qmcintdim17

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=17


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

N: 511        SOB: -7.05    NX: -1.18     HAl: -25.88   UNF: 1.92
N: 1023       SOB: -1.51    NX: -0.43     HAl: -15.52   UNF: -0.18
N: 2047       SOB: -0.22    NX: 2.4       HAl: -10.19   UNF: 2.99
N: 4095       SOB: 0.72     NX: 3.43      HAl: -6.46    UNF: 5.95
N: 8191       SOB: 0.57     NX: 3.29      HAl: -3.22    UNF: 2.28
N: 16383      SOB: 0.13     NX: 2.82      HAl: -1.6     UNF: 1.71
N: 32767      SOB: 0.21     NX: 3.24      HAl: -0.97    UNF: 0.09



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

N: 511        SOB: -11.44   NX: -10.28    HAl: -17.03   UNF: 13.33
N: 1023       SOB: -9.4     NX: -2.22     HAl: -10.54   UNF: 3.17
N: 2047       SOB: -2.95    NX: -1.92     HAl: -5.52    UNF: 2.64
N: 4095       SOB: -0.47    NX: 0.04      HAl: -2.83    UNF: 3.86
N: 8191       SOB: 0.35     NX: 0.07      HAl: -1.03    UNF: 3.67
N: 16383      SOB: 0.36     NX: 0.33      HAl: -0.8     UNF: 0.97
N: 32767      SOB: 0.23     NX: 0.13      HAl: -0.77    UNF: 0.58



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

N: 511        SOB: -8.71    NX: -2.17     HAl: -14.41   UNF: -7.63
N: 1023       SOB: -6.68    NX: -0.67     HAl: -7.58    UNF: 3.41
N: 2047       SOB: -2.38    NX: -0.94     HAl: -1.99    UNF: 0.74
N: 4095       SOB: -0.96    NX: -1.39     HAl: -1.88    UNF: 1.99
N: 8191       SOB: -0.7     NX: 0.25      HAl: -0.43    UNF: 1.25
N: 16383      SOB: -1.2     NX: 0.07      HAl: 0.21     UNF: 1.32
N: 32767      SOB: -0.93    NX: 0.52      HAl: 0.38     UNF: 0.93



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

N: 511        SOB: -75.77   NX: -84.72    HAl: -98.57   UNF: -89.77
N: 1023       SOB: -80.39   NX: -83.29    HAl: -92.68   UNF: -67.12
N: 2047       SOB: -71.13   NX: -68.37    HAl: -90.52   UNF: -79.0
N: 4095       SOB: -36.93   NX: -58.18    HAl: -90.32   UNF: -68.28
N: 8191       SOB: -57.57   NX: -64.67    HAl: -81.51   UNF: -58.24
N: 16383      SOB: -34.48   NX: -26.01    HAl: -75.61   UNF: -65.08
N: 32767      SOB: -31.12   NX: 18.23     HAl: -68.62   UNF: -64.12



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

N: 511        SOB: -93.64   NX: -86.68    HAl: -100.0   UNF: 363.64
N: 1023       SOB: -79.76   NX: -63.86    HAl: -100.0   UNF: 168.74
N: 2047       SOB: -62.11   NX: -42.64    HAl: -100.0   UNF: 40.24
N: 4095       SOB: 144.73   NX: -58.84    HAl: -100.0   UNF: -23.11
N: 8191       SOB: 76.57    NX: -67.65    HAl: -100.0   UNF: -47.34
N: 16383      SOB: 0.58     NX: 148.84    HAl: -100.0   UNF: -57.03
N: 32767      SOB: -26.97   NX: 56.3      HAl: -100.0   UNF: -28.93



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

N: 511        SOB: -76.83   NX: -87.58    HAl: -100.0   UNF: -82.5
N: 1023       SOB: -77.55   NX: 169.4     HAl: -100.0   UNF: -87.16
N: 2047       SOB: -68.74   NX: 48.32     HAl: -100.0   UNF: -87.5
N: 4095       SOB: -62.2    NX: 3.92      HAl: -100.0   UNF: -82.98
N: 8191       SOB: -63.05   NX: -21.04    HAl: -100.0   UNF: -77.32
N: 16383      SOB: -59.32   NX: 51.54     HAl: -100.0   UNF: -55.91
N: 32767      SOB: -44.08   NX: -1.01     HAl: -100.0   UNF: 6.35



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

N: 511        SOB: 51280.22 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 25590.11 NX: -100.0    HAl: 31.07    UNF: 31.07
N: 2047       SOB: 13007.2  NX: 31.07     HAl: 31.07    UNF: 31.07
N: 4095       SOB: 6584.67  NX: 31.07     HAl: 31.07    UNF: 31.07
N: 8191       SOB: 3307.87  NX: 31.07     HAl: 31.07    UNF: 31.07
N: 16383      SOB: 1603.94  NX: 31.07     HAl: 31.07    UNF: 31.07
N: 32767      SOB: 817.5    NX: 31.07     HAl: 31.07    UNF: 31.07



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

N: 511        SOB: 102791.52NX: 686.43    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 51280.22 NX: 424.29    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 25590.11 NX: 162.14    HAl: 31.07    UNF: -100.0
N: 4095       SOB: 12745.06 NX: 31.07     HAl: 31.07    UNF: -100.0
N: 8191       SOB: 6322.53  NX: 31.07     HAl: 31.07    UNF: 31.07
N: 16383      SOB: 3176.8   NX: 31.07     HAl: 31.07    UNF: 31.07
N: 32767      SOB: 1603.94  NX: 31.07     HAl: 31.07    UNF: 31.07



