Does cummax works in a sde solver?

I want to model a 2-dimension SDE where the drift and the diffusion are:

% solve the system
F =  @(t, X) [mu*pi(y-(X(2)-z)+(X(1)-v)+max(y, cummax(X(2)-X(1)))-y, X(2))-c(y-(X(2)-z) ...
    +(X(1)-v)+max(y, cummax(X(2)-X(1)))-y, X(2)); muz*X(2)];

G =  @(t, X) [sigma*pi(y-(X(2)-z)+(X(1)-v)+max(y, cummax(X(2)-X(1)))-y, X(2)) 0; 0 sigmaz*X(2)];
X = sde(F, G, "StartState", x);
[X, T] = simByEuler(X, n, 'DeltaTime', dt);

Here, pi is some function that I have defined before in the code. My question is: does here cummax works on all previous values of X(2)-X(1), or it takes it as a scalar at every iteration? In particular

max(y, cummax(X(2)-X(1)))

should represent the process .

Thank you so much for your attention, I hope I made myself clear.

Answer 1

[Torsten]

My question is: does here cummax works on all previous values of X(2)-X(1), or it takes it as a scalar at every iteration?

It takes it as a scalar in each iteration because X(2)-X(1) is a scalar.

I'm not an expert in SDEs, but it seems that your equation does not belong to the class that can be solved using "sde".

[Salvatore Bianco]

So, there is a method I can use that keeps track of all previous values?

[Torsten]

I think even if there were a way to keep track of them, your equation cannot be solved with "sde" because F and G in the underlying equation can only depend on X_t, not on X_s for s < t.

But as said: I'm not an expert in this field - I might be mistaken. (And I also don't know how to keep track of them.)