Finding a stochastic differential equation as limit of a discrete stochastic equation

Question

I'm dealing with the following problem:

Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation:

$Z_{k+1}-Z_k=P_k(1-2Z_k)$

where $P_k=0$ with probability $1-q_c+q_c m_k$ and $P_k=1$ with probability $q_c(1- m_k)$. Here $m_k$ is a certain function that depends on other variables.

I'm interested in the following limit: put $q_c=\frac{1}{N}$ and let $N \to +\infty$. By doing that, we see that $P_k \to 0$ in probability. I would like to change the time scale accordingly in order to obtain a SDE equivalent of the above equation. So my question is how can one find this SDE.

Answer 1

It looks like you're modeling a two-state Markov process with a (symmetric) transition probability that is inhomogeneous in $k$ and governed by the parameter $m_k$. Continuous time Markov chains describe a such a system in the continous limit if your transition probabilities are fixed. If $m(t)$ varies in $t$ in the continuous limit, the times between state transitions will not be exponentially distributed. In this case you will have to create a continuous-time semi-markov process also known as a renewal process.

Because you have only two states $[0,1]$ and your transition probabilities are symmetric, you might consider modeling the rates of "flipping" only and dispense with the $[0,1]$ state space. In this case you are interested in a point process that defines the rate of state transitions. The rates of flipping will be determined by your continuously varying parameter $m(t)$. In general you will have to find an equation for the intensity $\lambda(t)$:

$$ \Pr(\textrm{flip}\in [t,t+\Delta)) = \int_t^{t+\Delta} \lambda(t) dt $$

Check that $\lambda(t) = p_c (1-m(t))$ yields the continuous time analogue of your system, if the continuous time parameter $m(t)$ can be approximated by the locally constant value $m_t$ from the discrete system:

$$ \Pr(flip\in [t,t+1)) = \int_t^{t+1} p_c(1-m(t)) dt \approx \int_t^{t+1} p_c(1-m_t) dt = p_c(1-m_t) $$