qmcintdim10

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=10


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

N: 511        SOB: -1.45    NX: -0.06     HAl: -7.02    UNF: -4.51
N: 1023       SOB: -0.51    NX: -0.41     HAl: -4.58    UNF: 1.28
N: 2047       SOB: -0.05    NX: 0.37      HAl: -2.83    UNF: 1.08
N: 4095       SOB: 0.01     NX: -0.07     HAl: -1.5     UNF: -0.17
N: 8191       SOB: 0.1      NX: 0.01      HAl: -0.98    UNF: 2.0
N: 16383      SOB: 0.09     NX: 0.0       HAl: -0.57    UNF: -0.09
N: 32767      SOB: 0.06     NX: 0.0       HAl: -0.26    UNF: 0.25



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

N: 511        SOB: -2.43    NX: -1.17     HAl: -5.28    UNF: -5.17
N: 1023       SOB: -1.54    NX: -1.22     HAl: -3.34    UNF: -6.82
N: 2047       SOB: -0.02    NX: -0.83     HAl: -2.13    UNF: -3.18
N: 4095       SOB: -0.26    NX: -0.68     HAl: -1.04    UNF: -1.45
N: 8191       SOB: -0.05    NX: -0.13     HAl: -0.37    UNF: -0.88
N: 16383      SOB: -0.1     NX: -0.02     HAl: -0.16    UNF: -1.09
N: 32767      SOB: -0.07    NX: -0.01     HAl: -0.18    UNF: -0.36



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

N: 511        SOB: -4.11    NX: 2.07      HAl: -3.29    UNF: 2.87
N: 1023       SOB: -2.38    NX: 1.13      HAl: 0.66     UNF: 5.02
N: 2047       SOB: -0.94    NX: 0.37      HAl: 0.94     UNF: 2.68
N: 4095       SOB: -0.36    NX: 0.83      HAl: 0.25     UNF: 2.15
N: 8191       SOB: -0.08    NX: 0.23      HAl: 0.03     UNF: 1.29
N: 16383      SOB: -0.16    NX: -0.19     HAl: 0.13     UNF: 1.43
N: 32767      SOB: -0.19    NX: -0.21     HAl: 0.0      UNF: 0.86



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

N: 511        SOB: -56.56   NX: -61.68    HAl: -81.57   UNF: 40.04
N: 1023       SOB: -59.01   NX: -65.26    HAl: -79.51   UNF: 0.77
N: 2047       SOB: -32.98   NX: 41.02     HAl: -70.64   UNF: -26.2
N: 4095       SOB: -4.52    NX: 3.54      HAl: -64.46   UNF: -37.97
N: 8191       SOB: -14.2    NX: 1.23      HAl: -58.52   UNF: -21.52
N: 16383      SOB: -13.16   NX: -7.36     HAl: -51.34   UNF: -19.22
N: 32767      SOB: -12.42   NX: -13.62    HAl: -41.42   UNF: -32.15



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

N: 511        SOB: -41.66   NX: -62.98    HAl: -100.0   UNF: -5.55
N: 1023       SOB: -34.57   NX: -50.74    HAl: -100.0   UNF: -8.91
N: 2047       SOB: -30.64   NX: -40.99    HAl: -100.0   UNF: -21.66
N: 4095       SOB: -39.19   NX: 4.86      HAl: -100.0   UNF: 24.18
N: 8191       SOB: -40.13   NX: -10.28    HAl: -100.0   UNF: 15.45
N: 16383      SOB: -38.73   NX: 6.24      HAl: -100.0   UNF: -0.91
N: 32767      SOB: -31.09   NX: 4.12      HAl: -100.0   UNF: 15.59



