qmcintdim14

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=14


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

N: 511        SOB: -5.8     NX: 3.24      HAl: -17.14   UNF: 6.14
N: 1023       SOB: -1.59    NX: 2.5       HAl: -9.46    UNF: 5.53
N: 2047       SOB: -0.9     NX: 2.81      HAl: -6.45    UNF: 0.98
N: 4095       SOB: 0.48     NX: 1.68      HAl: -3.75    UNF: 4.32
N: 8191       SOB: 0.35     NX: 0.26      HAl: -2.23    UNF: 2.27
N: 16383      SOB: 0.17     NX: 0.23      HAl: -1.11    UNF: 0.68
N: 32767      SOB: 0.19     NX: 0.16      HAl: -0.68    UNF: 0.16



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

N: 511        SOB: -7.74    NX: -3.37     HAl: -11.23   UNF: -2.54
N: 1023       SOB: -6.66    NX: -2.11     HAl: -7.51    UNF: -4.03
N: 2047       SOB: -1.97    NX: 0.25      HAl: -4.04    UNF: -2.43
N: 4095       SOB: -0.49    NX: -0.17     HAl: -2.4     UNF: -2.62
N: 8191       SOB: 0.17     NX: 0.4       HAl: -0.79    UNF: -0.52
N: 16383      SOB: 0.0      NX: 0.18      HAl: -0.43    UNF: 0.93
N: 32767      SOB: 0.0      NX: -0.02     HAl: -0.5     UNF: 0.17



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

N: 511        SOB: -5.88    NX: 1.4       HAl: -9.3     UNF: 2.71
N: 1023       SOB: -4.76    NX: 2.08      HAl: -5.15    UNF: 1.45
N: 2047       SOB: -1.67    NX: -1.96     HAl: -0.79    UNF: -0.91
N: 4095       SOB: -1.26    NX: 0.09      HAl: -0.93    UNF: 1.62
N: 8191       SOB: -0.81    NX: -0.49     HAl: -0.45    UNF: 1.67
N: 16383      SOB: -0.81    NX: -0.1      HAl: -0.06    UNF: 0.53
N: 32767      SOB: -0.54    NX: 0.06      HAl: 0.15     UNF: 0.48



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

N: 511        SOB: -78.17   NX: -80.08    HAl: -95.0    UNF: -44.59
N: 1023       SOB: -63.51   NX: -68.74    HAl: -89.29   UNF: -57.8
N: 2047       SOB: -49.66   NX: 32.05     HAl: -87.15   UNF: -64.64
N: 4095       SOB: 118.72   NX: 62.3      HAl: -77.85   UNF: -24.61
N: 8191       SOB: 30.41    NX: -5.2      HAl: -64.95   UNF: -44.78
N: 16383      SOB: 8.8      NX: 3.15      HAl: -63.25   UNF: -16.9
N: 32767      SOB: 4.65     NX: -12.95    HAl: -60.88   UNF: -32.3



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

N: 511        SOB: -89.45   NX: -85.7     HAl: -100.0   UNF: -59.0
N: 1023       SOB: -90.73   NX: -70.39    HAl: -100.0   UNF: -50.33
N: 2047       SOB: -77.13   NX: -67.93    HAl: -100.0   UNF: 12.78
N: 4095       SOB: -53.51   NX: -51.96    HAl: -100.0   UNF: -12.71
N: 8191       SOB: -54.11   NX: 3.97      HAl: -100.0   UNF: -29.54
N: 16383      SOB: -54.51   NX: -15.55    HAl: -100.0   UNF: -41.33
N: 32767      SOB: -52.07   NX: -21.06    HAl: -100.0   UNF: -38.04



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

N: 511        SOB: -86.59   NX: -85.11    HAl: -100.0   UNF: -49.52
N: 1023       SOB: -86.79   NX: -70.75    HAl: -100.0   UNF: -44.02
N: 2047       SOB: -73.2    NX: -49.91    HAl: -100.0   UNF: -38.41
N: 4095       SOB: -66.62   NX: 92.06     HAl: -100.0   UNF: -38.89
N: 8191       SOB: -36.61   NX: 28.7      HAl: -100.0   UNF: -22.83
N: 16383      SOB: -49.86   NX: -14.44    HAl: -100.0   UNF: -42.84
N: 32767      SOB: -22.09   NX: -20.96    HAl: -100.0   UNF: -37.87



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

N: 511        SOB: 6338.91  NX: 47.46     HAl: -50.85   UNF: -50.85
N: 1023       SOB: 3127.65  NX: 80.22     HAl: -50.85   UNF: -34.46
N: 2047       SOB: 1800.54  NX: 31.07     HAl: -34.46   UNF: -34.46
N: 4095       SOB: 915.81   NX: 31.07     HAl: -18.08   UNF: -34.46
N: 8191       SOB: 506.21   NX: -1.7      HAl: 14.69    UNF: -18.08
N: 16383      SOB: 244.06   NX: 14.69     HAl: 14.69    UNF: -1.7
N: 32767      SOB: 112.99   NX: -1.7      HAl: -1.7     UNF: -1.7



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

N: 511        SOB: 12761.44 NX: -50.85    HAl: -67.23   UNF: 47.46
N: 1023       SOB: 6338.91  NX: -34.46    HAl: -67.23   UNF: -1.7
N: 2047       SOB: 3160.42  NX: -18.08    HAl: -34.46   UNF: -18.08
N: 4095       SOB: 1571.17  NX: -18.08    HAl: -34.46   UNF: 31.07
N: 8191       SOB: 801.12   NX: -18.08    HAl: -18.08   UNF: 31.07
N: 16383      SOB: 391.52   NX: -18.08    HAl: -18.08   UNF: -1.7
N: 32767      SOB: 194.91   NX: -18.08    HAl: -1.7     UNF: -1.7



