qmcintdim18

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=18


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

N: 511        SOB: -8.62    NX: -2.98     HAl: -27.32   UNF: 3.07
N: 1023       SOB: -2.92    NX: 0.15      HAl: -20.11   UNF: -2.12
N: 2047       SOB: -1.07    NX: 1.94      HAl: -10.79   UNF: -1.15
N: 4095       SOB: 1.1      NX: 3.27      HAl: -5.83    UNF: 3.28
N: 8191       SOB: 0.71     NX: 3.26      HAl: -3.2     UNF: 0.37
N: 16383      SOB: 0.25     NX: 2.79      HAl: -1.88    UNF: 1.17
N: 32767      SOB: 0.26     NX: 3.05      HAl: -1.03    UNF: 0.54



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

N: 511        SOB: -13.2    NX: -10.81    HAl: -20.36   UNF: -4.23
N: 1023       SOB: -12.21   NX: -3.1      HAl: -11.33   UNF: 0.96
N: 2047       SOB: -3.51    NX: -2.27     HAl: -5.25    UNF: -1.19
N: 4095       SOB: -1.84    NX: 0.02      HAl: -2.95    UNF: -1.34
N: 8191       SOB: -0.9     NX: -0.06     HAl: -1.56    UNF: -0.28
N: 16383      SOB: 0.1      NX: 0.3       HAl: -0.97    UNF: -0.57
N: 32767      SOB: 0.47     NX: 0.2       HAl: -0.89    UNF: -1.04



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

N: 511        SOB: 1.77     NX: -2.71     HAl: -19.16   UNF: -2.4
N: 1023       SOB: 2.25     NX: -0.66     HAl: -8.89    UNF: -0.87
N: 2047       SOB: -0.06    NX: -1.5      HAl: -2.25    UNF: 4.31
N: 4095       SOB: -0.67    NX: -1.09     HAl: -2.42    UNF: -0.49
N: 8191       SOB: -0.23    NX: 0.21      HAl: -0.93    UNF: 0.07
N: 16383      SOB: 0.11     NX: 0.05      HAl: -0.09    UNF: 0.15
N: 32767      SOB: -0.71    NX: 0.51      HAl: 0.24     UNF: -0.2



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

N: 511        SOB: -91.88   NX: -86.48    HAl: -99.3    UNF: -67.89
N: 1023       SOB: -86.44   NX: -77.64    HAl: -97.81   UNF: -82.22
N: 2047       SOB: -86.38   NX: -77.97    HAl: -90.56   UNF: -84.45
N: 4095       SOB: 484.75   NX: -63.23    HAl: -89.97   UNF: -15.0
N: 8191       SOB: 204.53   NX: -62.16    HAl: -86.14   UNF: -49.94
N: 16383      SOB: 80.24    NX: -39.7     HAl: -75.96   UNF: -67.94
N: 32767      SOB: 17.74    NX: -31.27    HAl: -75.09   UNF: -66.31



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

N: 511        SOB: -75.24   NX: -88.17    HAl: -100.0   UNF: -96.65
N: 1023       SOB: -85.52   NX: -78.43    HAl: -100.0   UNF: 194.59
N: 2047       SOB: -88.73   NX: -61.54    HAl: -100.0   UNF: 90.46
N: 4095       SOB: -84.48   NX: -62.64    HAl: -100.0   UNF: 27.38
N: 8191       SOB: -77.97   NX: -71.98    HAl: -100.0   UNF: -21.27
N: 16383      SOB: -68.59   NX: 57.42     HAl: -100.0   UNF: -21.86
N: 32767      SOB: -29.91   NX: 9.41      HAl: -100.0   UNF: -43.06



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

N: 511        SOB: -93.1    NX: -94.43    HAl: -100.0   UNF: -37.52
N: 1023       SOB: -94.7    NX: 98.92     HAl: -100.0   UNF: -64.37
N: 2047       SOB: -90.73   NX: 9.57      HAl: -100.0   UNF: -75.64
N: 4095       SOB: -80.21   NX: -18.35    HAl: -100.0   UNF: -74.33
N: 8191       SOB: -69.03   NX: -29.72    HAl: -100.0   UNF: -49.2
N: 16383      SOB: -69.05   NX: -36.69    HAl: -100.0   UNF: 14.96
N: 32767      SOB: 13.51    NX: -47.53    HAl: -100.0   UNF: -21.61



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

N: 511        SOB: 102660.45NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 51280.22 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 26376.54 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 13269.34 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 6715.74  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 3307.87  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 1735.01  NX: -100.0    HAl: -100.0   UNF: -100.0



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

N: 511        SOB: 205420.9 NX: 1210.72   HAl: -100.0   UNF: -100.0
N: 1023       SOB: 102398.3 NX: 686.43    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 51280.22 NX: 424.29    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 25590.11 NX: 162.14    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 12745.06 NX: 162.14    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 6453.6   NX: 162.14    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 3307.87  NX: -100.0    HAl: -100.0   UNF: -100.0



