qmcintdim8

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=8


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

N: 511        SOB: -0.54    NX: -0.32     HAl: -4.34    UNF: -5.07
N: 1023       SOB: -0.23    NX: 0.0       HAl: -2.68    UNF: 1.22
N: 2047       SOB: -0.1     NX: 0.07      HAl: -1.88    UNF: 2.38
N: 4095       SOB: -0.03    NX: 0.09      HAl: -0.83    UNF: 1.13
N: 8191       SOB: -0.01    NX: 0.02      HAl: -0.53    UNF: 1.77
N: 16383      SOB: -0.02    NX: 0.0       HAl: -0.25    UNF: 1.15
N: 32767      SOB: -0.01    NX: 0.0       HAl: -0.13    UNF: 0.56



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

N: 511        SOB: -2.01    NX: -0.56     HAl: -2.83    UNF: -2.91
N: 1023       SOB: -1.58    NX: -0.26     HAl: -1.86    UNF: -2.58
N: 2047       SOB: -0.12    NX: -0.24     HAl: -0.95    UNF: 0.35
N: 4095       SOB: -0.12    NX: -0.15     HAl: -0.5     UNF: 1.44
N: 8191       SOB: -0.07    NX: 0.04      HAl: -0.27    UNF: 0.33
N: 16383      SOB: -0.05    NX: 0.03      HAl: -0.18    UNF: -0.53
N: 32767      SOB: -0.03    NX: 0.01      HAl: -0.14    UNF: -0.5



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

N: 511        SOB: -3.33    NX: 0.11      HAl: -2.39    UNF: 1.12
N: 1023       SOB: -2.06    NX: -0.27     HAl: 0.68     UNF: -4.06
N: 2047       SOB: -0.83    NX: 0.02      HAl: 0.28     UNF: -4.11
N: 4095       SOB: -0.03    NX: -0.01     HAl: 0.31     UNF: -2.66
N: 8191       SOB: 0.08     NX: 0.01      HAl: 0.24     UNF: -0.48
N: 16383      SOB: -0.14    NX: -0.03     HAl: 0.26     UNF: 0.16
N: 32767      SOB: -0.15    NX: -0.03     HAl: -0.04    UNF: 0.72



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

N: 511        SOB: -44.99   NX: 218.47    HAl: -68.84   UNF: 23.96
N: 1023       SOB: -49.84   NX: 93.56     HAl: -64.73   UNF: -14.88
N: 2047       SOB: -29.95   NX: 35.4      HAl: -58.21   UNF: -29.27
N: 4095       SOB: 5.65     NX: 4.34      HAl: -48.42   UNF: -20.61
N: 8191       SOB: -4.94    NX: -0.39     HAl: -42.95   UNF: -25.19
N: 16383      SOB: -13.11   NX: -6.93     HAl: -30.52   UNF: -13.71
N: 32767      SOB: -11.41   NX: 14.26     HAl: -24.84   UNF: -14.26



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

N: 511        SOB: -58.2    NX: 27.95     HAl: -100.0   UNF: 54.93
N: 1023       SOB: -49.61   NX: -9.94     HAl: -100.0   UNF: 52.44
N: 2047       SOB: 15.72    NX: -25.47    HAl: -100.0   UNF: -0.92
N: 4095       SOB: -10.63   NX: -23.25    HAl: -100.0   UNF: -3.98
N: 8191       SOB: -11.18   NX: 4.54      HAl: -100.0   UNF: 3.76
N: 16383      SOB: 0.74     NX: -2.63     HAl: -100.0   UNF: -12.93
N: 32767      SOB: 5.08     NX: -3.23     HAl: -100.0   UNF: -2.57



