qmcintdim6

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=6


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

N: 511        SOB: -0.25    NX: 0.71      HAl: -2.11    UNF: -0.02
N: 1023       SOB: -0.11    NX: 0.05      HAl: -1.19    UNF: -1.03
N: 2047       SOB: -0.11    NX: 0.17      HAl: -0.74    UNF: 0.9
N: 4095       SOB: -0.03    NX: 0.02      HAl: -0.36    UNF: 0.39
N: 8191       SOB: -0.02    NX: 0.03      HAl: -0.24    UNF: 1.19
N: 16383      SOB: -0.01    NX: 0.0       HAl: -0.12    UNF: 1.59
N: 32767      SOB: 0.0      NX: 0.0       HAl: -0.06    UNF: 0.62



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

N: 511        SOB: -0.67    NX: 1.04      HAl: -2.16    UNF: -6.66
N: 1023       SOB: -0.28    NX: 0.63      HAl: -0.91    UNF: -4.57
N: 2047       SOB: -0.14    NX: 0.51      HAl: -0.42    UNF: 0.57
N: 4095       SOB: -0.14    NX: 0.07      HAl: -0.18    UNF: 0.17
N: 8191       SOB: -0.05    NX: 0.01      HAl: -0.09    UNF: 0.49
N: 16383      SOB: -0.02    NX: -0.01     HAl: -0.03    UNF: 0.33
N: 32767      SOB: -0.01    NX: -0.02     HAl: -0.01    UNF: -0.49



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

N: 511        SOB: -1.04    NX: -0.58     HAl: -1.58    UNF: -1.41
N: 1023       SOB: -0.74    NX: -0.12     HAl: -0.41    UNF: 2.0
N: 2047       SOB: -0.46    NX: -0.17     HAl: -0.38    UNF: 2.12
N: 4095       SOB: 0.16     NX: -0.03     HAl: 0.03     UNF: 1.74
N: 8191       SOB: 0.02     NX: 0.03      HAl: -0.01    UNF: 1.77
N: 16383      SOB: -0.02    NX: -0.02     HAl: 0.01     UNF: 1.13
N: 32767      SOB: -0.03    NX: -0.07     HAl: 0.0      UNF: 0.48



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

N: 511        SOB: -29.18   NX: 1.23      HAl: -44.43   UNF: -17.31
N: 1023       SOB: -25.4    NX: -16.67    HAl: -44.72   UNF: -22.51
N: 2047       SOB: -9.35    NX: -11.53    HAl: -36.82   UNF: 33.11
N: 4095       SOB: -12.11   NX: -10.01    HAl: -30.89   UNF: 24.15
N: 8191       SOB: 5.95     NX: -10.28    HAl: -25.49   UNF: 13.47
N: 16383      SOB: 5.99     NX: -6.13     HAl: -17.57   UNF: 13.56
N: 32767      SOB: 1.2      NX: -4.57     HAl: -14.62   UNF: 14.68



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

N: 511        SOB: -34.56   NX: 18.52     HAl: -100.0   UNF: -29.78
N: 1023       SOB: -27.53   NX: 3.53      HAl: -100.0   UNF: -27.89
N: 2047       SOB: -18.27   NX: 10.74     HAl: -100.0   UNF: -32.43
N: 4095       SOB: -2.37    NX: -11.02    HAl: -100.0   UNF: -21.1
N: 8191       SOB: 2.07     NX: -9.22     HAl: -100.0   UNF: -22.91
N: 16383      SOB: 6.39     NX: -5.01     HAl: -100.0   UNF: -19.92
N: 32767      SOB: 3.27     NX: -6.07     HAl: -100.0   UNF: -9.27



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

N: 511        SOB: -30.19   NX: -15.24    HAl: -100.0   UNF: -5.7
N: 1023       SOB: -22.34   NX: -11.34    HAl: -100.0   UNF: -14.99
N: 2047       SOB: -23.59   NX: -10.86    HAl: -100.0   UNF: -23.69
N: 4095       SOB: 8.78     NX: -9.29     HAl: -100.0   UNF: -2.16
N: 8191       SOB: -0.69    NX: -9.74     HAl: -100.0   UNF: -13.5
N: 16383      SOB: -6.25    NX: 0.41      HAl: -100.0   UNF: -8.71
N: 32767      SOB: -7.82    NX: -6.11     HAl: -100.0   UNF: -6.53



