function intgxtdw
% Fig. 4.2 Book code example for int[g(x(t),t)dw] on [t0,t] (3/2007) 
%   by RNG Simulation:
%   Generation is by summing g(X(i),t(i))dW(i) of even spacing dt for
%      i=0:n, but converted to from index base zero to base one:
%      matlab[G(X(i),T(i))DW(i);i=1:N+1] = math[g(X(i),t(i))dW(i);i=0:n].
%   Chain Rule Form: Int[gdW](t) = G(W,t)-G(0,0) - Int[(g_t+0.5*g_w)(w,t)dt];
%      G_w(w,t) = g(w,t), G_{ww}(w,t) = g_w(w,t).
%   Sample Test Code for various g's, here for g(x,t) = exp(x) and x = w.
clc % clear variables, but must come before globals, 
%     else clears globals too.
clf % clear figures
fprintf('\nfunction intgxtdw OutPut:')
nfig = 0; % figure counter.
TF= 2.0; T0 = 0; N = (TF-T0)*10000;  NI = N+1; dt = (TF-T0)/NI; 
% Set initial time grid: Fixed Delta{t}, Scaled to [T0,TF] with N.
sqrtdt = sqrt(dt); % Set standard Wiener increment time scale.
% Begin Sample Path Calculation:
kstate = 1;
randn('state',kstate);  % set randn state for repeatability.
dW = sqrtdt*randn(1,NI); % Generate normal random vector of N+1 
%                          samples for dW(t).
t = zeros(1,NI+1); t(1) = T0;  % set T initially.
W = zeros(1,NI+1);  % Set W(1) in place of W(0) = 0 wpo for base 1 vector.
X = zeros(1,NI+1);  % Set integral sum initially.
gv = zeros(1,NI+1); gv(1) = g(X(1),t(1)); % Set integrand initially.
sdw = zeros(1,NI+1); sdt = zeros(1,NI+1); % Set integral sum initially.
ev = zeros(1,NI+1); % Set Error initially.
for i = 1:NI % Simulated Sample paths by Increment Accumulation:
    t(i+1) = i*dt;
    W(i+1) = W(i) + dW(i);
    X(i+1) = W(i+1); % Set State for this g Example.
    gv(i+1) = g(X(i+1),t(i+1));
    sdw(i+1) = sdw(i) + gv(i)*dW(i);% integrand g in subfunction.
    sdt(i+1) = sdt(i) - gthgw(X(i+1),t(i+1))*dt;% gthgw in subfunction.
    ev(i+1) = sdw(i+1) - exact(X(i+1),t(i+1)) - sdt(i+1); 
    % CAUTION: For given g only!
end
t(NI+1) = TF;  % Correct for final cumulative time rounding errors.
% Begin Plot:
nfig = nfig + 1;
fprintf('\n\nFigure(%i):  int[g](t) versus t Simulations\n',nfig)
figure(nfig)
scrsize = get(0,'ScreenSize'); % figure spacing for target screen
ss = [5.0,4.0,3.5]; % figure spacing factors
plot(t,sdw,'k-',t,W,'k-.',t,ev,'k--','LineWidth',2);
title('\int g(X,t)dW(t) for g = exp(X), X = W'...
    ,'FontWeight','Bold','Fontsize',44);
ylabel('\int g(X,t)dW(t), X = W(t) and Error(t)'...
    ,'FontWeight','Bold','Fontsize',44);
xlabel('t, Time'...
    ,'FontWeight','Bold','Fontsize',44);
hlegend=legend('\int g(X,t)dW(t)','X = W(t)','Error(t)'...
    ,'Location','SouthWest');
set(hlegend,'Fontsize',36,'FontWeight','Bold');;
set(gca,'Fontsize',36,'FontWeight','Bold','linewidth',3);
set(gcf,'Color','White','Position' ...
   ,[scrsize(3)/ss(nfig) 60 scrsize(3)*0.60 scrsize(4)*0.80]); %[l,b,w,h]
% End Main
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function gv = g(x,t) % ignore caution that t might not be used here.
% Sample g(X(t),t) only, e.g.,
%1% gv = exp(x-t/2); % x = w.
%2% gv = exp(x); % x = w.
%3% gv = x; % x = w.
gv = exp(x);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function exactv = exact(x,t)
% Sample g(X(t),t) exact integrals only, e.g., 
%1% exactv = exp(x-t/2) - 1; % i.e., G(w,t)-G(0,0), x=w, G(w,t)=exp(w-t/2).
%2% exactv = exp(x) - 1; % i.e., G(w,t)-G(0,0), x=w, G(w,t)=exp(w).
%3% exactv = 0.5*(x^2-t);  % i.e., G(w,t)-G(0,0), x=w, G(w,t)=0.5*(w^2-t).
exactv = exp(x) - 1; % i.e., G(w,t)-G(0,0), x=w, G(w,t)=exp(w).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function gthgwv = gthgw(x,t)
% Reg. Correction Int. of (G_t+0.5*G_{ww})(X(t),t), G_w = g.
%1% gthgwv = 0; % i.e., g=exp(x-t/2)=G, G_t=-0.5*G, G_{ww}=G.
%2% gthgwv = 0.5*exp(x); % i.e., G=g=exp(w), G_t=0, G_{ww}=g_w=exp(w).
%3% gthgw 0; % i.e., g=x=w, G=0.5*(w^2-t), G_t=-0.5, G_{ww}=g_w=1;
gthgwv = 0.5*exp(x); % i.e., G=g=exp(w), G_t=0, G_{ww}=g_w=exp(w).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% End Code