I have an Ito SDE of the form
$$ dX_{t} = \mu(X_t)dt + \sigma(X_t)dW_{t} $$ that is, the mean and variance terms do not depend explicitly on time, and only on the position of the particle at each time point.
I want to examine the behaviour of this system without having to explicitly solve the Kolmogorov forward equation to find the density $p(x,t)$. This is because for the functions $\mu$ and $\sigma$ I have considered to far, this is very difficult. It is also because I would like to make statements about broad classes of functions.
My question is, if I state a family of functions $\mu$ and $\sigma$ for which I know they have some properties, then is it possible to do both of the following things
- Given a final point of $X$, $x_{t}$, can i interogate whether $\mathbf{E}_{p(x_{t}\mid X_{0})}[X_{0}] = x_{t}$? In other words, if I can write down a likelihood of a particle $x$ at time $t$ given all the possible starting locations $x_{0}$, can I find the expected started location, and is it equal to that given final location?
- Given a starting point of $X$, $x_{0}$, can I interogate wether $\mathbf{E}_{p(X_{t}\mid x_{0})}[X_{t}] \neq x_{0}$? In other words, if I can write down a conditional density of a particle $x$ at time $t$ given a fixed starting location $x_{0}$, can I find the expected end location, and is it different to the starting location?
I realise that in order to be able to do this, I probably need to make stronger assumptions on the functions $\mu(X_{t})$ and $\sigma(X_{t})$, but I don't really know what those assumptions might be, or if its worth trying to think about them if the problem I'm trying to solve has no chance of success. Please advise.