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

N: 511        SOB: 21.92    NX: -5.41     HAl: 8.42     UNF: -9.44
N: 1023       SOB: 1.44     NX: -6.05     HAl: -0.48    UNF: -6.56
N: 2047       SOB: 1.89     NX: -5.54     HAl: 1.7      UNF: -0.54
N: 4095       SOB: 0.48     NX: -0.42     HAl: -0.54    UNF: 0.54
N: 8191       SOB: 1.25     NX: -0.29     HAl: 0.29     UNF: -4.38
N: 16383      SOB: 1.25     NX: -0.29     HAl: 0.1      UNF: -0.35
N: 32767      SOB: -0.1     NX: 0.1       HAl: 0.61     UNF: -0.54



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

N: 511        SOB: 51.1     NX: -0.93     HAl: -18.27   UNF: 11.49
N: 1023       SOB: 18.53    NX: 0.35      HAl: -5.15    UNF: -1.18
N: 2047       SOB: 6.94     NX: 0.99      HAl: 2.14     UNF: -7.97
N: 4095       SOB: 6.69     NX: 0.54      HAl: 0.22     UNF: -0.03
N: 8191       SOB: 3.81     NX: -0.22     HAl: 1.63     UNF: 3.74
N: 16383      SOB: 3.87     NX: -0.1      HAl: 2.14     UNF: 2.78
N: 32767      SOB: 0.03     NX: -0.03     HAl: 1.06     UNF: 3.55



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

N: 511        SOB: 67.36    NX: -21.66    HAl: 4.0      UNF: 0.29
N: 1023       SOB: 23.46    NX: -19.42    HAl: -0.67    UNF: -3.68
N: 2047       SOB: 14.69    NX: -7.78     HAl: 0.54     UNF: -2.72
N: 4095       SOB: 4.64     NX: 1.06      HAl: -2.78    UNF: 0.61
N: 8191       SOB: 3.1      NX: 0.86      HAl: -1.7     UNF: 1.82
N: 16383      SOB: 2.46     NX: 0.35      HAl: -0.48    UNF: 1.57
N: 32767      SOB: 1.95     NX: 0.61      HAl: -0.61    UNF: 0.54



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

N: 511        SOB: 0.25     NX: -2.45     HAl: -1.24    UNF: 32.53
N: 1023       SOB: -9.28    NX: -2.99     HAl: -15.59   UNF: -1.48
N: 2047       SOB: -9.87    NX: 2.96      HAl: -7.72    UNF: -0.51
N: 4095       SOB: -0.44    NX: -0.14     HAl: -3.87    UNF: -6.79
N: 8191       SOB: -0.36    NX: 0.54      HAl: -3.95    UNF: -6.07
N: 16383      SOB: 0.97     NX: 0.29      HAl: -1.84    UNF: -2.06
N: 32767      SOB: 0.95     NX: 0.02      HAl: -1.53    UNF: -0.79



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

N: 511        SOB: -15.51   NX: 33.91     HAl: -44.3    UNF: 60.36
N: 1023       SOB: -13.19   NX: 54.46     HAl: -31.94   UNF: 56.02
N: 2047       SOB: -14.94   NX: 72.63     HAl: -21.25   UNF: 16.46
N: 4095       SOB: -22.83   NX: 8.93      HAl: 7.57     UNF: 25.42
N: 8191       SOB: 10.06    NX: 7.61      HAl: -2.48    UNF: 7.2
N: 16383      SOB: 16.12    NX: -0.51     HAl: -6.53    UNF: 1.66
N: 32767      SOB: 6.63     NX: -0.37     HAl: -1.07    UNF: -5.16



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

N: 511        SOB: -27.16   NX: -34.18    HAl: -59.19   UNF: 57.94
N: 1023       SOB: 3.95     NX: -18.6     HAl: -48.98   UNF: 52.77
N: 2047       SOB: -20.65   NX: -14.68    HAl: -41.94   UNF: 50.91
N: 4095       SOB: -3.3     NX: -22.47    HAl: 4.5      UNF: 2.51
N: 8191       SOB: -13.74   NX: 4.38      HAl: 22.93    UNF: 9.54
N: 16383      SOB: -6.9     NX: -6.78     HAl: -3.24    UNF: 13.9
N: 32767      SOB: -9.81    NX: -8.55     HAl: -12.7    UNF: 15.54



