qmcintdim5

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=5


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

N: 511        SOB: -0.2     NX: 0.38      HAl: -1.72    UNF: -0.7
N: 1023       SOB: -0.07    NX: 0.27      HAl: -0.8     UNF: -2.76
N: 2047       SOB: -0.06    NX: 0.21      HAl: -0.53    UNF: 0.82
N: 4095       SOB: -0.03    NX: 0.04      HAl: -0.27    UNF: 0.81
N: 8191       SOB: -0.01    NX: 0.01      HAl: -0.18    UNF: 0.28
N: 16383      SOB: 0.0      NX: 0.01      HAl: -0.09    UNF: 1.36
N: 32767      SOB: 0.0      NX: 0.0       HAl: -0.05    UNF: 0.5



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

N: 511        SOB: -0.5     NX: 0.48      HAl: -2.01    UNF: -0.32
N: 1023       SOB: -0.25    NX: 0.18      HAl: -0.75    UNF: -0.82
N: 2047       SOB: -0.12    NX: 0.16      HAl: -0.44    UNF: -0.87
N: 4095       SOB: -0.12    NX: 0.02      HAl: -0.27    UNF: -0.78
N: 8191       SOB: -0.04    NX: 0.01      HAl: -0.08    UNF: -0.41
N: 16383      SOB: -0.01    NX: 0.01      HAl: -0.05    UNF: -0.09
N: 32767      SOB: -0.01    NX: 0.01      HAl: -0.01    UNF: -0.19



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

N: 511        SOB: -0.54    NX: -0.5      HAl: -0.72    UNF: 0.01
N: 1023       SOB: -1.11    NX: 0.22      HAl: -0.34    UNF: 1.26
N: 2047       SOB: -0.45    NX: 0.2       HAl: -0.36    UNF: 0.33
N: 4095       SOB: 0.08     NX: -0.24     HAl: -0.11    UNF: 1.04
N: 8191       SOB: -0.01    NX: -0.09     HAl: -0.05    UNF: 1.38
N: 16383      SOB: -0.02    NX: 0.0       HAl: -0.03    UNF: 1.04
N: 32767      SOB: 0.0      NX: 0.02      HAl: -0.01    UNF: 0.96



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

N: 511        SOB: -24.47   NX: -7.47     HAl: -39.05   UNF: -26.71
N: 1023       SOB: -13.67   NX: -11.93    HAl: -31.22   UNF: 1.14
N: 2047       SOB: 1.5      NX: -7.71     HAl: -27.43   UNF: 5.32
N: 4095       SOB: -6.32    NX: -5.78     HAl: -20.86   UNF: -2.22
N: 8191       SOB: -4.36    NX: -5.12     HAl: -17.91   UNF: -5.77
N: 16383      SOB: -1.39    NX: 5.23      HAl: -12.28   UNF: -2.53
N: 32767      SOB: -2.54    NX: 2.52      HAl: -9.92    UNF: -0.59



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

N: 511        SOB: -22.61   NX: 21.92     HAl: -100.0   UNF: -7.78
N: 1023       SOB: -24.44   NX: -0.91     HAl: -100.0   UNF: -3.39
N: 2047       SOB: -6.7     NX: 86.98     HAl: -100.0   UNF: -7.56
N: 4095       SOB: -11.58   NX: 34.32     HAl: -100.0   UNF: -11.9
N: 8191       SOB: -6.63    NX: 16.45     HAl: -100.0   UNF: -7.42
N: 16383      SOB: 0.37     NX: 11.02     HAl: -100.0   UNF: 5.26
N: 32767      SOB: -1.12    NX: 6.84      HAl: -100.0   UNF: -3.32



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

N: 511        SOB: -34.46   NX: -38.03    HAl: -100.0   UNF: 4.46
N: 1023       SOB: -18.15   NX: -24.73    HAl: -100.0   UNF: -9.41
N: 2047       SOB: -13.46   NX: -7.69     HAl: -100.0   UNF: -0.86
N: 4095       SOB: 10.59    NX: -2.07     HAl: -100.0   UNF: -3.25
N: 8191       SOB: 5.57     NX: -0.79     HAl: -100.0   UNF: -9.16
N: 16383      SOB: -4.16    NX: 0.32      HAl: -100.0   UNF: -10.64
N: 32767      SOB: -2.93    NX: 0.74      HAl: -100.0   UNF: -3.89



