qmcintdim11

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=11


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

N: 511        SOB: -2.52    NX: 1.19      HAl: -9.01    UNF: -6.85
N: 1023       SOB: -0.93    NX: 0.24      HAl: -5.31    UNF: -0.31
N: 2047       SOB: 0.07     NX: -0.58     HAl: -3.75    UNF: 0.0
N: 4095       SOB: 0.22     NX: -0.38     HAl: -1.92    UNF: 1.24
N: 8191       SOB: 0.13     NX: -0.17     HAl: -1.3     UNF: 3.01
N: 16383      SOB: 0.13     NX: -0.11     HAl: -0.75    UNF: 0.96
N: 32767      SOB: 0.06     NX: 0.0       HAl: -0.33    UNF: 1.24



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

N: 511        SOB: -2.12    NX: 2.6       HAl: -5.91    UNF: -2.83
N: 1023       SOB: -1.44    NX: 0.5       HAl: -4.32    UNF: -2.87
N: 2047       SOB: -0.3     NX: -1.11     HAl: -2.37    UNF: -2.94
N: 4095       SOB: -0.26    NX: -0.31     HAl: -1.33    UNF: -0.88
N: 8191       SOB: -0.13    NX: -0.07     HAl: -0.34    UNF: 0.29
N: 16383      SOB: -0.01    NX: 0.02      HAl: -0.03    UNF: -0.84
N: 32767      SOB: 0.03     NX: 0.0       HAl: -0.19    UNF: 0.27



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

N: 511        SOB: -3.96    NX: -1.44     HAl: -4.79    UNF: -5.0
N: 1023       SOB: -2.68    NX: -3.29     HAl: -0.39    UNF: -2.53
N: 2047       SOB: -0.98    NX: -0.17     HAl: 0.99     UNF: -1.9
N: 4095       SOB: -0.62    NX: 0.03      HAl: 0.19     UNF: 1.39
N: 8191       SOB: -0.49    NX: -0.3      HAl: 0.02     UNF: 1.61
N: 16383      SOB: -0.46    NX: -0.08     HAl: 0.07     UNF: 1.48
N: 32767      SOB: -0.34    NX: -0.28     HAl: 0.01     UNF: 0.77



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

N: 511        SOB: -69.22   NX: -47.1     HAl: -86.47   UNF: -67.0
N: 1023       SOB: -67.4    NX: -54.38    HAl: -83.96   UNF: -68.19
N: 2047       SOB: -41.24   NX: -50.16    HAl: -75.68   UNF: 46.8
N: 4095       SOB: 276.5    NX: -24.7     HAl: -70.08   UNF: 8.87
N: 8191       SOB: 117.23   NX: -5.58     HAl: -64.6    UNF: -0.78
N: 16383      SOB: 59.71    NX: 1.24      HAl: -56.11   UNF: -19.95
N: 32767      SOB: 21.11    NX: -10.87    HAl: -48.2    UNF: -27.17



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

N: 511        SOB: -59.35   NX: -54.36    HAl: -100.0   UNF: -64.99
N: 1023       SOB: -44.22   NX: -52.35    HAl: -100.0   UNF: -59.73
N: 2047       SOB: -51.86   NX: -55.77    HAl: -100.0   UNF: -57.98
N: 4095       SOB: -26.84   NX: -34.36    HAl: -100.0   UNF: -45.97
N: 8191       SOB: -37.17   NX: -27.46    HAl: -100.0   UNF: -18.87
N: 16383      SOB: -39.07   NX: -26.56    HAl: -100.0   UNF: 26.17
N: 32767      SOB: -27.4    NX: -26.35    HAl: -100.0   UNF: 2.7



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

N: 511        SOB: -72.71   NX: -47.48    HAl: -100.0   UNF: -63.4
N: 1023       SOB: -71.77   NX: -59.31    HAl: -100.0   UNF: 2.51
N: 2047       SOB: -58.38   NX: -8.12     HAl: -100.0   UNF: 58.47
N: 4095       SOB: -43.54   NX: -28.43    HAl: -100.0   UNF: -0.98
N: 8191       SOB: -36.5    NX: -10.48    HAl: -100.0   UNF: -12.15
N: 16383      SOB: -38.45   NX: 10.56     HAl: -100.0   UNF: -7.56
N: 32767      SOB: -32.41   NX: -9.67     HAl: -100.0   UNF: 16.93



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

N: 511        SOB: 739.68   NX: -9.89     HAl: -13.98   UNF: 35.17
N: 1023       SOB: 340.32   NX: -36.51    HAl: -18.08   UNF: 12.64
N: 2047       SOB: 209.25   NX: 22.88     HAl: -5.79    UNF: -3.74
N: 4095       SOB: 119.14   NX: 12.64     HAl: -7.84    UNF: -11.94
N: 8191       SOB: 67.94    NX: 6.5       HAl: -5.79    UNF: -7.84
N: 16383      SOB: 26.98    NX: 0.35      HAl: -1.7     UNF: -3.74
N: 32767      SOB: 18.78    NX: -3.74     HAl: 0.35     UNF: -1.7



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

N: 511        SOB: 1548.64  NX: -48.8     HAl: -26.27   UNF: -54.94
N: 1023       SOB: 762.21   NX: 12.64     HAl: -22.18   UNF: -26.27
N: 2047       SOB: 366.94   NX: 22.88     HAl: -18.08   UNF: -5.79
N: 4095       SOB: 176.48   NX: 14.69     HAl: -11.94   UNF: 4.45
N: 8191       SOB: 102.75   NX: 6.5       HAl: -5.79    UNF: -3.74
N: 16383      SOB: 49.5     NX: 6.5       HAl: -5.79    UNF: 2.4
N: 32767      SOB: 24.93    NX: 0.35      HAl: -3.74    UNF: 0.35