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

N: 511        SOB: -56.74   NX: 209.26    HAl: -100.0   UNF: -42.79
N: 1023       SOB: -32.23   NX: 76.21     HAl: -100.0   UNF: -37.61
N: 2047       SOB: -34.02   NX: 26.32     HAl: -100.0   UNF: -39.82
N: 4095       SOB: -20.43   NX: 4.38      HAl: -100.0   UNF: -35.05
N: 8191       SOB: -13.94   NX: 11.35     HAl: -100.0   UNF: -3.5
N: 16383      SOB: -17.17   NX: -2.45     HAl: -100.0   UNF: -11.43
N: 32767      SOB: -18.93   NX: 0.47      HAl: -100.0   UNF: -17.42



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

N: 511        SOB: 75.87    NX: -14.5     HAl: 14.18    UNF: 5.47
N: 1023       SOB: 31.58    NX: -4.51     HAl: -1.7     UNF: -8.61
N: 2047       SOB: 19.04    NX: -4.0      HAl: 1.12     UNF: 1.89
N: 4095       SOB: 5.98     NX: -1.95     HAl: 6.5      UNF: 4.7
N: 8191       SOB: 5.47     NX: 0.86      HAl: 1.38     UNF: -0.16
N: 16383      SOB: -0.16    NX: 0.61      HAl: 0.1      UNF: -0.67
N: 32767      SOB: 0.61     NX: 1.38      HAl: 0.61     UNF: -0.16



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

N: 511        SOB: 193.89   NX: -18.85    HAl: -33.7    UNF: -3.74
N: 1023       SOB: 101.47   NX: -15.01    HAl: -23.2    UNF: -3.23
N: 2047       SOB: 45.41    NX: -3.49     HAl: -6.3     UNF: -10.91
N: 4095       SOB: 14.94    NX: -1.7      HAl: -1.7     UNF: -1.95
N: 8191       SOB: 7.52     NX: -2.46     HAl: 1.63     UNF: -3.23
N: 16383      SOB: 6.24     NX: -1.44     HAl: 2.91     UNF: -0.93
N: 32767      SOB: 2.66     NX: -0.42     HAl: 1.89     UNF: 0.1



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

N: 511        SOB: 140.9    NX: 24.42     HAl: 6.75     UNF: 21.6
N: 1023       SOB: 57.18    NX: 9.31      HAl: -4.77    UNF: 13.92
N: 2047       SOB: 39.01    NX: -1.95     HAl: 4.96     UNF: 0.35
N: 4095       SOB: 17.5     NX: 2.91      HAl: -0.16    UNF: -9.12
N: 8191       SOB: 8.29     NX: 0.35      HAl: 0.61     UNF: -2.21
N: 16383      SOB: 6.24     NX: 1.63      HAl: -0.42    UNF: 0.35
N: 32767      SOB: 5.22     NX: 0.86      HAl: -0.42    UNF: 2.4



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

N: 511        SOB: -17.46   NX: -10.21    HAl: -44.84   UNF: -23.12
N: 1023       SOB: -23.68   NX: -0.83     HAl: -39.19   UNF: -17.39
N: 2047       SOB: -16.81   NX: 1.65      HAl: -28.76   UNF: -25.71
N: 4095       SOB: 11.86    NX: -1.98     HAl: -12.0    UNF: -6.61
N: 8191       SOB: 5.31     NX: 3.59      HAl: -14.14   UNF: -10.21
N: 16383      SOB: -0.6     NX: 0.04      HAl: -4.3     UNF: -6.91
N: 32767      SOB: -0.18    NX: 1.33      HAl: -2.42    UNF: -6.8



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

N: 511        SOB: -65.29   NX: -80.82    HAl: -41.98   UNF: -62.51
N: 1023       SOB: -70.44   NX: -83.21    HAl: -11.43   UNF: -56.08
N: 2047       SOB: -36.46   NX: -4.04     HAl: -14.87   UNF: -24.21
N: 4095       SOB: -43.26   NX: -7.84     HAl: -14.61   UNF: -20.97
N: 8191       SOB: -41.86   NX: 22.34     HAl: -22.97   UNF: -22.26
N: 16383      SOB: -22.54   NX: 13.47     HAl: -20.7    UNF: 9.23
N: 32767      SOB: -21.14   NX: 11.64     HAl: -12.92   UNF: 4.29