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

N: 511        SOB: 5.63     NX: -6.34     HAl: 0.74     UNF: 10.18
N: 1023       SOB: 1.73     NX: -1.31     HAl: -0.45    UNF: 8.64
N: 2047       SOB: 2.46     NX: -3.1      HAl: 0.0      UNF: 2.21
N: 4095       SOB: 0.99     NX: -2.14     HAl: -1.38    UNF: 1.92
N: 8191       SOB: 1.22     NX: -0.06     HAl: -0.16    UNF: -2.14
N: 16383      SOB: 1.22     NX: 0.0       HAl: -0.06    UNF: -2.27
N: 32767      SOB: -0.16    NX: 0.0       HAl: 0.0      UNF: -1.15



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

N: 511        SOB: 18.21    NX: -3.81     HAl: 1.31     UNF: 3.52
N: 1023       SOB: 9.5      NX: -3.36     HAl: 1.86     UNF: -1.6
N: 2047       SOB: 0.74     NX: -0.77     HAl: 4.0      UNF: 2.91
N: 4095       SOB: -0.38    NX: -0.26     HAl: 2.34     UNF: 0.8
N: 8191       SOB: 0.26     NX: -0.1      HAl: 1.44     UNF: 4.16
N: 16383      SOB: 0.35     NX: 0.0       HAl: 1.44     UNF: 2.18
N: 32767      SOB: 0.1      NX: -0.03     HAl: 0.22     UNF: 0.7



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

N: 511        SOB: 42.02    NX: 4.8       HAl: 0.83     UNF: -8.16
N: 1023       SOB: 9.73     NX: -0.99     HAl: -0.19    UNF: -2.14
N: 2047       SOB: 6.4      NX: 0.38      HAl: -0.99    UNF: -0.77
N: 4095       SOB: 1.44     NX: 0.96      HAl: -0.77    UNF: -1.41
N: 8191       SOB: 0.83     NX: 0.83      HAl: -1.06    UNF: -2.59
N: 16383      SOB: 0.42     NX: 0.42      HAl: -0.16    UNF: -1.79
N: 32767      SOB: 0.29     NX: 0.22      HAl: 0.03     UNF: -1.41



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

N: 511        SOB: 2.96     NX: 2.89      HAl: -12.79   UNF: -17.54
N: 1023       SOB: 1.57     NX: 2.79      HAl: -8.37    UNF: -25.13
N: 2047       SOB: 1.08     NX: 2.65      HAl: -5.82    UNF: -1.62
N: 4095       SOB: -0.16    NX: 0.21      HAl: -2.8     UNF: -5.27
N: 8191       SOB: -0.18    NX: -0.03     HAl: -2.86    UNF: -0.37
N: 16383      SOB: 0.02     NX: 0.06      HAl: -0.73    UNF: 0.72
N: 32767      SOB: -0.06    NX: 0.04      HAl: -0.52    UNF: -0.98



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

N: 511        SOB: -4.14    NX: 70.25     HAl: -34.62   UNF: -4.73
N: 1023       SOB: 0.14     NX: 24.74     HAl: -14.04   UNF: -6.76
N: 2047       SOB: 2.26     NX: 40.17     HAl: -14.69   UNF: -7.95
N: 4095       SOB: -4.97    NX: 9.3       HAl: -1.73    UNF: -9.08
N: 8191       SOB: 1.45     NX: 5.32      HAl: -3.68    UNF: -0.91
N: 16383      SOB: 3.69     NX: 1.17      HAl: -6.31    UNF: -3.01
N: 32767      SOB: 1.57     NX: 1.19      HAl: -1.37    UNF: 0.17



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

N: 511        SOB: -45.55   NX: -53.34    HAl: -21.53   UNF: 73.28
N: 1023       SOB: -29.52   NX: -48.74    HAl: -23.04   UNF: 32.49
N: 2047       SOB: -35.18   NX: -2.98     HAl: 0.65     UNF: 9.79
N: 4095       SOB: 12.15    NX: -16.19    HAl: 14.45    UNF: -3.31
N: 8191       SOB: 4.32     NX: -4.61     HAl: 25.18    UNF: -4.89
N: 16383      SOB: -6.85    NX: -4.04     HAl: 8.84     UNF: -12.42
N: 32767      SOB: -1.99    NX: -6.83     HAl: 0.84     UNF: -7.67