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

N: 511        SOB: 6388.06  NX: -50.85    HAl: -50.85   UNF: -34.46
N: 1023       SOB: 3176.8   NX: -34.46    HAl: -34.46   UNF: 80.22
N: 2047       SOB: 1718.62  NX: -50.85    HAl: -50.85   UNF: 14.69
N: 4095       SOB: 850.27   NX: -1.7      HAl: -1.7     UNF: 31.07
N: 8191       SOB: 424.29   NX: 14.69     HAl: -1.7     UNF: 14.69
N: 16383      SOB: 211.3    NX: 14.69     HAl: -18.08   UNF: -1.7
N: 32767      SOB: 112.99   NX: -1.7      HAl: -1.7     UNF: -1.7



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

N: 511        SOB: -77.33   NX: -42.33    HAl: -91.26   UNF: 189.71
N: 1023       SOB: -22.22   NX: -45.35    HAl: -50.6    UNF: 49.35
N: 2047       SOB: -36.24   NX: -37.06    HAl: -44.59   UNF: 8.76
N: 4095       SOB: 2.32     NX: -3.34     HAl: -22.97   UNF: -3.7
N: 8191       SOB: -6.18    NX: -28.98    HAl: -3.95    UNF: 22.5
N: 16383      SOB: -11.22   NX: 23.44     HAl: -16.6    UNF: 13.05
N: 32767      SOB: 26.25    NX: 16.24     HAl: -2.05    UNF: 0.37



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

N: 511        SOB: -99.32   NX: -95.39    HAl: -97.49   UNF: -82.61
N: 1023       SOB: -99.59   NX: -85.78    HAl: -96.87   UNF: -79.53
N: 2047       SOB: -82.38   NX: -79.48    HAl: -91.11   UNF: -84.48
N: 4095       SOB: -88.41   NX: -80.42    HAl: -86.83   UNF: -48.99
N: 8191       SOB: -87.41   NX: -47.57    HAl: -82.98   UNF: 794.43
N: 16383      SOB: -3.84    NX: 113.62    HAl: -56.55   UNF: 352.33
N: 32767      SOB: -38.75   NX: 56.76     HAl: -57.69   UNF: 141.22



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

N: 511        SOB: -99.95   NX: -99.46    HAl: -99.98   UNF: -99.93
N: 1023       SOB: -99.8    NX: -99.63    HAl: -99.25   UNF: -99.96
N: 2047       SOB: -97.99   NX: -99.41    HAl: -97.69   UNF: -99.91
N: 4095       SOB: -97.74   NX: 1333.31   HAl: -97.87   UNF: -83.22
N: 8191       SOB: -97.79   NX: 622.65    HAl: -96.88   UNF: -91.36
N: 16383      SOB: -97.32   NX: 262.39    HAl: -97.57   UNF: -68.04
N: 32767      SOB: 1564.21  NX: 87.61     HAl: 11.5     UNF: -78.14



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

N: 511        SOB: -110.33  NX: 448.62    HAl: 42.07    UNF: 23.31
N: 1023       SOB: -66.57   NX: 441.32    HAl: 62.47    UNF: 144.1
N: 2047       SOB: -65.99   NX: 202.68    HAl: -38.38   UNF: 42.76
N: 4095       SOB: 34.33    NX: 135.84    HAl: 6.16     UNF: 31.59
N: 8191       SOB: -2.12    NX: 130.15    HAl: -98.51   UNF: 43.89
N: 16383      SOB: 60.85    NX: 68.49     HAl: -7.11    UNF: 5.63
N: 32767      SOB: 25.35    NX: 20.83     HAl: -30.95   UNF: -17.05



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

N: 511        SOB: 6377.29  NX: -85.07    HAl: -164.92  UNF: 195.14
N: 1023       SOB: 3020.23  NX: 163.42    HAl: -3.73    UNF: 7.78
N: 2047       SOB: 2120.06  NX: -31.78    HAl: 30.67    UNF: 10.32
N: 4095       SOB: 960.31   NX: 30.26     HAl: -105.72  UNF: 46.4
N: 8191       SOB: 546.37   NX: -55.55    HAl: 55.93    UNF: 17.18
N: 16383      SOB: 272.02   NX: 31.59     HAl: 53.18    UNF: 31.6
N: 32767      SOB: 183.58   NX: 11.92     HAl: 33.08    UNF: -31.61



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

N: 511        SOB: -117.09  NX: -262.31   HAl: 196.27   UNF: 152.6
N: 1023       SOB: -250.99  NX: 22.93     HAl: 22.72    UNF: -10.43
N: 2047       SOB: -146.06  NX: 29.88     HAl: 56.0     UNF: -80.31
N: 4095       SOB: -133.95  NX: 22.09     HAl: 39.79    UNF: 7.76
N: 8191       SOB: -71.26   NX: 57.85     HAl: 52.21    UNF: -10.53
N: 16383      SOB: -52.22   NX: 3.92      HAl: 33.02    UNF: -47.35
N: 32767      SOB: -54.33   NX: -16.95    HAl: 4.55     UNF: 15.38



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: 1875574.12NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 935901.76NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 467443.4 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 233557.5 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 116700.22NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 58292.98 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 29094.71 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: -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