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

N: 511        SOB: 51280.22 NX: -100.0    HAl: -100.0   UNF: 31.07
N: 1023       SOB: 25721.18 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 12745.06 NX: -100.0    HAl: -100.0   UNF: 31.07
N: 4095       SOB: 6453.6   NX: 31.07     HAl: 31.07    UNF: 31.07
N: 8191       SOB: 3176.8   NX: 31.07     HAl: 31.07    UNF: 31.07
N: 16383      SOB: 1603.94  NX: 31.07     HAl: 31.07    UNF: 31.07
N: 32767      SOB: 817.5    NX: 31.07     HAl: 31.07    UNF: 31.07



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

N: 511        SOB: -36.01   NX: -61.65    HAl: -97.75   UNF: 116.54
N: 1023       SOB: -61.02   NX: -70.55    HAl: 1.4      UNF: 15.02
N: 2047       SOB: -65.46   NX: -13.36    HAl: 27.74    UNF: 83.65
N: 4095       SOB: 38.25    NX: -39.4     HAl: -27.9    UNF: 12.22
N: 8191       SOB: 32.22    NX: -40.46    HAl: -44.3    UNF: -12.01
N: 16383      SOB: -8.32    NX: 3.13      HAl: -44.67   UNF: 82.13
N: 32767      SOB: -26.97   NX: 29.55     HAl: -19.06   UNF: 92.06



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

N: 511        SOB: -99.99   NX: -99.77    HAl: -98.98   UNF: -99.93
N: 1023       SOB: -99.99   NX: -37.96    HAl: -97.47   UNF: 548.95
N: 2047       SOB: -99.64   NX: -53.93    HAl: -83.94   UNF: 226.78
N: 4095       SOB: -99.53   NX: -75.18    HAl: -91.2    UNF: 65.22
N: 8191       SOB: -94.36   NX: -84.95    HAl: -86.78   UNF: -15.79
N: 16383      SOB: -79.11   NX: -70.17    HAl: -80.57   UNF: -54.39
N: 32767      SOB: -86.56   NX: -74.9     HAl: -84.69   UNF: -10.48



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

N: 511        SOB: -100.0   NX: -99.98    HAl: -100.0   UNF: -100.0
N: 1023       SOB: -100.0   NX: -97.91    HAl: -98.72   UNF: -99.99
N: 2047       SOB: -99.9    NX: -98.58    HAl: -99.34   UNF: -99.99
N: 4095       SOB: -99.79   NX: -99.27    HAl: -99.43   UNF: -99.76
N: 8191       SOB: -99.79   NX: -99.2     HAl: -99.6    UNF: -98.1
N: 16383      SOB: -99.41   NX: -23.44    HAl: -99.26   UNF: -98.36
N: 32767      SOB: -95.21   NX: -60.51    HAl: -49.05   UNF: -93.51



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

N: 511        SOB: 205.67   NX: -153.63   HAl: 196.5    UNF: -703.19
N: 1023       SOB: 76.12    NX: -271.69   HAl: 264.98   UNF: 120.32
N: 2047       SOB: 143.1    NX: -347.64   HAl: 29.99    UNF: 634.88
N: 4095       SOB: 156.56   NX: -179.68   HAl: 251.24   UNF: 124.28
N: 8191       SOB: 31.61    NX: 17.82     HAl: -49.83   UNF: -67.88
N: 16383      SOB: 84.26    NX: 68.7      HAl: 24.05    UNF: -242.66
N: 32767      SOB: 93.86    NX: -1.68     HAl: -112.58  UNF: -213.46



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

N: 511        SOB: 51268.58 NX: 490.48    HAl: 335.24   UNF: 78.74
N: 1023       SOB: 25329.93 NX: 95.78     HAl: 611.55   UNF: 602.28
N: 2047       SOB: 13808.27 NX: 383.96    HAl: 514.62   UNF: 426.64
N: 4095       SOB: 6704.74  NX: 12.93     HAl: 198.01   UNF: 232.6
N: 8191       SOB: 3346.5   NX: 157.11    HAl: 141.64   UNF: 65.33
N: 16383      SOB: 1614.27  NX: -1.88     HAl: -97.24   UNF: 37.82
N: 32767      SOB: 871.36   NX: -2.93     HAl: -95.31   UNF: 64.81



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

N: 511        SOB: 886.43   NX: -197.94   HAl: 213.67   UNF: -206.77
N: 1023       SOB: 540.5    NX: -315.09   HAl: 86.73    UNF: -166.82
N: 2047       SOB: 40.98    NX: -185.52   HAl: 143.75   UNF: -56.18
N: 4095       SOB: 18.83    NX: -372.13   HAl: 58.36    UNF: 7.14
N: 8191       SOB: -145.88  NX: -114.8    HAl: 48.6     UNF: 126.8
N: 16383      SOB: -99.97   NX: -47.93    HAl: 18.22    UNF: 318.26
N: 32767      SOB: -136.44  NX: -54.64    HAl: 0.44     UNF: 183.81



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: 5.064310118E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 2.527194755E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 1.262357185E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 6308652.47NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 3153505.93NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 1576510.46NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 788157.11NX: -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