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

N: 511        SOB: 10.37    NX: 266.02    HAl: -91.56   UNF: -75.43
N: 1023       SOB: -10.09   NX: 91.78     HAl: -25.52   UNF: -71.74
N: 2047       SOB: -44.32   NX: 16.37     HAl: -41.65   UNF: -75.92
N: 4095       SOB: -53.55   NX: 3.81      HAl: -44.45   UNF: -39.32
N: 8191       SOB: -17.52   NX: 31.22     HAl: 45.24    UNF: -52.31
N: 16383      SOB: -36.29   NX: -4.06     HAl: 2.08     UNF: -8.08
N: 32767      SOB: -37.55   NX: -19.22    HAl: -16.96   UNF: -26.17



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

N: 511        SOB: -24.52   NX: 15.69     HAl: 12.88    UNF: -3.25
N: 1023       SOB: 5.33     NX: -10.26    HAl: 21.22    UNF: -46.74
N: 2047       SOB: 6.61     NX: 2.83      HAl: -11.33   UNF: -38.11
N: 4095       SOB: -6.83    NX: -11.56    HAl: -5.19    UNF: -2.12
N: 8191       SOB: -14.39   NX: -0.8      HAl: -5.62    UNF: 11.6
N: 16383      SOB: -8.88    NX: -1.15     HAl: -8.02    UNF: 1.13
N: 32767      SOB: -2.24    NX: -1.4      HAl: -1.64    UNF: -7.14



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

N: 511        SOB: 40.17    NX: -49.79    HAl: 51.41    UNF: 25.03
N: 1023       SOB: 15.18    NX: -20.3     HAl: 23.76    UNF: 24.82
N: 2047       SOB: 8.05     NX: -10.04    HAl: 0.63     UNF: 2.37
N: 4095       SOB: -6.35    NX: -10.2     HAl: 7.67     UNF: 10.91
N: 8191       SOB: 11.41    NX: -4.82     HAl: -1.54    UNF: 3.59
N: 16383      SOB: -3.12    NX: 0.02      HAl: 3.43     UNF: -0.09
N: 32767      SOB: -0.33    NX: 4.93      HAl: -0.18    UNF: -3.02



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

N: 511        SOB: 16.7     NX: -38.67    HAl: 59.35    UNF: -44.46
N: 1023       SOB: -13.4    NX: -21.87    HAl: 16.5     UNF: 38.54
N: 2047       SOB: 2.93     NX: -4.76     HAl: -5.13    UNF: 7.6
N: 4095       SOB: -8.08    NX: 1.54      HAl: -0.77    UNF: -1.57
N: 8191       SOB: -6.31    NX: -6.46     HAl: -10.93   UNF: -4.29
N: 16383      SOB: 2.11     NX: -7.79     HAl: 1.31     UNF: -8.45
N: 32767      SOB: 4.22     NX: -0.33     HAl: -0.64    UNF: -11.5



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: -19.89   NX: -19.89    HAl: -100.0   UNF: -100.0
N: 16383      SOB: -59.95   NX: -19.9     HAl: -59.95   UNF: -59.95
N: 32767      SOB: 0.12     NX: -19.9     HAl: -19.9    UNF: 20.14



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

N: 511        SOB: 2472.94  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 1183.95  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 541.35   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 220.52   NX: 60.26     HAl: -100.0   UNF: -100.0
N: 8191       SOB: 60.22    NX: -19.89    HAl: -19.89   UNF: -100.0
N: 16383      SOB: 60.2     NX: -59.95    HAl: -59.95   UNF: 60.2
N: 32767      SOB: 20.14    NX: 20.14     HAl: 20.14    UNF: 0.12



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: 220.67   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 60.26    NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: -19.89   NX: -100.0    HAl: -19.89   UNF: 140.33
N: 16383      SOB: -19.9    NX: -19.9     HAl: -19.9    UNF: 140.3
N: 32767      SOB: 60.19    NX: 20.14     HAl: 0.12     UNF: 80.21