Little experiment to validate that absolute variation tends to infinity, mean squared variation tends to 1 (for standard normal), and the cubic, quartic etc variations tend to 0 (as delta t tends towards 0). This, if I am not mistaken, is why we can do all the tricks we do in Ito calculus.
MATLAB code (my first ever):
function drawExpectationConvergence = drawExpectationConvergence(nstepsmax, moment) figure; hold on; results = zeros(0, nstepsmax); parfor n = 1:nstepsmax results(n) = expectation(n, moment); end plot(results); hold off; end function expectation = expectation(nsteps, moment) T = 1.0; dt = T / nsteps; sqrt_dt = dt^0.5; V = 0; for i = 0:nsteps V = V + abs((normrnd(0,1)*sqrt_dt)^moment); end expectation = V; end
Maybe marissa killed it