Extract the values for some parameters from stochastic differential equation solution - matlab

Question

I am solving stochastic differential equation in matlab.
For example:
consider the stochastic differential equation
dx=k A(x,t)dt+ B(x,t)dW(t)
where k is constants, A and B are functions, and dW(t) is Wiener process.
I plot the solution for all t in [0,20]. We know that dW(t) is randomly generated. My question is: I want to know the value of A(x,t), B(x,t), dW(t) for a particular value of t and for particular sub-interval, say [3,6]. What command in Matlab I can use?
Here is the code I used based on a paper by D.Higham:
clear all
close all
t0 = 0; % start time of simulation
tend = 20; % end time
m=2^9; %number of steps in each Brownian path
deltat= tend/m; % time increment for each Brownian path
D=0.1; %diffsuion
R=4;
dt = R*deltat;
dW=sqrt( deltat)*randn(2,m);
theta0=pi*rand(1);
phi0=2*pi*rand(1);
P_initial=[ theta0; phi0];
L = m/ R;
pem=zeros(2,L);
EM_rescale=zeros(2,L);
ptemp=P_initial;
for j=1:L
Winc = sum(dW(:,[ R*(j-1)+1: R*j]),2);
theta=ptemp(1);% updating theta
phi=ptemp(2); % updating phi
%psi=ptemp(3); % updating psi
A=[ D.*cot(theta);...
0];% updating the drift
B=[sqrt(D) 0 ;...
0 sqrt(D)./sin(theta) ]; %% updating the diffusion function
ptemp=ptemp+ dt*A+B*Winc;
pem(1,j)=ptemp(1);%store theta
pem(2,j)=ptemp(2);%store phi
EM_rescale(1,j)=mod(pem(1,j),pi); % re-scale theta
EM_rescale(2,j)=mod(pem(2,j),2*pi); % re-scale phi
end
plot([0:dt:tend],[P_initial,EM_rescale],'--*')
Suppose I want to know all parameters (including random: Brownian) at each specific time point or for any time interval. How to do that?

Answer 1

I'm doing my best to understand your question here, but it's still a bit unclear to me.
Change the loop to:
for ii=1:L
Winc = sum(dW(:,[ R*(ii-1)+1: R*ii]),2);
theta=ptemp(1);% updating theta
phi=ptemp(2); % updating phi
A{ii}=[ D.*cot(theta);...
0];% updating the drift
B{ii}=[sqrt(D) 0 ;...
0 sqrt(D)./sin(theta) ]; %% updating the diffusion function
ptemp = ptemp + dt*A{ii}+B{ii}*Winc;
pem(:,ii) = ptemp;
EM_rescale(1,ii) = mod(pem(1,ii),pi); % re-scale theta
EM_rescale(2,ii) = mod(pem(2,ii),2*pi); % re-scale phi
end
Now, you can get the values of A and B this way:
t = 3;
t_num = round(m/tend*t);
A{t_num}
B{t_num}
ans =
0.0690031455719538
0
ans =
0.316227766016838 0
0 0.38420611784333