Suppose the SDE $$\mathsf dX_t =b(t,X_t)\mathsf dt + \sigma(t,X_t)\mathsf dW_t,\; X_0 = x$$ where $t\in[0,T]$ has a strong solution. I know in general we can't find an explicit formula for the solution, but is there a way to calculate the mean and variance of $\{X_t\mid t\in[0,T]\}$ or give an upper bound to $\{\textrm{Var}(X_t)\mid t\in[0,T]\}$? Or do we need to impose some other assumptions to $b,\sigma$ besides the requirements for existence of strong solution?
If it's too long to explain in an answer, may you give me some references so that I can study it? Thank you.
You could try a Monte Carlo approach. Basically, you can simulate a large number of strong solutions and then evaluate the sample mean and variance of the specific instant of interest.
Depending on the structure of the diffusion coefficient, it is possible to perform exact simulation. In this case, no approximation error will be propagated to your inference. A great reference is the seminal work of Beskos and Roberts: https://projecteuclid.org/euclid.aoap/1133965767 .
If the diffusion coefficient is not "regular" enough, you could use traditional discretization schemes. They perform very well estimating moments once it is a problem of weak convergence. See Kloden and Platen, 1992: http://www.springer.com/us/book/9783540540625 .