# Discretization formula for a system of two differential equations. “Solution to one of these is the initial condition of the other”. In which sense?

Consider the following stochastic differential equation $$$$dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1}$$$$ where $$A$$, $$B$$ and $$C$$ are parameters and $$dW$$ is a Wiener increment.
Equation $$(1)$$ will be our point of reference in what follows.

Now, first let us consider a "method" for equation $$\left(1 \right)$$ which can be described by the following one-step discretization scheme: $$$$y_{n+1}=y_n+\left(A-\left(A+B\right)y_n\right)\Delta t +C\sqrt{y_n\left(1-y_n\right)}\Delta W_n + D\left(y_n\right)\left(y_n-y_{n+1}\right)\tag{2}$$$$ where $$\Delta t$$ is the length of the time discretization interval, $$\Delta W_n$$ is a Wiener increment and $$D(y_n)$$ is the system of control functions and takes the form $$D(y_n)=d^0(y_n)\Delta t + d^1\left(y_n\right)|\Delta W_n|$$ with $$d^1(y)= \begin{cases} C\sqrt{\frac{1-\varepsilon}{\varepsilon}}\hspace{0.5cm}\text{if }y<\varepsilon\\ C\sqrt{\frac{1-y}{y}}\hspace{0.5cm}\text{if }\varepsilon\le y<\frac{1}{2}\\ C\sqrt{\frac{y}{1-y}}\hspace{0.5cm}\text{if }\frac{1}{2}\le y\le 1-\varepsilon\\ C\sqrt{\frac{1-\varepsilon}{\varepsilon}}\hspace{0.5cm}\text{if }y>1-\varepsilon \end{cases}$$ At this point, let us consider a "method" which decomposes $$\left(1\right)$$ into two equations. Specifically, the first equation is a stochastic one, that consists of the diffusion term of $$\left(1\right)$$ only (see eqtn $$\left(3\right)$$), while the second one is an ordinary differential equation (see eqtn $$\left(4\right)$$) that consists of the drift part of $$\left(1\right)$$. We have:

$$$$dy_1=C\sqrt{y_1\left(1-y_1\right)}dW\tag{3}$$$$ $$$$dy_2=\left(A-\left(A+B\right)y_2\right)dt\tag{4}$$$$

This last method approximates the solution to $$\left(3\right)$$ at each time step using $$\left(2\right)$$ (and numerical solution to $$\left(3\right)$$ is used as the initial condition in $$\left(4\right)$$), while $$\left(4\right)$$ can be solved using the Euler method. Thus, such a method can be described by the following one step discretization formula: $$y_{n+1}=y_n+\left(A-\left(A+B\right)y_n\right)\Delta t + \dfrac{C\sqrt{y_n\left(1-y_n\right)}\Delta W_n}{1+d^1\left(y_n\right)|\Delta W_n|}\left(1-\left(A+B\right)\Delta t\right)\tag{5}$$

My doubts:

1. I cannot understand in which way the last method approximates solution to $$\left(3\right)$$ at each time step using $$\left(2\right)$$. Could you please explicit such an approximation? How is it obtained by means of $$\left(2\right)$$?
2. In which sense numerical solution to $$\left(3\right)$$ is used as the initial condition in $$\left(4\right)$$? Which is such an initial condition?
3. Could you please explicit the way in which solution to $$\left(3\right)$$ and solution to $$\left(4\right)$$ are combined so as to obtain discretization formula $$\left(5\right)$$?