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

N: 511        SOB: -68.32   NX: -38.94    HAl: -100.0   UNF: 18.75
N: 1023       SOB: -61.59   NX: -51.46    HAl: -100.0   UNF: -1.93
N: 2047       SOB: -50.68   NX: -32.97    HAl: -100.0   UNF: -11.2
N: 4095       SOB: -27.11   NX: -20.74    HAl: -100.0   UNF: -3.42
N: 8191       SOB: -21.36   NX: -30.5     HAl: -100.0   UNF: -26.32
N: 16383      SOB: -25.35   NX: -23.33    HAl: -100.0   UNF: 52.82
N: 32767      SOB: -19.69   NX: -23.7     HAl: -100.0   UNF: 18.11



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

N: 511        SOB: 344.42   NX: -27.3     HAl: 30.05    UNF: -22.18
N: 1023       SOB: 154.98   NX: 12.64     HAl: 9.57     UNF: 65.89
N: 2047       SOB: 108.9    NX: 12.64     HAl: -0.67    UNF: 24.93
N: 4095       SOB: 62.82    NX: 2.4       HAl: 3.42     UNF: 18.78
N: 8191       SOB: 41.31    NX: -1.7      HAl: -2.72    UNF: 19.81
N: 16383      SOB: 11.62    NX: 0.35      HAl: -1.7     UNF: 10.59
N: 32767      SOB: 9.57     NX: 0.35      HAl: 0.35     UNF: 1.38



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

N: 511        SOB: 741.73   NX: 28.0      HAl: -15.01   UNF: 50.53
N: 1023       SOB: 384.35   NX: 22.88     HAl: -22.18   UNF: 38.24
N: 2047       SOB: 192.86   NX: 9.57      HAl: -3.74    UNF: 29.02
N: 4095       SOB: 93.54    NX: 11.62     HAl: -5.79    UNF: 7.52
N: 8191       SOB: 63.84    NX: 2.4       HAl: 1.38     UNF: 1.38
N: 16383      SOB: 26.98    NX: 4.45      HAl: 0.35     UNF: 5.47
N: 32767      SOB: 13.66    NX: 1.38      HAl: 0.35     UNF: 1.38



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

N: 511        SOB: 401.76   NX: 73.06     HAl: 7.52     UNF: 21.86
N: 1023       SOB: 184.67   NX: 26.98     HAl: -9.89    UNF: 16.74
N: 2047       SOB: 114.02   NX: 18.78     HAl: -9.89    UNF: 4.45
N: 4095       SOB: 58.72    NX: 4.45      HAl: 1.38     UNF: 13.66
N: 8191       SOB: 31.07    NX: -0.67     HAl: 1.38     UNF: 9.57
N: 16383      SOB: 21.86    NX: -2.72     HAl: -1.7     UNF: 3.42
N: 32767      SOB: 16.74    NX: -2.72     HAl: 0.35     UNF: -3.74



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

N: 511        SOB: -35.03   NX: -30.42    HAl: -57.57   UNF: 62.12
N: 1023       SOB: -31.83   NX: -43.48    HAl: -60.18   UNF: 10.91
N: 2047       SOB: -16.2    NX: 59.11     HAl: -41.38   UNF: 68.52
N: 4095       SOB: 30.51    NX: 23.96     HAl: -33.72   UNF: 48.0
N: 8191       SOB: 31.02    NX: 8.08      HAl: -30.66   UNF: 24.68
N: 16383      SOB: 22.02    NX: -2.05     HAl: -26.74   UNF: 0.72
N: 32767      SOB: 15.83    NX: -5.18     HAl: -13.5    UNF: -0.27



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

N: 511        SOB: -77.56   NX: -41.81    HAl: -47.3    UNF: -64.38
N: 1023       SOB: -63.09   NX: -43.74    HAl: -56.1    UNF: -28.07
N: 2047       SOB: 26.9     NX: -28.3     HAl: -65.3    UNF: -54.71
N: 4095       SOB: -11.01   NX: -18.93    HAl: -69.26   UNF: -11.61
N: 8191       SOB: -17.92   NX: -36.08    HAl: -64.22   UNF: 0.92
N: 16383      SOB: -12.62   NX: 31.61     HAl: -54.67   UNF: 30.38
N: 32767      SOB: -13.58   NX: 16.79     HAl: -14.67   UNF: -11.46



