If I'm interpreting correctly, your v (aka S^2) is recursively defined. So presumably you have some way to determine u(0) and S(0)^2. Based on that, you can write a function to determine v(k) either as a matrix calculation or using a loop.
v = @(x) (diag(ones(n,1),0)+diag(-x(3)*ones(n-1,1),-1))\(x(1)+[x(2)*u0^2+x(3)*v0;x(2)*u(1:end-1).^2]);
Now define your objective function
f = @(x) sum(-log(v(x)) - u.^2./v(x));
Then apply a minimization routine
were x0 is your initial guess for x = [w;A;B].
EDIT TO ADD Based on comments below, here's a fuller solution:
n = 123;
v0 = 0;
u =
u0 = u(1,1);
v = @(x) (diag(ones(n,1),0)+diag(-x(3)*ones(n-1,1),-1))\(x(1)+[x(2)*u0^2+x(3)*v0;x(2)*u(1:end-1).^2]);
f = @(x) -sum(-log(v(x)) - u.^2./v(x));
x0 = [0.001; 0.0626; 0.8976];
xmin = fmincon(f,x0,[0,1,1],1,[],[],[-Inf;0;0],[Inf;1;1],[],optimset('Algorithm','interior-point'))
-f(xmin)
If you have Global Optimization Toolbox, you can do this multiple times:
ms = MultiStart;
problem = createOptimProblem('fmincon','x0',x0,...
'objective',f,'lb',[-Inf;0;0],'ub',[Inf;1;1],'Aineq',[0,1,1],'bineq',1,'options',optimset('Algorithm','interior-point'));
xmin = run(ms,problem,30)
-f(xmin)
(change the 30 to whatever you consider sufficient). If you don't have Global Opt TB, you could program it yourself. Make a grid of initial points, loop over them, and keep the best solution. Or do a loop and use a random x0 each time.
Best Answer