Question

我正在考虑使用Sobol序列作为方差减少技术来加速模拟。然而，QMC方法的收敛似乎很弱。

有谁知道这是怎么回事？ QMC的优势是否在这些高维度情况下丢失了？我得到了布朗桥的类似结果。

下面的代码试图在Julia中模拟以下

$$ \ mathbb {E} \ int B_t dt $$

其中$ B_t $是标准布朗运动。真正的价值是$ \ 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

这是一个评论，而不是一个答案，但我的SO分数不够高，无法发表评论用tic（）... toc（）计时 - 在3秒内完成。

tic()
dt = 0.5;
T = 10;
...
...
p = plot(linspace(0, M, M), estims);
toc()

elapsed time: 2.928828918 seconds

准蒙特卡罗在布朗运动的路径模拟中表现不佳。朱莉娅

1 个答案: