qmcintdim19

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=19


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

N: 511        SOB: -9.67    NX: 0.3       HAl: -29.74   UNF: 5.22
N: 1023       SOB: -3.96    NX: 1.76      HAl: -21.22   UNF: 4.55
N: 2047       SOB: -2.11    NX: 2.94      HAl: -12.85   UNF: 0.54
N: 4095       SOB: 0.72     NX: 3.72      HAl: -6.23    UNF: 6.14
N: 8191       SOB: 0.41     NX: 3.34      HAl: -3.83    UNF: 1.95
N: 16383      SOB: 0.25     NX: 3.34      HAl: -2.38    UNF: 2.53
N: 32767      SOB: 0.28     NX: 3.74      HAl: -1.27    UNF: 0.74



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

N: 511        SOB: -14.84   NX: -12.85    HAl: -22.55   UNF: -2.35
N: 1023       SOB: -7.19    NX: -2.58     HAl: -13.08   UNF: -4.48
N: 2047       SOB: -1.53    NX: -1.77     HAl: -5.63    UNF: -5.49
N: 4095       SOB: 1.38     NX: 0.2       HAl: -3.55    UNF: 1.43
N: 8191       SOB: 1.56     NX: 0.08      HAl: -2.02    UNF: -1.72
N: 16383      SOB: 1.8      NX: 0.4       HAl: -1.1     UNF: -0.98
N: 32767      SOB: 0.93     NX: 0.3       HAl: -1.04    UNF: -0.85



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

N: 511        SOB: -12.46   NX: 0.44      HAl: -18.86   UNF: -0.76
N: 1023       SOB: -9.3     NX: 0.14      HAl: -9.9     UNF: 0.9
N: 2047       SOB: -4.34    NX: -0.48     HAl: -2.78    UNF: 0.82
N: 4095       SOB: -2.0     NX: -0.54     HAl: -2.37    UNF: 1.21
N: 8191       SOB: -1.02    NX: 0.3       HAl: -0.84    UNF: -0.08
N: 16383      SOB: -0.74    NX: 0.19      HAl: -0.58    UNF: -1.21
N: 32767      SOB: 0.15     NX: 0.52      HAl: 0.02     UNF: -0.18



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

N: 511        SOB: -95.08   NX: -72.17    HAl: -99.28   UNF: -96.35
N: 1023       SOB: -88.54   NX: -72.83    HAl: -98.06   UNF: -94.31
N: 2047       SOB: 101.72   NX: -78.36    HAl: -96.65   UNF: -75.91
N: 4095       SOB: 16.34    NX: -66.92    HAl: -92.99   UNF: -80.75
N: 8191       SOB: -34.61   NX: -66.49    HAl: -85.49   UNF: -82.88
N: 16383      SOB: -26.4    NX: -41.42    HAl: -82.68   UNF: -76.7
N: 32767      SOB: 13.06    NX: -26.92    HAl: -82.13   UNF: -47.52



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

N: 511        SOB: -66.36   NX: -93.53    HAl: -100.0   UNF: -79.14
N: 1023       SOB: -81.13   NX: -74.4     HAl: -100.0   UNF: -86.96
N: 2047       SOB: -86.95   NX: -77.4     HAl: -100.0   UNF: -68.69
N: 4095       SOB: -88.62   NX: -77.3     HAl: -100.0   UNF: -55.25
N: 8191       SOB: -81.8    NX: -79.33    HAl: -100.0   UNF: -72.78
N: 16383      SOB: -72.26   NX: 46.41     HAl: -100.0   UNF: -52.45
N: 32767      SOB: -66.61   NX: 0.15      HAl: -100.0   UNF: -59.8



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

N: 511        SOB: -97.96   NX: -94.19    HAl: -100.0   UNF: -31.33
N: 1023       SOB: -96.01   NX: -25.43    HAl: -100.0   UNF: -34.96
N: 2047       SOB: -89.96   NX: -55.4     HAl: -100.0   UNF: 1527.32
N: 4095       SOB: -75.14   NX: -57.62    HAl: -100.0   UNF: 718.22
N: 8191       SOB: -80.78   NX: -62.37    HAl: -100.0   UNF: 325.65
N: 16383      SOB: -75.4    NX: -56.66    HAl: -100.0   UNF: 160.57
N: 32767      SOB: -66.27   NX: -61.73    HAl: -100.0   UNF: 65.32



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

N: 511        SOB: 205420.9 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 102660.45NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 51280.22 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 25590.11 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 13007.2  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 6191.46  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 3045.73  NX: -100.0    HAl: -100.0   UNF: -100.0



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

N: 511        SOB: 411466.08NX: 424.29    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 205420.9 NX: 424.29    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 102660.45NX: 424.29    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 51280.22 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 25590.11 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 13007.2  NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 6191.46  NX: -100.0    HAl: -100.0   UNF: -100.0



