qmcintdim16

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=16


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

N: 511        SOB: -6.8     NX: 2.58      HAl: -23.07   UNF: 3.65
N: 1023       SOB: -1.36    NX: 3.55      HAl: -12.75   UNF: 2.53
N: 2047       SOB: -0.13    NX: 4.09      HAl: -9.45    UNF: -2.07
N: 4095       SOB: 0.72     NX: 2.15      HAl: -5.76    UNF: 1.51
N: 8191       SOB: 0.44     NX: 0.42      HAl: -2.92    UNF: 0.64
N: 16383      SOB: 0.04     NX: 0.27      HAl: -1.5     UNF: 0.16
N: 32767      SOB: 0.1      NX: 0.14      HAl: -1.08    UNF: -0.22



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

N: 511        SOB: -11.25   NX: -4.35     HAl: -13.26   UNF: 0.89
N: 1023       SOB: -7.79    NX: -1.69     HAl: -9.45    UNF: -1.97
N: 2047       SOB: -2.17    NX: 0.49      HAl: -4.79    UNF: 3.59
N: 4095       SOB: -1.27    NX: -0.08     HAl: -2.54    UNF: 3.05
N: 8191       SOB: -0.23    NX: 0.11      HAl: -1.14    UNF: 0.82
N: 16383      SOB: 0.14     NX: -0.28     HAl: -1.1     UNF: -0.04
N: 32767      SOB: 0.2      NX: 0.03      HAl: -0.81    UNF: -0.18



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

N: 511        SOB: -6.76    NX: 1.32      HAl: -12.78   UNF: -7.39
N: 1023       SOB: -6.24    NX: -0.53     HAl: -7.01    UNF: -6.18
N: 2047       SOB: -0.77    NX: -0.7      HAl: -1.42    UNF: -5.15
N: 4095       SOB: -1.08    NX: -0.14     HAl: -1.49    UNF: -2.9
N: 8191       SOB: -0.54    NX: 0.16      HAl: -0.61    UNF: -1.52
N: 16383      SOB: -0.87    NX: 0.1       HAl: -0.05    UNF: -0.96
N: 32767      SOB: -0.34    NX: 0.34      HAl: 0.01     UNF: -1.35



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

N: 511        SOB: -73.33   NX: -86.97    HAl: -96.91   UNF: -60.34
N: 1023       SOB: -49.83   NX: -63.77    HAl: -90.39   UNF: -69.7
N: 2047       SOB: -37.91   NX: 534.68    HAl: -90.74   UNF: -73.55
N: 4095       SOB: -33.07   NX: 245.31    HAl: -87.69   UNF: -79.28
N: 8191       SOB: -24.6    NX: 83.93     HAl: -59.54   UNF: -67.02
N: 16383      SOB: 32.85    NX: 15.34     HAl: -62.36   UNF: -50.29
N: 32767      SOB: -1.89    NX: -5.19     HAl: -59.67   UNF: -56.24



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

N: 511        SOB: -95.29   NX: -86.32    HAl: -100.0   UNF: -84.33
N: 1023       SOB: -83.41   NX: -55.63    HAl: -100.0   UNF: -82.41
N: 2047       SOB: -81.05   NX: -56.25    HAl: -100.0   UNF: -74.32
N: 4095       SOB: -72.82   NX: -25.35    HAl: -100.0   UNF: -77.16
N: 8191       SOB: 14.69    NX: -13.05    HAl: -100.0   UNF: -79.75
N: 16383      SOB: -24.62   NX: -37.72    HAl: -100.0   UNF: -61.36
N: 32767      SOB: -32.14   NX: -17.07    HAl: -100.0   UNF: -49.64



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

N: 511        SOB: -91.35   NX: -77.49    HAl: -100.0   UNF: 793.57
N: 1023       SOB: -68.82   NX: -78.49    HAl: -100.0   UNF: 378.03
N: 2047       SOB: -64.53   NX: -25.23    HAl: -100.0   UNF: 147.12
N: 4095       SOB: -60.19   NX: -20.14    HAl: -100.0   UNF: 42.56
N: 8191       SOB: -68.94   NX: -31.17    HAl: -100.0   UNF: -21.03
N: 16383      SOB: -70.09   NX: -38.84    HAl: -100.0   UNF: -50.14
N: 32767      SOB: -70.14   NX: -49.59    HAl: -100.0   UNF: -29.55



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

N: 511        SOB: 25590.11 NX: 31.07     HAl: -34.46   UNF: 162.14
N: 1023       SOB: 12745.06 NX: 31.07     HAl: -34.46   UNF: 31.07
N: 2047       SOB: 6912.35  NX: -34.46    HAl: -34.46   UNF: 96.61
N: 4095       SOB: 3438.94  NX: 31.07     HAl: -34.46   UNF: 31.07
N: 8191       SOB: 1735.01  NX: -34.46    HAl: 31.07    UNF: 31.07
N: 16383      SOB: 883.04   NX: 31.07     HAl: -34.46   UNF: -34.46
N: 32767      SOB: 424.29   NX: 31.07     HAl: -34.46   UNF: 31.07



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

N: 511        SOB: 51345.76 NX: -100.0    HAl: -100.0   UNF: -34.46
N: 1023       SOB: 25590.11 NX: -34.46    HAl: -34.46   UNF: -34.46
N: 2047       SOB: 12745.06 NX: 31.07     HAl: -34.46   UNF: -34.46
N: 4095       SOB: 6388.06  NX: -34.46    HAl: -34.46   UNF: -34.46
N: 8191       SOB: 3176.8   NX: -34.46    HAl: -34.46   UNF: -34.46
N: 16383      SOB: 1538.4   NX: -34.46    HAl: -34.46   UNF: -34.46
N: 32767      SOB: 751.97   NX: -34.46    HAl: -34.46   UNF: -34.46



