qmcintdim7

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=7


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

N: 511        SOB: -0.41    NX: 1.03      HAl: -2.82    UNF: -1.73
N: 1023       SOB: -0.13    NX: 0.2       HAl: -1.72    UNF: -0.13
N: 2047       SOB: -0.13    NX: 0.27      HAl: -1.04    UNF: 2.26
N: 4095       SOB: -0.03    NX: 0.04      HAl: -0.43    UNF: 0.82
N: 8191       SOB: -0.02    NX: 0.02      HAl: -0.34    UNF: 1.75
N: 16383      SOB: -0.01    NX: 0.0       HAl: -0.17    UNF: 0.92
N: 32767      SOB: -0.01    NX: 0.0       HAl: -0.09    UNF: 0.45



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

N: 511        SOB: -0.79    NX: 1.12      HAl: -2.31    UNF: 2.41
N: 1023       SOB: -0.22    NX: 0.65      HAl: -0.75    UNF: -0.05
N: 2047       SOB: 0.01     NX: 0.55      HAl: -0.06    UNF: 0.3
N: 4095       SOB: -0.07    NX: 0.08      HAl: 0.01     UNF: -1.04
N: 8191       SOB: -0.03    NX: 0.04      HAl: 0.03     UNF: -0.56
N: 16383      SOB: -0.05    NX: -0.01     HAl: 0.03     UNF: -0.55
N: 32767      SOB: -0.04    NX: -0.02     HAl: -0.03    UNF: -0.68



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

N: 511        SOB: -2.16    NX: -0.48     HAl: -1.93    UNF: -0.46
N: 1023       SOB: -1.14    NX: -0.74     HAl: 0.0      UNF: 1.54
N: 2047       SOB: -0.47    NX: -0.08     HAl: -0.26    UNF: -0.63
N: 4095       SOB: 0.05     NX: 0.18      HAl: 0.14     UNF: 1.01
N: 8191       SOB: 0.04     NX: 0.1       HAl: 0.09     UNF: 1.85
N: 16383      SOB: -0.12    NX: 0.03      HAl: 0.15     UNF: 0.26
N: 32767      SOB: -0.12    NX: 0.0       HAl: -0.03    UNF: 0.49



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

N: 511        SOB: -39.25   NX: 32.58     HAl: -51.17   UNF: -8.61
N: 1023       SOB: -33.1    NX: 1.79      HAl: -48.06   UNF: -0.88
N: 2047       SOB: -19.87   NX: -10.33    HAl: -40.3    UNF: -0.93
N: 4095       SOB: -5.39    NX: -15.49    HAl: -33.21   UNF: -20.24
N: 8191       SOB: -4.59    NX: -12.24    HAl: -33.76   UNF: -11.61
N: 16383      SOB: -7.64    NX: -6.19     HAl: -23.16   UNF: -8.48
N: 32767      SOB: -6.14    NX: -6.3      HAl: -18.37   UNF: -6.56



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

N: 511        SOB: -41.64   NX: -6.91     HAl: -100.0   UNF: -30.81
N: 1023       SOB: -25.86   NX: 0.23      HAl: -100.0   UNF: -25.6
N: 2047       SOB: -4.19    NX: 7.34      HAl: -100.0   UNF: -19.1
N: 4095       SOB: -14.82   NX: -11.41    HAl: -100.0   UNF: -21.8
N: 8191       SOB: -17.2    NX: -5.96     HAl: -100.0   UNF: 15.92
N: 16383      SOB: -9.53    NX: -10.12    HAl: -100.0   UNF: -5.01
N: 32767      SOB: -5.28    NX: -3.78     HAl: -100.0   UNF: 3.15



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

N: 511        SOB: -47.88   NX: -13.97    HAl: -100.0   UNF: -9.44
N: 1023       SOB: -26.84   NX: -18.04    HAl: -100.0   UNF: -1.35
N: 2047       SOB: -29.39   NX: -7.42     HAl: -100.0   UNF: -21.4
N: 4095       SOB: -11.12   NX: -9.53     HAl: -100.0   UNF: 8.03
N: 8191       SOB: -10.51   NX: -11.92    HAl: -100.0   UNF: 25.96
N: 16383      SOB: -15.27   NX: -9.76     HAl: -100.0   UNF: 13.31
N: 32767      SOB: -9.93    NX: -12.24    HAl: -100.0   UNF: 0.77



