qmcintdim4

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=4


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

N: 511        SOB: -0.16    NX: 0.03      HAl: -0.87    UNF: 0.11
N: 1023       SOB: -0.04    NX: 0.14      HAl: -0.46    UNF: -1.51
N: 2047       SOB: -0.03    NX: 0.02      HAl: -0.3     UNF: 0.48
N: 4095       SOB: -0.01    NX: 0.0       HAl: -0.16    UNF: 0.78
N: 8191       SOB: -0.01    NX: 0.0       HAl: -0.09    UNF: 0.21
N: 16383      SOB: 0.0      NX: 0.01      HAl: -0.04    UNF: 0.83
N: 32767      SOB: 0.0      NX: 0.0       HAl: -0.02    UNF: 0.66



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

N: 511        SOB: -0.29    NX: 0.08      HAl: -1.05    UNF: 1.23
N: 1023       SOB: -0.14    NX: 0.04      HAl: -0.29    UNF: -0.6
N: 2047       SOB: -0.08    NX: 0.07      HAl: -0.25    UNF: -0.53
N: 4095       SOB: -0.04    NX: 0.02      HAl: -0.16    UNF: 0.03
N: 8191       SOB: -0.02    NX: 0.02      HAl: -0.01    UNF: -0.2
N: 16383      SOB: -0.01    NX: 0.01      HAl: -0.02    UNF: -0.35
N: 32767      SOB: 0.0      NX: 0.0       HAl: 0.0      UNF: -0.18



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

N: 511        SOB: -0.43    NX: 1.16      HAl: -0.23    UNF: -6.3
N: 1023       SOB: -0.58    NX: 0.58      HAl: -0.14    UNF: -1.13
N: 2047       SOB: -0.24    NX: -0.19     HAl: -0.3     UNF: 1.76
N: 4095       SOB: 0.05     NX: -0.02     HAl: 0.06     UNF: 0.65
N: 8191       SOB: 0.04     NX: -0.02     HAl: -0.02    UNF: 0.47
N: 16383      SOB: 0.01     NX: 0.01      HAl: -0.03    UNF: 0.46
N: 32767      SOB: 0.02     NX: 0.01      HAl: 0.01     UNF: 0.62



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

N: 511        SOB: -15.4    NX: -5.99     HAl: -27.85   UNF: -19.17
N: 1023       SOB: -10.59   NX: -5.28     HAl: -22.42   UNF: -28.61
N: 2047       SOB: 21.15    NX: 4.36      HAl: -18.2    UNF: -22.67
N: 4095       SOB: 5.96     NX: 3.44      HAl: -13.33   UNF: -6.22
N: 8191       SOB: 1.32     NX: 0.74      HAl: -9.63    UNF: -6.83
N: 16383      SOB: -0.59    NX: 0.71      HAl: -7.14    UNF: -2.1
N: 32767      SOB: -0.32    NX: -0.17     HAl: -5.0     UNF: 0.43



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

N: 511        SOB: -4.48    NX: -19.6     HAl: -100.0   UNF: -23.83
N: 1023       SOB: -12.37   NX: -17.73    HAl: -100.0   UNF: -18.78
N: 2047       SOB: 2.25     NX: 19.2      HAl: -100.0   UNF: 78.07
N: 4095       SOB: 4.08     NX: 8.44      HAl: -100.0   UNF: 46.54
N: 8191       SOB: 2.99     NX: 2.83      HAl: -100.0   UNF: 27.4
N: 16383      SOB: -2.24    NX: 4.23      HAl: -100.0   UNF: 24.03
N: 32767      SOB: -2.42    NX: 0.4       HAl: -100.0   UNF: 12.81



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

N: 511        SOB: -20.48   NX: 1.92      HAl: -100.0   UNF: -19.82
N: 1023       SOB: -1.28    NX: -5.25     HAl: -100.0   UNF: -8.76
N: 2047       SOB: 7.3      NX: -12.98    HAl: -100.0   UNF: 7.71
N: 4095       SOB: 7.22     NX: -6.24     HAl: -100.0   UNF: 0.63
N: 8191       SOB: 5.23     NX: -6.88     HAl: -100.0   UNF: 3.84
N: 16383      SOB: -1.08    NX: -1.98     HAl: -100.0   UNF: -0.59
N: 32767      SOB: 0.61     NX: -2.15     HAl: -100.0   UNF: -1.2



