Table 1.

The method ContDetApprox (left) and the main procedure sWindow (right)

ALGORITHM ContDetApprox (z, τ, b+, b-)	ALGORITHM sWindow(y, t, ϵ, δ)
Input: initial state z, length τ of time interval, old boundaries	Input: initial state y, times t = (t0, ..., tr), error ϵ > 0, threshold δ > 0
Output: new boundaries	Output: probability vectors p0, ..., pr
1 for each branch i ∈ {1, ..., 2n} do	1 p0 = (1y)T; //start with probability one in y
2    x⟨i⟩:= z; //z is initial state of all branches	2 for j := 1 to r do
3 end	3    τj := tj - tj-1; //length of next time step
4 time := 0;	//define Sjfor construction of Wj
5 Δ := step_size; //choose length of time steps	4    Sj := {x | pj-1(x) >δ};
6 while time <τ do	5    numStates := min(10, size(Sj));
7    for each branch i ∈ {1, ..., 2n} do	//choose numStates random states from Sj
//compare current state variables with boundaries	6    Aj := rand(Sj, numStates);
8       for d = 1 to n do	7    b+: -∞; b- : +∞; //initial boundaries
9          if then	8    for each z in Aj do
10             ; //adjust upper bound	//run continuous determ. approximation
11          end	//on z and update boundaries
12          if then	9    (b+, b-) := ContDetApprox (z, τj, b+, b-);
13             ; //adjust lower bound	10    end
14          end	11    Qj := generator (Sj, b+, b-); //construct Qj
15       end	//construct diagonal matrix for Wj (cf. Eq. (7))
16       for m := 1 to k do	12    Dj := diag(Sj, b+, b-);
//choose more/fewer transitions of type Rm	13    ; //solve Eq. (7)
//depending on branch i	14 end
17          κm := choose(αm(x⟨i⟩)·Δ, i);
18       end
19       ; //update state (cf. Eq. (11))
20    end
21    time := time + Δ;
22 end