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

N: 511        SOB: 772.45   NX: -11.94    HAl: 43.36    UNF: -9.89
N: 1023       SOB: 368.99   NX: 0.35      HAl: 12.64    UNF: -7.84
N: 2047       SOB: 186.72   NX: -11.94    HAl: -3.74    UNF: -7.84
N: 4095       SOB: 88.42    NX: 6.5       HAl: 0.35     UNF: -16.03
N: 8191       SOB: 43.36    NX: 2.4       HAl: -1.7     UNF: -11.94
N: 16383      SOB: 24.93    NX: 2.4       HAl: -5.79    UNF: -13.98
N: 32767      SOB: 16.74    NX: -1.7      HAl: 2.4      UNF: -7.84



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

N: 511        SOB: -68.27   NX: -40.62    HAl: -73.14   UNF: 156.87
N: 1023       SOB: -48.57   NX: -50.02    HAl: -69.94   UNF: 123.28
N: 2047       SOB: -26.79   NX: -57.35    HAl: -50.35   UNF: 47.02
N: 4095       SOB: 100.01   NX: -30.61    HAl: -40.29   UNF: 11.67
N: 8191       SOB: 48.26    NX: 2.35      HAl: -35.31   UNF: -6.01
N: 16383      SOB: 39.46    NX: -7.26     HAl: -29.32   UNF: 3.27
N: 32767      SOB: 23.93    NX: -7.86     HAl: -12.16   UNF: 10.26



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

N: 511        SOB: -93.15   NX: -71.59    HAl: -93.54   UNF: -37.38
N: 1023       SOB: -52.6    NX: -76.41    HAl: -88.63   UNF: -59.72
N: 2047       SOB: -50.99   NX: -80.96    HAl: -83.12   UNF: -50.12
N: 4095       SOB: -64.65   NX: -71.27    HAl: -72.89   UNF: -47.04
N: 8191       SOB: -64.8    NX: -52.27    HAl: -44.86   UNF: -37.34
N: 16383      SOB: -42.96   NX: -34.79    HAl: -42.42   UNF: -17.96
N: 32767      SOB: -22.81   NX: -39.58    HAl: -39.9    UNF: -0.28



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

N: 511        SOB: -93.16   NX: -86.45    HAl: -93.06   UNF: 180.86
N: 1023       SOB: -94.48   NX: -92.52    HAl: -80.68   UNF: 44.17
N: 2047       SOB: -95.64   NX: 19.94     HAl: -64.3    UNF: 13.65
N: 4095       SOB: -92.36   NX: -30.91    HAl: -73.15   UNF: -41.06
N: 8191       SOB: -82.85   NX: -45.76    HAl: -77.61   UNF: -45.57
N: 16383      SOB: -88.78   NX: 10.15     HAl: -81.44   UNF: -60.16
N: 32767      SOB: -30.04   NX: -34.81    HAl: -76.67   UNF: -53.49



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

N: 511        SOB: -24.53   NX: -70.12    HAl: -1.65    UNF: 225.56
N: 1023       SOB: 26.2     NX: -84.76    HAl: 61.24    UNF: 33.59
N: 2047       SOB: 40.8     NX: -49.95    HAl: 37.84    UNF: 16.08
N: 4095       SOB: 7.43     NX: -39.37    HAl: 42.53    UNF: 32.48
N: 8191       SOB: 4.82     NX: -31.27    HAl: -2.68    UNF: 8.64
N: 16383      SOB: -10.97   NX: -14.18    HAl: -10.95   UNF: 17.69
N: 32767      SOB: 1.73     NX: 1.55      HAl: 0.06     UNF: 5.62



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

N: 511        SOB: 683.94   NX: -56.44    HAl: 16.89    UNF: 2.73
N: 1023       SOB: 308.84   NX: -113.22   HAl: 17.7     UNF: 16.97
N: 2047       SOB: 197.78   NX: 75.68     HAl: -1.76    UNF: -36.34
N: 4095       SOB: 139.21   NX: 49.35     HAl: -1.71    UNF: 11.89
N: 8191       SOB: 87.31    NX: 26.99     HAl: -2.52    UNF: 2.01
N: 16383      SOB: 24.43    NX: 7.49      HAl: 2.13     UNF: -11.18
N: 32767      SOB: 50.23    NX: -10.14    HAl: -5.5     UNF: -26.47



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

N: 511        SOB: -143.46  NX: 243.7     HAl: 158.69   UNF: -48.39
N: 1023       SOB: -145.49  NX: 271.86    HAl: 96.47    UNF: -103.29
N: 2047       SOB: -98.05   NX: 130.35    HAl: 48.42    UNF: -70.13
N: 4095       SOB: -78.87   NX: 81.11     HAl: 35.65    UNF: -34.42
N: 8191       SOB: -56.96   NX: 60.51     HAl: 6.77     UNF: -12.23
N: 16383      SOB: -38.32   NX: 33.46     HAl: 24.91    UNF: 1.88
N: 32767      SOB: -10.07   NX: 23.85     HAl: 2.19     UNF: -9.36



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: 981.35
N: 32767      SOB: 440.64   NX: 440.64    HAl: 440.64   UNF: 981.29



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

N: 511        SOB: 69369.41 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 34566.73 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 17216.42 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 8553.98  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 4225.93  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 2062.7   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 981.29   NX: -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: 981.35   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 440.64   NX: -100.0    HAl: -100.0   UNF: -100.0