qmcintdim13

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=13


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

N: 511        SOB: -5.13    NX: 4.36      HAl: -13.78   UNF: -0.69
N: 1023       SOB: -1.33    NX: -0.05     HAl: -8.47    UNF: 0.16
N: 2047       SOB: -0.64    NX: -0.37     HAl: -6.06    UNF: 0.61
N: 4095       SOB: 0.55     NX: -0.12     HAl: -3.54    UNF: 1.93
N: 8191       SOB: 0.39     NX: -0.58     HAl: -2.14    UNF: 3.39
N: 16383      SOB: 0.18     NX: 0.03      HAl: -1.15    UNF: 1.29
N: 32767      SOB: 0.11     NX: 0.03      HAl: -0.67    UNF: 0.54



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

N: 511        SOB: -9.01    NX: -0.14     HAl: -8.71    UNF: -1.76
N: 1023       SOB: -6.83    NX: 0.32      HAl: -6.26    UNF: 3.57
N: 2047       SOB: -2.17    NX: 0.16      HAl: -3.7     UNF: 2.52
N: 4095       SOB: -0.76    NX: -0.1      HAl: -1.92    UNF: -0.42
N: 8191       SOB: -0.03    NX: -0.2      HAl: -0.56    UNF: -0.35
N: 16383      SOB: -0.02    NX: -0.02     HAl: -0.39    UNF: 0.79
N: 32767      SOB: 0.09     NX: -0.08     HAl: -0.34    UNF: 0.34



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

N: 511        SOB: -3.9     NX: 1.81      HAl: -6.82    UNF: -6.41
N: 1023       SOB: -3.79    NX: 0.14      HAl: -3.12    UNF: -2.9
N: 2047       SOB: -0.82    NX: -1.57     HAl: 0.12     UNF: -1.98
N: 4095       SOB: -0.73    NX: -1.33     HAl: -0.3     UNF: 0.54
N: 8191       SOB: -0.68    NX: -1.55     HAl: -0.33    UNF: 0.91
N: 16383      SOB: -0.87    NX: -1.18     HAl: 0.06     UNF: 0.35
N: 32767      SOB: -0.5     NX: -0.83     HAl: 0.03     UNF: 0.5



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

N: 511        SOB: -78.44   NX: 16.91     HAl: -91.53   UNF: -76.86
N: 1023       SOB: -65.14   NX: -19.47    HAl: -89.38   UNF: -23.71
N: 2047       SOB: -62.3    NX: -37.13    HAl: -85.79   UNF: -28.12
N: 4095       SOB: 157.77   NX: -14.36    HAl: -79.36   UNF: -46.57
N: 8191       SOB: 55.24    NX: -39.25    HAl: -65.23   UNF: -54.71
N: 16383      SOB: 29.89    NX: -38.43    HAl: -63.29   UNF: -43.0
N: 32767      SOB: 2.38     NX: -29.81    HAl: -56.54   UNF: -43.55



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

N: 511        SOB: -89.04   NX: -66.57    HAl: -100.0   UNF: -51.28
N: 1023       SOB: -88.56   NX: 35.59     HAl: -100.0   UNF: -60.13
N: 2047       SOB: -58.5    NX: -16.83    HAl: -100.0   UNF: -70.49
N: 4095       SOB: -38.93   NX: -4.49     HAl: -100.0   UNF: -40.36
N: 8191       SOB: -49.0    NX: -25.68    HAl: -100.0   UNF: 56.29
N: 16383      SOB: -50.77   NX: -27.21    HAl: -100.0   UNF: 12.21
N: 32767      SOB: -43.43   NX: -36.86    HAl: -100.0   UNF: -19.68



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

N: 511        SOB: -85.47   NX: -37.67    HAl: -100.0   UNF: -28.8
N: 1023       SOB: -84.15   NX: -59.48    HAl: -100.0   UNF: -59.39
N: 2047       SOB: -69.44   NX: -58.05    HAl: -100.0   UNF: -57.17
N: 4095       SOB: -63.52   NX: -60.71    HAl: -100.0   UNF: -8.18
N: 8191       SOB: -52.42   NX: -29.66    HAl: -100.0   UNF: 39.26
N: 16383      SOB: -57.57   NX: -27.41    HAl: -100.0   UNF: 29.65
N: 32767      SOB: -30.81   NX: -31.82    HAl: -100.0   UNF: 1.7



