GIVEN: A discrete time-inhomogenous MC (TIMC):
where
$0\leq\lambda\leq1$, $\lambda=\frac{1}{\mu L}$, $\mu=\textrm{constant}>0$,
$L$ is the total time,
$l$ is the time step,
$\gamma_h^{(l)}=\frac{1}{1+e^{\delta_l/T_h}}$,
$\delta_l = \delta_a - \frac{\Delta l}{L}$,
$\delta_a>0$, $\Delta>0$, $T_h>0$,
QUESTION(S):
Can I obtain a continuous limit of the same? More generally, can one always obtain a continuous MC from a discrete one?
Next, the above discrete TIMC is related to a random variable $W$ such that at each time step if the state is $1$ $W$ increases by a value of $\epsilon_L=\frac{\Delta}{L}$ modulo the last step (see table below). I'm interested in the $\textbf{variance}$ of the distribution of $\textbf{W}$ for both the $\textbf{discrete}$ and the $\textbf{continuous}$ ($L\rightarrow\infty$) cases.
The distribution for $L=2$ can be obtained from the table below:
where $a$ is the starting point, $b$ is the ending point, $e_1$ is the mid point as there are two steps involved here. Each row denotes a possible path and $p_a$ is the probability of state $1$ at $a$.
Looking forward to the discussion.
Thanks.