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

N: 511        SOB: -11.34   NX: -5.36     HAl: -5.2     UNF: 3.57
N: 1023       SOB: -6.45    NX: -0.59     HAl: 4.64     UNF: 6.84
N: 2047       SOB: 1.49     NX: 1.64      HAl: 0.13     UNF: 5.13
N: 4095       SOB: 0.97     NX: 2.13      HAl: 2.93     UNF: 3.41
N: 8191       SOB: 1.02     NX: 0.93      HAl: 1.09     UNF: 6.41
N: 16383      SOB: 1.08     NX: 0.54      HAl: 0.13     UNF: 2.87
N: 32767      SOB: 1.17     NX: 0.05      HAl: 0.11     UNF: 0.89



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

N: 511        SOB: 14.75    NX: -15.66    HAl: -8.96    UNF: -26.7
N: 1023       SOB: 5.42     NX: -10.8     HAl: -4.57    UNF: -11.3
N: 2047       SOB: 4.22     NX: -15.13    HAl: -2.3     UNF: -12.63
N: 4095       SOB: 2.12     NX: -5.94     HAl: -4.4     UNF: -14.41
N: 8191       SOB: 1.33     NX: 0.08      HAl: -0.76    UNF: -6.76
N: 16383      SOB: 3.67     NX: 0.05      HAl: 0.48     UNF: -5.98
N: 32767      SOB: -0.25    NX: -0.01     HAl: 0.19     UNF: -3.67



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

N: 511        SOB: 5.28     NX: -1.9      HAl: 6.82     UNF: 24.27
N: 1023       SOB: -3.37    NX: 4.6       HAl: 8.29     UNF: 25.46
N: 2047       SOB: -1.11    NX: 0.69      HAl: 3.75     UNF: 23.77
N: 4095       SOB: -0.37    NX: 2.18      HAl: 1.56     UNF: 8.96
N: 8191       SOB: 0.31     NX: -0.48     HAl: -1.99    UNF: 3.28
N: 16383      SOB: 0.55     NX: 0.13      HAl: 0.49     UNF: 2.78
N: 32767      SOB: 0.65     NX: -0.06     HAl: 0.21     UNF: 0.07



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

N: 511        SOB: -4.71    NX: -4.71     HAl: 42.94    UNF: -4.71
N: 1023       SOB: -28.67   NX: -4.89     HAl: 42.66    UNF: -52.45
N: 2047       SOB: 18.77    NX: 18.77     HAl: 30.64    UNF: -28.74
N: 4095       SOB: 12.78    NX: 6.84      HAl: 6.84     UNF: 0.9
N: 8191       SOB: 18.68    NX: -2.09     HAl: 9.78     UNF: -5.05
N: 16383      SOB: 6.8      NX: -2.1      HAl: 9.77     UNF: 3.83
N: 32767      SOB: 3.83     NX: 2.34      HAl: 2.34     UNF: 8.28



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

N: 511        SOB: -4.71    NX: -52.35    HAl: 138.24   UNF: 42.94
N: 1023       SOB: -52.45   NX: -28.67    HAl: 18.88    UNF: 18.88
N: 2047       SOB: -52.49   NX: -40.62    HAl: 30.64    UNF: -16.86
N: 4095       SOB: -52.52   NX: -46.58    HAl: 0.9      UNF: -22.84
N: 8191       SOB: -22.86   NX: -19.89    HAl: -10.99   UNF: 3.85
N: 16383      SOB: 6.8      NX: -9.52     HAl: -3.58    UNF: 0.87
N: 32767      SOB: -2.11    NX: -2.11     HAl: -1.36    UNF: 0.12



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

N: 511        SOB: -52.35   NX: 90.59     HAl: -100.0   UNF: -100.0
N: 1023       SOB: -28.67   NX: 18.88     HAl: -52.45   UNF: 18.88
N: 2047       SOB: -28.74   NX: 6.89      HAl: -4.98    UNF: -16.86
N: 4095       SOB: -22.84   NX: -16.9     HAl: -22.84   UNF: -16.9
N: 8191       SOB: -8.02    NX: -19.89    HAl: -22.86   UNF: -22.86
N: 16383      SOB: -8.03    NX: -11.0     HAl: -9.52    UNF: -18.42
N: 32767      SOB: 3.83     NX: -5.81     HAl: -4.33    UNF: -9.52