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

N: 511        SOB: 40.29    NX: -4.38     HAl: 14.82    UNF: -1.95
N: 1023       SOB: 8.16     NX: -2.59     HAl: -4.13    UNF: 0.35
N: 2047       SOB: 7.65     NX: -6.43     HAl: 1.63     UNF: 2.27
N: 4095       SOB: 1.76     NX: -0.67     HAl: 0.99     UNF: 3.17
N: 8191       SOB: 2.91     NX: -0.8      HAl: 1.12     UNF: -4.77
N: 16383      SOB: 1.38     NX: -0.16     HAl: 0.35     UNF: -2.98
N: 32767      SOB: 0.1      NX: 0.1       HAl: 0.74     UNF: -1.57



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

N: 511        SOB: 85.09    NX: -5.92     HAl: -31.65   UNF: -3.36
N: 1023       SOB: 36.06    NX: -3.74     HAl: -13.6    UNF: -19.87
N: 2047       SOB: 18.02    NX: 1.25      HAl: -2.59    UNF: 1.89
N: 4095       SOB: 12.26    NX: 0.1       HAl: -1.18    UNF: 4.19
N: 8191       SOB: 5.86     NX: -0.67     HAl: 1.38     UNF: 3.04
N: 16383      SOB: 5.73     NX: -0.54     HAl: 1.76     UNF: -1.06
N: 32767      SOB: 2.4      NX: -0.03     HAl: 1.38     UNF: 1.25



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

N: 511        SOB: 99.94    NX: -21.02    HAl: 7.01     UNF: 8.42
N: 1023       SOB: 31.33    NX: -17.7     HAl: -2.98    UNF: -8.86
N: 2047       SOB: 15.97    NX: -3.74     HAl: 0.35     UNF: 5.86
N: 4095       SOB: 3.17     NX: 2.14      HAl: -1.95    UNF: 6.5
N: 8191       SOB: 0.74     NX: 1.76      HAl: -2.46    UNF: 7.78
N: 16383      SOB: 4.32     NX: 1.38      HAl: -1.31    UNF: 3.17
N: 32767      SOB: 3.3      NX: 0.35      HAl: -0.93    UNF: 2.78



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

N: 511        SOB: -15.9    NX: 20.39     HAl: 1.97     UNF: -41.91
N: 1023       SOB: -19.78   NX: 16.29     HAl: -5.46    UNF: -39.2
N: 2047       SOB: -15.73   NX: 4.54      HAl: -2.34    UNF: 13.61
N: 4095       SOB: 3.52     NX: 0.27      HAl: 3.35     UNF: 3.98
N: 8191       SOB: -0.12    NX: -1.32     HAl: -5.09    UNF: 11.63
N: 16383      SOB: -0.22    NX: -0.29     HAl: -0.7     UNF: -0.89
N: 32767      SOB: 0.26     NX: -0.3      HAl: -1.23    UNF: 0.71



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

N: 511        SOB: -22.35   NX: 13.55     HAl: -35.97   UNF: 12.17
N: 1023       SOB: -3.2     NX: 36.55     HAl: -19.74   UNF: 64.79
N: 2047       SOB: -5.17    NX: 64.14     HAl: 12.24    UNF: -0.63
N: 4095       SOB: -23.87   NX: 2.31      HAl: 16.57    UNF: 1.77
N: 8191       SOB: -15.3    NX: 1.63      HAl: 6.12     UNF: -20.07
N: 16383      SOB: -18.16   NX: -7.82     HAl: 2.45     UNF: -27.13
N: 32767      SOB: -14.81   NX: 0.36      HAl: 2.5      UNF: -6.32



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

N: 511        SOB: -29.21   NX: 5.92      HAl: -71.5    UNF: -16.97
N: 1023       SOB: 133.92   NX: -20.49    HAl: -38.78   UNF: -38.01
N: 2047       SOB: 33.49    NX: 6.02      HAl: -40.14   UNF: -36.99
N: 4095       SOB: -8.95    NX: -17.37    HAl: -31.36   UNF: -44.23
N: 8191       SOB: 15.17    NX: 18.76     HAl: 116.03   UNF: -20.58
N: 16383      SOB: -4.72    NX: -17.72    HAl: 35.04    UNF: -24.63
N: 32767      SOB: -8.58    NX: -9.57     HAl: -0.24    UNF: -4.5