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

N: 511        SOB: -92.69   NX: -84.8     HAl: -97.56   UNF: -35.52
N: 1023       SOB: -88.81   NX: -87.51    HAl: -33.18   UNF: -56.19
N: 2047       SOB: -92.35   NX: -86.19    HAl: -46.89   UNF: -70.75
N: 4095       SOB: -86.76   NX: 70.6      HAl: -65.85   UNF: -46.87
N: 8191       SOB: -57.35   NX: -10.99    HAl: -74.47   UNF: 159.81
N: 16383      SOB: -67.43   NX: -37.82    HAl: -27.43   UNF: 68.94
N: 32767      SOB: 13.87    NX: -16.39    HAl: -52.93   UNF: 46.6



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

N: 511        SOB: 0.78     NX: 11.8      HAl: -34.69   UNF: 24.26
N: 1023       SOB: 25.21    NX: 2.16      HAl: 91.83    UNF: 73.1
N: 2047       SOB: 21.23    NX: 9.11      HAl: 63.63    UNF: 37.45
N: 4095       SOB: -2.74    NX: -8.22     HAl: 35.24    UNF: -2.85
N: 8191       SOB: -14.19   NX: 4.87      HAl: -0.99    UNF: 0.51
N: 16383      SOB: -17.95   NX: -12.4     HAl: -25.89   UNF: -2.74
N: 32767      SOB: -7.54    NX: 0.04      HAl: -1.4     UNF: -5.33



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

N: 511        SOB: 224.06   NX: -70.64    HAl: 165.27   UNF: -60.44
N: 1023       SOB: 78.39    NX: 58.45     HAl: 108.5    UNF: -27.22
N: 2047       SOB: 77.74    NX: 53.09     HAl: 46.2     UNF: -65.02
N: 4095       SOB: 56.68    NX: 7.19      HAl: 32.69    UNF: -43.91
N: 8191       SOB: 55.34    NX: -13.08    HAl: 6.16     UNF: -21.99
N: 16383      SOB: -4.93    NX: -1.52     HAl: 2.98     UNF: -10.6
N: 32767      SOB: 19.67    NX: -4.49     HAl: -0.83    UNF: -15.79



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

N: 511        SOB: -19.17   NX: 76.66     HAl: 103.35   UNF: 58.05
N: 1023       SOB: -42.09   NX: 26.15     HAl: 40.46    UNF: -35.61
N: 2047       SOB: -53.7    NX: -33.85    HAl: 33.73    UNF: 9.6
N: 4095       SOB: -13.67   NX: -35.17    HAl: 19.51    UNF: 10.32
N: 8191       SOB: -12.44   NX: -24.17    HAl: 16.65    UNF: -2.85
N: 16383      SOB: -1.0     NX: 6.52      HAl: 14.74    UNF: 15.1
N: 32767      SOB: 6.08     NX: -0.56     HAl: -1.42    UNF: 13.85



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: 5677.79
N: 2047       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: 2786.07
N: 4095       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: 1342.33
N: 8191       SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: 620.99
N: 16383      SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: 260.45
N: 32767      SOB: 80.21    NX: -100.0    HAl: 80.21    UNF: 80.21



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

N: 511        SOB: 23056.47 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 11455.58 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 5672.14  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 2784.66  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 1341.98  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 620.9    NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 260.43   NX: -100.0    HAl: 80.21    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: 620.99   UNF: 620.99
N: 16383      SOB: 260.45   NX: -100.0    HAl: 260.45   UNF: 260.45
N: 32767      SOB: 260.43   NX: -100.0    HAl: 80.21    UNF: 80.21