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

Mathematics Asked by Strictly_increasing on July 27, 2020

Consider the following stochastic differential equation
$$begin{equation} dy=left(A-left(A+Bright)yright)dt+Csqrt{yleft(1-yright)}dWtag{1} end{equation}$$
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:
$$begin{equation} y_{n+1}=y_n+left(A-left(A+Bright)y_nright)Delta t +Csqrt{y_nleft(1-y_nright)}Delta W_n + Dleft(y_nright)left(y_n-y_{n+1}right)tag{2} end{equation}$$
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
$$D(y_n)=d^0(y_n)Delta t + d^1left(y_nright)|Delta W_n|$$
with
$$d^1(y)= begin{cases} Csqrt{frac{1-varepsilon}{varepsilon}}hspace{0.5cm}text{if }y1-varepsilon end{cases}$$
At this point, let us consider a "method" which decomposes $$left(1right)$$ into two equations. Specifically, the first equation is a stochastic one, that consists of the diffusion term of $$left(1right)$$ only (see eqtn $$left(3right)$$), while the second one is an ordinary differential equation (see eqtn $$left(4right)$$) that consists of the drift part of $$left(1right)$$. We have:

$$begin{equation} dy_1=Csqrt{y_1left(1-y_1right)}dWtag{3} end{equation}$$
$$begin{equation} dy_2=left(A-left(A+Bright)y_2right)dttag{4} end{equation}$$

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

My doubts:

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