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

N: 511        SOB: 25590.11 NX: -34.46    HAl: -100.0   UNF: -34.46
N: 1023       SOB: 12876.13 NX: -34.46    HAl: -34.46   UNF: -34.46
N: 2047       SOB: 6584.67  NX: -34.46    HAl: -34.46   UNF: -34.46
N: 4095       SOB: 3242.34  NX: -34.46    HAl: 31.07    UNF: -34.46
N: 8191       SOB: 1603.94  NX: -34.46    HAl: 31.07    UNF: -34.46
N: 16383      SOB: 817.5    NX: -34.46    HAl: 31.07    UNF: -34.46
N: 32767      SOB: 424.29   NX: -34.46    HAl: 31.07    UNF: -34.46



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

N: 511        SOB: -80.67   NX: -51.75    HAl: -94.04   UNF: -8.18
N: 1023       SOB: -68.99   NX: -45.99    HAl: -15.57   UNF: -16.51
N: 2047       SOB: -69.14   NX: 32.32     HAl: -21.78   UNF: -17.83
N: 4095       SOB: 60.27    NX: 5.05      HAl: -43.84   UNF: -5.49
N: 8191       SOB: 36.27    NX: -28.83    HAl: 33.08    UNF: -39.78
N: 16383      SOB: 197.66   NX: -17.77    HAl: -4.69    UNF: -40.32
N: 32767      SOB: 84.4     NX: -22.37    HAl: -3.56    UNF: -8.54



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

N: 511        SOB: -99.76   NX: -97.71    HAl: -98.81   UNF: -99.7
N: 1023       SOB: -99.82   NX: -58.84    HAl: -98.37   UNF: -77.42
N: 2047       SOB: -98.97   NX: -75.63    HAl: -87.17   UNF: -87.93
N: 4095       SOB: -93.14   NX: -63.98    HAl: -70.52   UNF: -92.03
N: 8191       SOB: -71.31   NX: -67.88    HAl: -80.78   UNF: -90.36
N: 16383      SOB: -81.97   NX: -68.56    HAl: -82.61   UNF: -89.93
N: 32767      SOB: -81.05   NX: -59.28    HAl: -82.95   UNF: -87.19



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

N: 511        SOB: -99.89   NX: -99.66    HAl: -100.0   UNF: -100.0
N: 1023       SOB: -99.93   NX: -99.8     HAl: -99.45   UNF: -87.33
N: 2047       SOB: -99.94   NX: -99.39    HAl: -99.2    UNF: -93.66
N: 4095       SOB: -99.41   NX: -97.64    HAl: -99.53   UNF: -96.35
N: 8191       SOB: -99.66   NX: -96.72    HAl: -98.1    UNF: -97.5
N: 16383      SOB: -96.44   NX: -97.27    HAl: -98.46   UNF: -98.32
N: 32767      SOB: -56.43   NX: -97.86    HAl: -80.21   UNF: -98.48



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

N: 511        SOB: -292.69  NX: 734.55    HAl: 161.49   UNF: -17.13
N: 1023       SOB: -536.77  NX: 1020.4    HAl: 267.6    UNF: -31.77
N: 2047       SOB: -302.05  NX: 616.64    HAl: -28.57   UNF: -123.69
N: 4095       SOB: -151.47  NX: 233.72    HAl: 101.44   UNF: -36.31
N: 8191       SOB: 12.52    NX: 228.62    HAl: -86.33   UNF: 24.93
N: 16383      SOB: 27.7     NX: 84.87     HAl: 30.61    UNF: -12.89
N: 32767      SOB: 41.63    NX: 33.54     HAl: -67.18   UNF: -44.82



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

N: 511        SOB: 26120.71 NX: -508.22   HAl: 95.86    UNF: -150.78
N: 1023       SOB: 12771.1  NX: -99.17    HAl: 226.22   UNF: -335.11
N: 2047       SOB: 7064.67  NX: -215.37   HAl: 238.39   UNF: -165.22
N: 4095       SOB: 3411.62  NX: -124.19   HAl: 116.93   UNF: -100.74
N: 8191       SOB: 2040.54  NX: -68.2     HAl: 136.11   UNF: -25.33
N: 16383      SOB: 999.94   NX: 19.65     HAl: -62.35   UNF: -64.72
N: 32767      SOB: 603.95   NX: -0.78     HAl: -93.31   UNF: -88.41



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

N: 511        SOB: -264.43  NX: -139.85   HAl: 350.52   UNF: -70.27
N: 1023       SOB: -305.77  NX: 32.62     HAl: 235.29   UNF: -86.3
N: 2047       SOB: -180.87  NX: 96.86     HAl: 220.86   UNF: -193.7
N: 4095       SOB: -94.42   NX: -12.4     HAl: 87.86    UNF: -79.3
N: 8191       SOB: -83.72   NX: 133.22    HAl: 151.15   UNF: -87.41
N: 16383      SOB: -140.87  NX: 155.11    HAl: 84.87    UNF: 15.24
N: 32767      SOB: -43.34   NX: 142.45    HAl: 63.37    UNF: 16.99



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: -100.0   UNF: -100.0



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

N: 511        SOB: 1.688096706E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 8423915.85NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 4207790.62NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 2102817.49NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 1051101.98NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 525436.82NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 262652.37NX: -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: -100.0