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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
$$
d^1(y)=
begin{cases}
Csqrt{frac{1-varepsilon}{varepsilon}}hspace{0.5cm}text{if }y<varepsilon\
Csqrt{frac{1-y}{y}}hspace{0.5cm}text{if }varepsilonle y<frac{1}{2}\
Csqrt{frac{y}{1-y}}hspace{0.5cm}text{if }frac{1}{2}le yle 1-varepsilon\
Csqrt{frac{1-varepsilon}{varepsilon}}hspace{0.5cm}text{if }y>1-varepsilon
end{cases}
$$

At this point, let us consider a "method" which decomposes $left(1right)$ 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 $left(5right)$?

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