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

N: 511        SOB: -26.94   NX: -13.04    HAl: -16.92   UNF: 28.17
N: 1023       SOB: 1.57     NX: -2.17     HAl: 5.44     UNF: 12.0
N: 2047       SOB: -0.4     NX: 4.36      HAl: 1.68     UNF: 1.79
N: 4095       SOB: 0.51     NX: 0.18      HAl: 4.54     UNF: -6.09
N: 8191       SOB: 1.29     NX: -0.25     HAl: 0.01     UNF: -5.81
N: 16383      SOB: 1.46     NX: -0.71     HAl: -0.19    UNF: -5.57
N: 32767      SOB: 1.46     NX: -0.45     HAl: 0.12     UNF: -2.36



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

N: 511        SOB: 46.19    NX: -32.55    HAl: 10.4     UNF: -39.49
N: 1023       SOB: 4.13     NX: -22.56    HAl: 7.54     UNF: -4.67
N: 2047       SOB: -0.35    NX: -21.99    HAl: 6.45     UNF: -12.83
N: 4095       SOB: -1.06    NX: -6.37     HAl: -1.94    UNF: -9.9
N: 8191       SOB: 1.73     NX: -6.87     HAl: 1.22     UNF: -9.52
N: 16383      SOB: 5.87     NX: -6.58     HAl: 2.71     UNF: -3.78
N: 32767      SOB: 1.41     NX: -5.85     HAl: 2.53     UNF: -2.42



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

N: 511        SOB: 5.56     NX: 0.95      HAl: -3.19    UNF: 19.28
N: 1023       SOB: 1.81     NX: 1.23      HAl: -1.05    UNF: 18.78
N: 2047       SOB: 4.72     NX: 3.61      HAl: 1.43     UNF: 12.34
N: 4095       SOB: -0.23    NX: -0.48     HAl: -2.83    UNF: 10.31
N: 8191       SOB: 0.23     NX: 0.19      HAl: -6.51    UNF: 4.89
N: 16383      SOB: 0.61     NX: 1.74      HAl: -1.2     UNF: 4.17
N: 32767      SOB: 1.72     NX: -0.59     HAl: 0.84     UNF: -3.84



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

N: 511        SOB: -100.0   NX: -100.0    HAl: 42.94    UNF: -100.0
N: 1023       SOB: -100.0   NX: -28.67    HAl: 113.99   UNF: -100.0
N: 2047       SOB: -28.74   NX: -28.74    HAl: 42.52    UNF: -28.74
N: 4095       SOB: -10.97   NX: -10.97    HAl: 60.26    UNF: -28.77
N: 8191       SOB: 15.71    NX: -2.09     HAl: 33.52    UNF: -10.99
N: 16383      SOB: 11.25    NX: 2.35      HAl: 15.7     UNF: -19.9
N: 32767      SOB: 11.24    NX: -6.56     HAl: 2.34     UNF: 2.34



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

N: 511        SOB: 185.88   NX: -100.0    HAl: 185.88   UNF: 42.94
N: 1023       SOB: 42.66    NX: -100.0    HAl: 42.66    UNF: 42.66
N: 2047       SOB: 42.52    NX: -28.74    HAl: 78.15    UNF: 6.89
N: 4095       SOB: 6.84     NX: 42.45     HAl: 60.26    UNF: 24.65
N: 8191       SOB: -10.99   NX: 42.42     HAl: 6.81     UNF: -2.09
N: 16383      SOB: 15.7     NX: 37.95     HAl: 6.8      UNF: -6.55
N: 32767      SOB: 20.14    NX: 46.84     HAl: 15.69    UNF: 11.24



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: -28.67   NX: 42.66     HAl: -100.0   UNF: 42.66
N: 2047       SOB: 42.52    NX: -28.74    HAl: -100.0   UNF: -28.74
N: 4095       SOB: 24.65    NX: -46.58    HAl: -100.0   UNF: 6.84
N: 8191       SOB: -2.09    NX: -28.79    HAl: -73.3    UNF: 33.52
N: 16383      SOB: -11.0    NX: -19.9     HAl: -33.25   UNF: 29.05
N: 32767      SOB: 22.37    NX: -8.78     HAl: -15.45   UNF: 15.69