Quasi-monte-carlo underperforms in path simulation of Brownian motion. Julia - math

Question

I'm looking at using Sobol sequences as a variance reduction technique to speed up a simulation. However, the convergence of the QMC method seems very weak.
Does anyone know how or why this is the case? Is the advantage of QMC lost on these high dimensional cases? I get similar results for the Brownian bridge.
Code below attempts to simulate in Julia the following
$$ \mathbb{E} \int B_t dt $$
where $B_t$ is a standard Brownian motion. The true value is $\frac{1}{2}T^2$.
using PyPlot
import Distributions: Normal
import PyPlot: plt
using Sobol
function pseudo_path_variation(dt, T, M)
# The pseudo random approach
N = round(Int, T/dt);
Z = zeros(N, M);
d = Normal(0, 1.0);
sum = 0.0;
for j in 1:M, i in 1:N-1
Z[i+1, j] = Z[i, j] + sqrt(dt)*rand(d, 1)[1];
end
# Calculate sum
return sumabs2(Z)*dt/M;
end
function quasi_path_variation(dt, T, M)
# An attempt at the above using a N dimensional sobol sequence
N = round(Int, T/dt);
Z = zeros(N, M);
d = Normal(0, 1.0);
sum = 0.0;
# The N dimensional sobol sequence
s = SobolSeq(N);
# Burn in the sequence
for i in 1:10
next(s);
end
for j in 1:M
B = next(s);
for i in 1:N-1
Z[i+1, j] = Z[i, j] + sqrt(dt)*quantile(d, B[i]);
end
end
# Calculate sum
return sumabs2(Z)*dt/M;
end
dt = 0.5;
T = 10;
M = 1000;
estims = zeros(M);
for N = 1:M-1
estims[N+1] = quasi_path_variation(dt, T, N)
end
p = plot(linspace(0, M, M), estims);
estims = zeros(M);
for N = 1:M-1
estims[N+1] = pseudo_path_variation(dt, T, N)
end
p = plot(linspace(0, M, M), estims);

Answer 1

This is a comment, not an answer, but my SO score is not high enough for comments.
Timing with tic()..toc() - completed in 3 seconds.
tic()
dt = 0.5;
T = 10;
...
...
p = plot(linspace(0, M, M), estims);
toc()
elapsed time: 2.928828918 seconds