_{t}=\mu (X_{t},t)\,\mathrm {d} t+\sigma (X_{t},t)\,\mathrm {d} B_{t}$$
I understand that it is analytical and computational convenient, and also many time-varying phenomena in various fields in science and engineering can be modeled as the above form. But I still wonder
- What is the exact reason that we just assume SDE with this form?
- Can you give me some examples of SDE that do not fall into the above form? Are they solvable?
- Are there any textbooks that consider other forms?
Remaind that $\sigma(t,X_t)dW_t$ is a notation for the stochastic integral $$ \int_0^t \sigma(s,X_s)dW_s $$ There is no good way to define the term $d^2 W_t$.
SPDES instead are the infinite dimensional generalization of the formula that you have written.
The most famous example is the stochastic heat equation
$$ \partial_t u= \Delta u + \xi $$ where $\xi$ is the white noise
More generally one can define as SPDE as
$$ dx_t=L x_t +F(x_t) dt +Q dW_t $$ where $L$ is the generator of a $C_0$ semigroup on a separable Banach space $B$, $F$ is measurable in $\mathcal{D}(F) \subseteq B$, $Q : K \to B$ is a bounded linear operator form an Hilbert space $K$ to $B$ and $W_t$ is a cylindrical B.M. see e.g these lecture notes on the subject.
Many of these condition can be probably lifted as well e.g you can consider multiplicative noise $Q(x_t) dW_t$, or you can consider integration w.r. to non Gaussian measures, but these are actually research topics and I'm not expecting to see that in any textbook (so far at least)