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

N: 511        SOB: 205420.9 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 102660.45NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 51280.22 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 25590.11 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 12482.91 NX: -100.0    HAl: -100.0   UNF: 424.29
N: 16383      SOB: 6191.46  NX: -100.0    HAl: -100.0   UNF: 424.29
N: 32767      SOB: 3045.73  NX: -100.0    HAl: -100.0   UNF: -100.0



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

N: 511        SOB: -97.6    NX: 10.94     HAl: -99.27   UNF: -96.34
N: 1023       SOB: -83.56   NX: -35.04    HAl: -93.24   UNF: -26.69
N: 2047       SOB: -83.67   NX: -52.38    HAl: -93.03   UNF: -54.1
N: 4095       SOB: -85.64   NX: -60.83    HAl: -84.57   UNF: -59.15
N: 8191       SOB: -45.73   NX: -63.15    HAl: -75.58   UNF: -56.39
N: 16383      SOB: -38.62   NX: -16.53    HAl: -57.6    UNF: -56.48
N: 32767      SOB: -50.6    NX: -30.59    HAl: -55.35   UNF: -47.37



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

N: 511        SOB: -99.97   NX: -99.98    HAl: -99.99   UNF: -99.97
N: 1023       SOB: -99.98   NX: -87.9     HAl: -99.89   UNF: -99.98
N: 2047       SOB: -98.31   NX: -93.71    HAl: -95.04   UNF: 17.28
N: 4095       SOB: -98.86   NX: -96.06    HAl: -96.5    UNF: -41.19
N: 8191       SOB: -99.1    NX: -96.71    HAl: -97.43   UNF: -70.42
N: 16383      SOB: -98.86   NX: -88.61    HAl: -94.89   UNF: 839.54
N: 32767      SOB: -97.35   NX: -46.07    HAl: -93.52   UNF: 385.13



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

N: 511        SOB: -100.0   NX: -99.99    HAl: -100.0   UNF: -99.86
N: 1023       SOB: -100.0   NX: -99.96    HAl: -100.0   UNF: -99.88
N: 2047       SOB: -99.99   NX: -99.19    HAl: -99.96   UNF: -99.94
N: 4095       SOB: -78.96   NX: -99.58    HAl: -99.74   UNF: -99.89
N: 8191       SOB: -89.47   NX: -99.65    HAl: -99.81   UNF: -99.93
N: 16383      SOB: -94.53   NX: -99.7     HAl: -99.87   UNF: -98.25
N: 32767      SOB: -96.98   NX: -97.17    HAl: -85.44   UNF: -98.66



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

N: 511        SOB: -717.11  NX: -1092.21  HAl: 789.07   UNF: -280.82
N: 1023       SOB: -270.95  NX: -772.86   HAl: 46.86    UNF: -452.07
N: 2047       SOB: 20.41    NX: -500.2    HAl: 156.09   UNF: -414.4
N: 4095       SOB: -107.71  NX: -8.81     HAl: 414.96   UNF: -422.34
N: 8191       SOB: 292.7    NX: 271.46    HAl: 144.41   UNF: -243.26
N: 16383      SOB: 126.57   NX: 94.99     HAl: 148.89   UNF: -99.61
N: 32767      SOB: 17.27    NX: 5.26      HAl: -105.84  UNF: -84.14



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

N: 511        SOB: 204968.91NX: -294.64   HAl: -119.75  UNF: -123.84
N: 1023       SOB: 101731.13NX: -378.65   HAl: 327.17   UNF: -192.74
N: 2047       SOB: 52052.72 NX: -523.16   HAl: 555.24   UNF: 231.25
N: 4095       SOB: 25208.15 NX: -640.02   HAl: 18.77    UNF: 134.98
N: 8191       SOB: 12698.1  NX: -486.8    HAl: 177.13   UNF: 161.69
N: 16383      SOB: 6117.93  NX: -500.6    HAl: -122.5   UNF: -12.38
N: 32767      SOB: 3147.07  NX: -242.38   HAl: -161.25  UNF: -120.06



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

N: 511        SOB: -446.71  NX: 365.42    HAl: -1168.66 UNF: 4373.85
N: 1023       SOB: -193.81  NX: -243.59   HAl: -152.03  UNF: 1556.39
N: 2047       SOB: -194.7   NX: -220.16   HAl: 454.57   UNF: 242.1
N: 4095       SOB: 79.67    NX: -344.3    HAl: 140.53   UNF: 72.41
N: 8191       SOB: -244.04  NX: -116.89   HAl: 211.69   UNF: -92.59
N: 16383      SOB: -92.62   NX: -160.56   HAl: -97.8    UNF: -19.29
N: 32767      SOB: -127.02  NX: -190.37   HAl: -71.12   UNF: 78.05



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: 4.5578871059E8NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 2.2744832798E8NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 1.1361294663E8NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 5.67786722E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 2.838235341E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 1.418939416E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 7094214.03NX: -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