qmcintdim12

金融资产定价,随机过程,MONTE CARLO 模拟 JAVA 程序和文档资料

QUASI MONTE CARLO COMPUTATION OF TEST INTEGRALS:

We compute Int_Q F(x)dx, where Q=(0,1)^d (unit cube), d=dimension
and F(x)=h(x_1)h(x_2)...h(x_d), where h(u)=g(m*u-[m*u]).

Here [t] is the greatest integer <= t as usual. The function h(u)
is periodic with m periods on (0,1) repeating the function g(u)
m times on (0,1). If g(u) has one peak on (0,1), h(u) has m peaks
and F(x) has m^d (very narrow) peaks on Q.

DIMENSION=12


g(u)=u+0.5, periods=1, QMC relative error (%):

N: 511        SOB: -4.1     NX: 0.64      HAl: -12.58   UNF: -3.26
N: 1023       SOB: -0.81    NX: -0.12     HAl: -7.07    UNF: 1.32
N: 2047       SOB: -0.01    NX: -0.35     HAl: -4.64    UNF: 0.32
N: 4095       SOB: 0.19     NX: -0.28     HAl: -2.33    UNF: 3.22
N: 8191       SOB: 0.22     NX: -0.63     HAl: -1.62    UNF: 3.76
N: 16383      SOB: 0.14     NX: -0.01     HAl: -0.91    UNF: 1.03
N: 32767      SOB: 0.09     NX: 0.01      HAl: -0.42    UNF: 1.08



g(u)=u+0.5, periods=2, QMC relative error (%):

N: 511        SOB: -5.69    NX: -0.14     HAl: -7.92    UNF: 2.29
N: 1023       SOB: -4.31    NX: -0.12     HAl: -5.43    UNF: 3.93
N: 2047       SOB: -1.52    NX: -0.08     HAl: -2.98    UNF: -1.85
N: 4095       SOB: -0.89    NX: -0.12     HAl: -1.45    UNF: 0.58
N: 8191       SOB: -0.04    NX: -0.14     HAl: -0.42    UNF: -1.18
N: 16383      SOB: -0.09    NX: 0.04      HAl: -0.21    UNF: 0.31
N: 32767      SOB: 0.04     NX: -0.04     HAl: -0.24    UNF: 0.24



g(u)=u+0.5, periods=3, QMC relative error (%):

N: 511        SOB: -3.05    NX: 2.14      HAl: -4.57    UNF: 4.71
N: 1023       SOB: -2.23    NX: 0.91      HAl: -1.08    UNF: -2.3
N: 2047       SOB: -0.68    NX: -0.6      HAl: 1.18     UNF: 1.92
N: 4095       SOB: -0.85    NX: -0.74     HAl: 0.2      UNF: 2.68
N: 8191       SOB: -0.66    NX: -1.3      HAl: 0.02     UNF: 1.28
N: 16383      SOB: -0.74    NX: -0.97     HAl: 0.14     UNF: 1.5
N: 32767      SOB: -0.49    NX: -0.65     HAl: 0.07     UNF: 1.0



g(u)=exp(N_Inverse(u)), periods=1, QMC relative error (%):

N: 511        SOB: -78.53   NX: -13.0     HAl: -91.42   UNF: -56.03
N: 1023       SOB: -64.84   NX: -21.24    HAl: -87.91   UNF: -52.32
N: 2047       SOB: -55.81   NX: -37.55    HAl: -77.39   UNF: -60.88
N: 4095       SOB: 83.78    NX: -27.24    HAl: -74.9    UNF: -62.77
N: 8191       SOB: 23.11    NX: -42.65    HAl: -68.02   UNF: -55.93
N: 16383      SOB: 8.93     NX: -32.49    HAl: -61.5    UNF: -18.17
N: 32767      SOB: 0.16     NX: -27.73    HAl: -51.73   UNF: -31.89



g(u)=exp(N_Inverse(u)), periods=2, QMC relative error (%):

N: 511        SOB: -81.3    NX: -55.7     HAl: -100.0   UNF: 2452.64
N: 1023       SOB: -81.38   NX: -51.18    HAl: -100.0   UNF: 1194.05
N: 2047       SOB: -66.27   NX: -56.97    HAl: -100.0   UNF: 568.57
N: 4095       SOB: -53.9    NX: -40.66    HAl: -100.0   UNF: 253.2
N: 8191       SOB: -48.89   NX: -39.73    HAl: -100.0   UNF: 113.29
N: 16383      SOB: -51.02   NX: -19.11    HAl: -100.0   UNF: 53.38
N: 32767      SOB: -32.96   NX: 9.59      HAl: -100.0   UNF: 12.28



g(u)=exp(N_Inverse(u)), periods=3, QMC relative error (%):