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

N: 511        SOB: -13.39   NX: -7.82     HAl: 35.84    UNF: 8.08
N: 1023       SOB: 11.84    NX: -6.8      HAl: 18.07    UNF: -26.91
N: 2047       SOB: 7.62     NX: 6.34      HAl: -3.49    UNF: -33.28
N: 4095       SOB: 2.49     NX: 1.22      HAl: -0.99    UNF: -3.3
N: 8191       SOB: -7.13    NX: 0.25      HAl: -6.31    UNF: -4.67
N: 16383      SOB: -4.08    NX: -0.39     HAl: -2.94    UNF: 3.91
N: 32767      SOB: -2.79    NX: 0.05      HAl: -0.83    UNF: 1.42



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

N: 511        SOB: 51.32    NX: -44.63    HAl: 33.89    UNF: -26.57
N: 1023       SOB: 1.86     NX: -13.57    HAl: 5.92     UNF: 9.0
N: 2047       SOB: -2.57    NX: -20.78    HAl: 5.07     UNF: 24.86
N: 4095       SOB: -2.34    NX: -8.0      HAl: -3.5     UNF: -1.62
N: 8191       SOB: 5.8      NX: -6.51     HAl: 1.68     UNF: 8.1
N: 16383      SOB: 4.42     NX: -5.1      HAl: 2.28     UNF: 8.64
N: 32767      SOB: 1.99     NX: -4.84     HAl: 2.67     UNF: 2.89



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

N: 511        SOB: -10.78   NX: -25.19    HAl: 8.49     UNF: 26.72
N: 1023       SOB: -0.14    NX: -27.18    HAl: -9.49    UNF: -2.1
N: 2047       SOB: 9.47     NX: -10.86    HAl: -2.9     UNF: 5.81
N: 4095       SOB: 4.05     NX: -1.96     HAl: -3.34    UNF: 2.41
N: 8191       SOB: -1.69    NX: -2.06     HAl: -12.5    UNF: -4.67
N: 16383      SOB: 5.5      NX: 0.41      HAl: -2.47    UNF: -4.07
N: 32767      SOB: 5.68     NX: -2.08     HAl: -0.14    UNF: -8.97



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

N: 511        SOB: -100.0   NX: -100.0    HAl: 328.82   UNF: -100.0
N: 1023       SOB: -100.0   NX: -100.0    HAl: 113.99   UNF: 113.99
N: 2047       SOB: 6.89     NX: 6.89      HAl: 6.89     UNF: 113.78
N: 4095       SOB: 6.84     NX: 6.84      HAl: -46.58   UNF: 6.84
N: 8191       SOB: 6.81     NX: 6.81      HAl: -19.89   UNF: -46.59
N: 16383      SOB: -6.55    NX: 33.5      HAl: -19.9    UNF: -6.55
N: 32767      SOB: 13.47    NX: 26.82     HAl: -6.56    UNF: -13.23



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

N: 511        SOB: 757.65   NX: -100.0    HAl: 328.82   UNF: -100.0
N: 1023       SOB: 327.98   NX: -100.0    HAl: 113.99   UNF: -100.0
N: 2047       SOB: 327.57   NX: -100.0    HAl: 220.67   UNF: -100.0
N: 4095       SOB: 113.68   NX: 6.84      HAl: 113.68   UNF: -46.58
N: 8191       SOB: 6.81     NX: 6.81      HAl: 33.52    UNF: 6.81
N: 16383      SOB: 6.8      NX: 46.85     HAl: 6.8      UNF: 6.8
N: 32767      SOB: 20.14    NX: 40.17     HAl: 13.47    UNF: 46.84



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: 113.99    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 6.89     NX: 6.89      HAl: -100.0   UNF: -100.0
N: 4095       SOB: -46.58   NX: -46.58    HAl: -100.0   UNF: 6.84
N: 8191       SOB: -73.3    NX: -46.59    HAl: -73.3    UNF: -46.59
N: 16383      SOB: -6.55    NX: -33.25    HAl: -33.25   UNF: -59.95
N: 32767      SOB: 66.86    NX: -33.25    HAl: -26.58   UNF: -46.6