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

N: 511        SOB: 4.83     NX: -0.13     HAl: 0.21     UNF: -1.71
N: 1023       SOB: 1.92     NX: 1.06      HAl: 0.88     UNF: -4.27
N: 2047       SOB: 1.98     NX: 1.01      HAl: 0.48     UNF: -4.3
N: 4095       SOB: 1.25     NX: 0.18      HAl: -0.22    UNF: -6.51
N: 8191       SOB: 1.31     NX: 0.02      HAl: -0.06    UNF: -1.87
N: 16383      SOB: 1.36     NX: -0.03     HAl: -0.22    UNF: -1.38
N: 32767      SOB: 0.0      NX: -0.05     HAl: -0.13    UNF: -1.26



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

N: 511        SOB: 6.3      NX: 2.5       HAl: 1.52     UNF: 3.47
N: 1023       SOB: 6.69     NX: 0.77      HAl: 0.62     UNF: 2.32
N: 2047       SOB: 0.9      NX: -0.83     HAl: 1.6      UNF: 2.85
N: 4095       SOB: -0.51    NX: 0.02      HAl: 1.23     UNF: -1.98
N: 8191       SOB: -0.21    NX: 0.03      HAl: 1.38     UNF: -0.83
N: 16383      SOB: -0.21    NX: 0.02      HAl: 1.02     UNF: -1.01
N: 32767      SOB: 0.0      NX: 0.0       HAl: 0.18     UNF: 0.46



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

N: 511        SOB: 24.74    NX: -7.28     HAl: -5.26    UNF: -3.02
N: 1023       SOB: 3.1      NX: -3.71     HAl: -0.42    UNF: -3.82
N: 2047       SOB: 2.54     NX: -3.55     HAl: -0.54    UNF: -2.08
N: 4095       SOB: 0.21     NX: -4.24     HAl: -0.62    UNF: 0.94
N: 8191       SOB: 0.72     NX: -0.43     HAl: -0.35    UNF: 0.29
N: 16383      SOB: 0.32     NX: -0.43     HAl: 0.05     UNF: 3.14
N: 32767      SOB: 0.0      NX: -0.32     HAl: -0.05    UNF: 3.04



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

N: 511        SOB: -0.89    NX: -0.18     HAl: -4.22    UNF: 8.38
N: 1023       SOB: -0.95    NX: -0.11     HAl: -3.8     UNF: 0.89
N: 2047       SOB: 0.12     NX: 0.04      HAl: -3.06    UNF: -5.94
N: 4095       SOB: -0.18    NX: 0.01      HAl: -0.93    UNF: -7.67
N: 8191       SOB: -0.13    NX: -0.01     HAl: -0.52    UNF: -0.38
N: 16383      SOB: -0.06    NX: 0.02      HAl: -0.29    UNF: -0.23
N: 32767      SOB: -0.03    NX: -0.01     HAl: -0.11    UNF: 0.31



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

N: 511        SOB: -0.98    NX: -17.29    HAl: -14.68   UNF: -0.08
N: 1023       SOB: -8.5     NX: -15.03    HAl: -2.22    UNF: 12.31
N: 2047       SOB: -4.64    NX: 3.85      HAl: -12.83   UNF: -1.69
N: 4095       SOB: 1.61     NX: 3.49      HAl: -2.2     UNF: 5.98
N: 8191       SOB: 0.85     NX: 2.36      HAl: -1.3     UNF: 6.46
N: 16383      SOB: -0.07    NX: 2.36      HAl: -0.86    UNF: 6.73
N: 32767      SOB: 0.0      NX: 0.31      HAl: -0.71    UNF: 1.23



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

N: 511        SOB: -21.11   NX: 58.33     HAl: -18.02   UNF: 47.56
N: 1023       SOB: -5.3     NX: 10.69     HAl: -15.14   UNF: 42.76
N: 2047       SOB: -7.8     NX: -13.05    HAl: 5.16     UNF: 5.88
N: 4095       SOB: -2.93    NX: 2.05      HAl: 12.47    UNF: 19.41
N: 8191       SOB: 0.5      NX: 0.69      HAl: 17.35    UNF: 3.31
N: 16383      SOB: -1.88    NX: 0.6       HAl: 5.18     UNF: -9.1
N: 32767      SOB: 1.02     NX: 0.27      HAl: 3.65     UNF: -4.39