N: 511        SOB: -83.57   NX: 29.12     HAl: -100.0   UNF: -75.62
N: 1023       SOB: -76.14   NX: -25.83    HAl: -100.0   UNF: -8.72
N: 2047       SOB: -67.14   NX: -43.45    HAl: -100.0   UNF: -34.45
N: 4095       SOB: -60.4    NX: -51.52    HAl: -100.0   UNF: 22.9
N: 8191       SOB: -42.88   NX: -34.14    HAl: -100.0   UNF: 94.67
N: 16383      SOB: -43.13   NX: -42.52    HAl: -100.0   UNF: 53.49
N: 32767      SOB: -30.44   NX: -46.61    HAl: -100.0   UNF: 32.91



g(u)=sin^2(2pi*u), periods=1, QMC relative error (%):

N: 511        SOB: 1534.3   NX: -22.18    HAl: -13.98   UNF: -50.85
N: 1023       SOB: 731.49   NX: -13.98    HAl: -13.98   UNF: 39.26
N: 2047       SOB: 424.29   NX: -30.37    HAl: -18.08   UNF: 6.5
N: 4095       SOB: 239.97   NX: -34.46    HAl: -22.18   UNF: -1.7
N: 8191       SOB: 129.38   NX: -30.37    HAl: -9.89    UNF: -1.7
N: 16383      SOB: 59.74    NX: -1.7      HAl: -1.7     UNF: 6.5
N: 32767      SOB: 31.07    NX: -1.7      HAl: -1.7     UNF: -1.7



g(u)=sin^2(2pi*u), periods=2, QMC relative error (%):

N: 511        SOB: 3156.32  NX: -67.23    HAl: -38.56   UNF: 22.88
N: 1023       SOB: 1538.4   NX: -13.98    HAl: -38.56   UNF: 35.17
N: 2047       SOB: 768.35   NX: 14.69     HAl: -34.46   UNF: 31.07
N: 4095       SOB: 383.33   NX: 6.5       HAl: -22.18   UNF: 31.07
N: 8191       SOB: 194.91   NX: 6.5       HAl: -1.7     UNF: 2.4
N: 16383      SOB: 92.51    NX: 2.4       HAl: -5.79    UNF: 2.4
N: 32767      SOB: 43.36    NX: 2.4       HAl: -5.79    UNF: -1.7



g(u)=sin^2(2pi*u), periods=3, QMC relative error (%):

N: 511        SOB: 1579.36  NX: -26.27    HAl: -5.79    UNF: 129.38
N: 1023       SOB: 768.35   NX: 22.88     HAl: 2.4      UNF: 39.26
N: 2047       SOB: 407.9    NX: 18.78     HAl: -13.98   UNF: 67.94
N: 4095       SOB: 194.91   NX: 10.59     HAl: -5.79    UNF: 22.88
N: 8191       SOB: 88.42    NX: 22.88     HAl: 2.4      UNF: 14.69
N: 16383      SOB: 43.36    NX: 6.5       HAl: -5.79    UNF: 6.5
N: 32767      SOB: 26.98    NX: 2.4       HAl: -1.7     UNF: -5.79



g(u)=3u^2, periods=1, QMC relative error (%):

N: 511        SOB: -73.62   NX: 65.76     HAl: -83.63   UNF: 66.6
N: 1023       SOB: -24.52   NX: 10.56     HAl: -75.28   UNF: -9.4
N: 2047       SOB: -22.72   NX: 2.38      HAl: -31.61   UNF: -23.54
N: 4095       SOB: 31.1     NX: 26.1      HAl: -27.93   UNF: -18.91
N: 8191       SOB: 15.26    NX: -13.45    HAl: -25.58   UNF: -27.83
N: 16383      SOB: 34.92    NX: 0.3       HAl: -28.64   UNF: -35.22
N: 32767      SOB: 33.34    NX: 0.38      HAl: -12.74   UNF: -25.05



g(u)=4u^3, periods=2, QMC relative error (%):