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

N: 511        SOB: 102660.45NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 51280.22 NX: -100.0    HAl: -100.0   UNF: 424.29
N: 2047       SOB: 25590.11 NX: -100.0    HAl: -100.0   UNF: 162.14
N: 4095       SOB: 12745.06 NX: -100.0    HAl: -100.0   UNF: 162.14
N: 8191       SOB: 6453.6   NX: -100.0    HAl: 162.14   UNF: 162.14
N: 16383      SOB: 3307.87  NX: -100.0    HAl: -100.0   UNF: 162.14
N: 32767      SOB: 1472.86  NX: -100.0    HAl: -100.0   UNF: 162.14



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

N: 511        SOB: -77.97   NX: -50.57    HAl: -99.4    UNF: -29.95
N: 1023       SOB: -78.58   NX: -59.27    HAl: -94.74   UNF: -29.28
N: 2047       SOB: -79.49   NX: -61.76    HAl: 54.49    UNF: -28.78
N: 4095       SOB: -53.3    NX: -69.43    HAl: -13.34   UNF: -34.47
N: 8191       SOB: -30.76   NX: -60.32    HAl: -41.19   UNF: -38.91
N: 16383      SOB: -18.01   NX: -30.47    HAl: -31.61   UNF: -35.61
N: 32767      SOB: 70.07    NX: -43.13    HAl: -34.18   UNF: 6.36



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

N: 511        SOB: -99.74   NX: -99.91    HAl: -99.96   UNF: -54.23
N: 1023       SOB: -99.64   NX: -95.61    HAl: -96.44   UNF: -76.93
N: 2047       SOB: -99.46   NX: -87.47    HAl: -82.96   UNF: -88.44
N: 4095       SOB: -98.19   NX: -90.46    HAl: -90.5    UNF: -90.07
N: 8191       SOB: -37.47   NX: -93.96    HAl: -93.64   UNF: -83.34
N: 16383      SOB: -56.69   NX: -88.56    HAl: -94.43   UNF: -90.33
N: 32767      SOB: -71.25   NX: -74.05    HAl: -91.53   UNF: -75.52



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

N: 511        SOB: -99.97   NX: -100.0    HAl: -100.0   UNF: -99.97
N: 1023       SOB: -99.98   NX: -98.87    HAl: -97.45   UNF: -99.74
N: 2047       SOB: -99.79   NX: -98.46    HAl: -98.71   UNF: -99.86
N: 4095       SOB: -91.88   NX: -99.22    HAl: -99.28   UNF: -99.92
N: 8191       SOB: -91.06   NX: -98.91    HAl: -99.6    UNF: -83.33
N: 16383      SOB: -95.19   NX: -99.35    HAl: -99.17   UNF: 39.65
N: 32767      SOB: -97.03   NX: -97.92    HAl: -90.51   UNF: -29.05



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

N: 511        SOB: -819.66  NX: -586.86   HAl: 308.89   UNF: -401.42
N: 1023       SOB: -266.38  NX: -449.23   HAl: -35.08   UNF: -161.68
N: 2047       SOB: 182.72   NX: -235.76   HAl: -187.84  UNF: -99.57
N: 4095       SOB: -6.75    NX: -3.77     HAl: 109.11   UNF: 225.23
N: 8191       SOB: -144.5   NX: 113.94    HAl: -82.72   UNF: -14.45
N: 16383      SOB: -97.0    NX: 99.57     HAl: -1.81    UNF: -146.05
N: 32767      SOB: -56.81   NX: 10.72     HAl: -200.68  UNF: -69.76



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

N: 511        SOB: 102620.44NX: 234.76    HAl: -37.52   UNF: -1090.23
N: 1023       SOB: 50809.41 NX: 32.59     HAl: 279.47   UNF: -1037.25
N: 2047       SOB: 28763.25 NX: 180.14    HAl: 488.81   UNF: -297.87
N: 4095       SOB: 14274.6  NX: -191.6    HAl: 32.6     UNF: 73.16
N: 8191       SOB: 7385.33  NX: 176.23    HAl: 246.42   UNF: 354.01
N: 16383      SOB: 3720.12  NX: -103.06   HAl: -51.1    UNF: 43.95
N: 32767      SOB: 2033.52  NX: -39.5     HAl: -100.48  UNF: -9.4



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

N: 511        SOB: -41.21   NX: -321.14   HAl: -412.05  UNF: 52.02
N: 1023       SOB: -605.21  NX: -109.81   HAl: -119.2   UNF: 330.17
N: 2047       SOB: -174.93  NX: -85.32    HAl: 167.47   UNF: 59.29
N: 4095       SOB: -244.31  NX: -369.46   HAl: 31.49    UNF: -183.5
N: 8191       SOB: -189.39  NX: -194.67   HAl: 38.33    UNF: -60.35
N: 16383      SOB: -194.67  NX: 18.99     HAl: -87.71   UNF: 18.54
N: 32767      SOB: -104.24  NX: -55.38    HAl: -44.31   UNF: 91.79



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: -100.0
N: 32767      SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0



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

N: 511        SOB: 1.5192950353E8NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 7.581604266E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 3.787091554E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 1.89261574E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 9460717.8NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 4729731.39NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 2364671.34NX: -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: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: -100.0   NX: -100.0    HAl: -100.0   UNF: -100.0