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

N: 511        SOB: -5.59    NX: -6.41     HAl: 0.2      UNF: -11.93
N: 1023       SOB: -2.38    NX: 1.38      HAl: 0.05     UNF: 1.25
N: 2047       SOB: -0.39    NX: 0.67      HAl: -0.87    UNF: -5.65
N: 4095       SOB: -0.01    NX: 0.7       HAl: 0.36     UNF: -5.47
N: 8191       SOB: -0.26    NX: 0.73      HAl: 0.1      UNF: -4.21
N: 16383      SOB: -0.14    NX: -0.03     HAl: -0.23    UNF: -3.42
N: 32767      SOB: -0.06    NX: 0.0       HAl: 0.04     UNF: -0.58



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

N: 511        SOB: 11.64    NX: 0.57      HAl: -1.24    UNF: 5.22
N: 1023       SOB: 5.15     NX: -0.24     HAl: 2.74     UNF: 0.88
N: 2047       SOB: 3.88     NX: -0.65     HAl: 1.89     UNF: -4.43
N: 4095       SOB: 2.06     NX: -0.78     HAl: 0.39     UNF: -4.99
N: 8191       SOB: 1.79     NX: -0.16     HAl: -0.02    UNF: -2.51
N: 16383      SOB: 3.94     NX: -0.12     HAl: 0.09     UNF: -1.11
N: 32767      SOB: -0.02    NX: -0.14     HAl: -0.21    UNF: -0.49



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

N: 511        SOB: -0.8     NX: -2.61     HAl: 6.34     UNF: -22.33
N: 1023       SOB: -1.84    NX: -0.59     HAl: 7.31     UNF: -9.2
N: 2047       SOB: -1.75    NX: -1.38     HAl: 5.12     UNF: -4.89
N: 4095       SOB: -0.78    NX: -1.09     HAl: 1.84     UNF: -1.09
N: 8191       SOB: -0.62    NX: -1.89     HAl: -1.69    UNF: 1.01
N: 16383      SOB: -0.45    NX: -0.33     HAl: -0.23    UNF: 3.05
N: 32767      SOB: 0.08     NX: -0.02     HAl: 0.04     UNF: 1.01



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

N: 511        SOB: 42.94    NX: -36.47    HAl: 11.18    UNF: -4.71
N: 1023       SOB: 10.96    NX: -4.89     HAl: -20.74   UNF: -28.67
N: 2047       SOB: 10.85    NX: -4.98     HAl: -8.94    UNF: -4.98
N: 4095       SOB: 8.82     NX: 0.9       HAl: -3.05    UNF: -7.01
N: 8191       SOB: 5.82     NX: 1.87      HAl: 1.87     UNF: 0.88
N: 16383      SOB: 0.37     NX: -0.12     HAl: -0.12    UNF: -2.1
N: 32767      SOB: 0.37     NX: 0.37      HAl: -0.13    UNF: 0.12



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

N: 511        SOB: -4.71    NX: -36.47    HAl: 42.94    UNF: 11.18
N: 1023       SOB: -4.89    NX: 10.96     HAl: -12.82   UNF: 10.96
N: 2047       SOB: -20.82   NX: -1.03     HAl: 10.85    UNF: -4.98
N: 4095       SOB: -16.9    NX: -3.05     HAl: -7.01    UNF: -7.01
N: 8191       SOB: -7.03    NX: -1.1      HAl: -7.03    UNF: 5.82
N: 16383      SOB: 13.72    NX: -0.62     HAl: -4.57    UNF: 0.87
N: 32767      SOB: -0.62    NX: -0.62     HAl: -2.6     UNF: -4.58



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

N: 511        SOB: -20.59   NX: -52.35    HAl: -4.71    UNF: 42.94
N: 1023       SOB: -12.82   NX: -28.67    HAl: 3.03     UNF: 26.81
N: 2047       SOB: -16.86   NX: -12.9     HAl: -8.94    UNF: 26.69
N: 4095       SOB: -16.9    NX: -16.9     HAl: -8.99    UNF: 0.9
N: 8191       SOB: -9.01    NX: -1.1      HAl: -10.99   UNF: 1.87
N: 16383      SOB: -12.48   NX: 5.32      HAl: -7.04    UNF: -1.11
N: 32767      SOB: -1.12    NX: 0.37      HAl: -3.34    UNF: -4.58