N: 511        SOB: -96.38   NX: -66.45    HAl: -94.32   UNF: -95.32
N: 1023       SOB: -97.74   NX: -78.88    HAl: -83.16   UNF: 149.89
N: 2047       SOB: -57.95   NX: -82.74    HAl: -79.39   UNF: 33.51
N: 4095       SOB: -76.38   NX: -17.67    HAl: -73.24   UNF: -15.23
N: 8191       SOB: -58.04   NX: -21.64    HAl: -68.25   UNF: -20.38
N: 16383      SOB: -58.51   NX: 10.36     HAl: -59.17   UNF: -27.67
N: 32767      SOB: 11.65    NX: -30.5     HAl: -34.9    UNF: 21.34



g(u)=5u^4, periods=3, QMC relative error (%):

N: 511        SOB: -99.47   NX: -87.08    HAl: -77.48   UNF: -91.74
N: 1023       SOB: -95.15   NX: -93.34    HAl: -83.06   UNF: -79.76
N: 2047       SOB: -95.93   NX: -96.13    HAl: -71.47   UNF: -83.66
N: 4095       SOB: -95.94   NX: -91.88    HAl: -57.28   UNF: -66.15
N: 8191       SOB: -94.53   NX: -92.02    HAl: -76.62   UNF: -14.23
N: 16383      SOB: -95.0    NX: -81.05    HAl: -85.0    UNF: -39.18
N: 32767      SOB: 83.65    NX: -85.06    HAl: -79.04   UNF: -61.71



g(u)=2-12*(u-0.5)^2, periods=1, QMC relative error (%):

N: 511        SOB: -90.38   NX: 296.48    HAl: -92.52   UNF: 180.94
N: 1023       SOB: -17.04   NX: 109.26    HAl: -4.12    UNF: 92.84
N: 2047       SOB: 11.91    NX: 70.84     HAl: 4.28     UNF: 91.53
N: 4095       SOB: -2.68    NX: -20.13    HAl: -14.61   UNF: 3.54
N: 8191       SOB: -12.99   NX: -39.63    HAl: -41.81   UNF: 35.23
N: 16383      SOB: -20.51   NX: -31.85    HAl: -30.28   UNF: 43.46
N: 32767      SOB: -1.8     NX: -32.23    HAl: -3.75    UNF: 39.29



g(u)=2-12*(u-0.5)^2, periods=2, QMC relative error (%):

N: 511        SOB: 1358.07  NX: -142.51   HAl: 10.06    UNF: 135.21
N: 1023       SOB: 589.62   NX: -78.14    HAl: 38.39    UNF: -56.33
N: 2047       SOB: 356.89   NX: -87.91    HAl: -30.15   UNF: 61.57
N: 4095       SOB: 246.68   NX: -39.44    HAl: -47.89   UNF: 19.21
N: 8191       SOB: 133.68   NX: -42.33    HAl: -18.3    UNF: 5.37
N: 16383      SOB: 44.97    NX: 1.98      HAl: -7.4     UNF: -44.63
N: 32767      SOB: 51.03    NX: -2.1      HAl: -3.44    UNF: -4.94



g(u)=2-12*(u-0.5)^2, periods=3, QMC relative error (%):

N: 511        SOB: -245.7   NX: 115.39    HAl: 164.26   UNF: -97.81
N: 1023       SOB: -306.8   NX: 83.67     HAl: 79.71    UNF: -96.43
N: 2047       SOB: -151.99  NX: 50.08     HAl: 41.7     UNF: -106.97
N: 4095       SOB: -96.37   NX: 15.32     HAl: 50.02    UNF: -81.64
N: 8191       SOB: -73.54   NX: 9.99      HAl: 23.53    UNF: -69.24
N: 16383      SOB: -49.37   NX: 9.83      HAl: 28.11    UNF: -62.68
N: 32767      SOB: -23.65   NX: -10.52    HAl: 0.44     UNF: -52.66



g(u)=3*I_[1/3,2/3](u), periods=1, QMC relative error (%):

N: 511        SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: -100.0   NX: -100.0    HAl: 1521.93  UNF: -100.0



g(u)=3*I_[1/3,2/3](u), periods=2, QMC relative error (%):

N: 511        SOB: 208308.24NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 103900.2 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 51849.27 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 25861.94 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 12877.8  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 6388.11  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 3143.86  NX: -100.0    HAl: -100.0   UNF: -100.0



g(u)=3*I_[1/3,2/3](u), periods=3, QMC relative error (%):

N: 511        SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: 1521.93