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

N: 511        SOB: 3127.65  NX: -9.89     HAl: -59.04   UNF: -50.85
N: 1023       SOB: 1530.21  NX: 6.5       HAl: -50.85   UNF: -50.85
N: 2047       SOB: 874.85   NX: -18.08    HAl: -34.46   UNF: -50.85
N: 4095       SOB: 440.67   NX: -34.46    HAl: -18.08   UNF: -26.27
N: 8191       SOB: 244.06   NX: -26.27    HAl: -1.7     UNF: -1.7
N: 16383      SOB: 121.18   NX: -1.7      HAl: -1.7     UNF: -9.89
N: 32767      SOB: 63.84    NX: -1.7      HAl: -1.7     UNF: -9.89



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

N: 511        SOB: 6371.68  NX: -67.23    HAl: -67.23   UNF: -9.89
N: 1023       SOB: 3144.03  NX: -9.89     HAl: -59.04   UNF: -42.66
N: 2047       SOB: 1562.98  NX: 31.07     HAl: -42.66   UNF: -9.89
N: 4095       SOB: 776.54   NX: 22.88     HAl: -34.46   UNF: 14.69
N: 8191       SOB: 383.33   NX: 14.69     HAl: -9.89    UNF: 6.5
N: 16383      SOB: 186.72   NX: 6.5       HAl: -9.89    UNF: 6.5
N: 32767      SOB: 80.22    NX: 6.5       HAl: -9.89    UNF: 6.5



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

N: 511        SOB: 3209.57  NX: -42.66    HAl: -50.85   UNF: -50.85
N: 1023       SOB: 1579.36  NX: 22.88     HAl: -9.89    UNF: -34.46
N: 2047       SOB: 850.27   NX: 39.26     HAl: -18.08   UNF: -18.08
N: 4095       SOB: 416.1    NX: 22.88     HAl: -1.7     UNF: -18.08
N: 8191       SOB: 203.1    NX: 14.69     HAl: 14.69    UNF: -18.08
N: 16383      SOB: 104.8    NX: 6.5       HAl: -1.7     UNF: -9.89
N: 32767      SOB: 55.65    NX: -1.7      HAl: -1.7     UNF: -9.89



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

N: 511        SOB: -75.77   NX: 86.62     HAl: -84.77   UNF: -76.97
N: 1023       SOB: -39.04   NX: 5.6       HAl: -69.33   UNF: -50.68
N: 2047       SOB: -47.19   NX: 6.91      HAl: -58.93   UNF: -66.8
N: 4095       SOB: 70.92    NX: 82.0      HAl: -44.97   UNF: -64.3
N: 8191       SOB: 29.25    NX: 24.53     HAl: -28.58   UNF: -32.08
N: 16383      SOB: 44.61    NX: 33.97     HAl: -21.66   UNF: 2.81
N: 32767      SOB: 31.58    NX: 23.83     HAl: -14.3    UNF: 105.58



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

N: 511        SOB: -99.51   NX: -67.92    HAl: -93.7    UNF: -84.13
N: 1023       SOB: -99.44   NX: -81.04    HAl: -94.8    UNF: -81.87
N: 2047       SOB: -61.24   NX: -85.41    HAl: -91.95   UNF: -66.19
N: 4095       SOB: -78.15   NX: 67.24     HAl: -76.88   UNF: -68.39
N: 8191       SOB: -79.72   NX: 9.06      HAl: -63.9    UNF: -40.82
N: 16383      SOB: -46.06   NX: -1.64     HAl: -64.56   UNF: 18.2
N: 32767      SOB: -53.78   NX: -38.76    HAl: -58.64   UNF: 7.97



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

N: 511        SOB: -99.74   NX: -98.79    HAl: -32.37   UNF: -99.72
N: 1023       SOB: -99.68   NX: -99.33    HAl: -65.41   UNF: 137.13
N: 2047       SOB: -97.17   NX: -99.37    HAl: -79.19   UNF: 22.06
N: 4095       SOB: -97.37   NX: -98.26    HAl: -88.43   UNF: -32.53
N: 8191       SOB: -97.14   NX: -91.99    HAl: -89.88   UNF: -56.41
N: 16383      SOB: -96.54   NX: -42.73    HAl: -92.66   UNF: -75.23
N: 32767      SOB: 338.0    NX: -65.67    HAl: -41.63   UNF: -74.18



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

N: 511        SOB: -143.43  NX: 238.74    HAl: 8.83     UNF: 120.6
N: 1023       SOB: -117.25  NX: -4.4      HAl: 74.2     UNF: 32.46
N: 2047       SOB: -11.82   NX: -35.82    HAl: -12.24   UNF: -27.83
N: 4095       SOB: -29.6    NX: -87.3     HAl: -4.1     UNF: -33.56
N: 8191       SOB: -37.52   NX: -119.15   HAl: -53.49   UNF: -17.06
N: 16383      SOB: -4.16    NX: -80.04    HAl: -35.81   UNF: -49.53
N: 32767      SOB: 11.98    NX: -46.65    HAl: -41.56   UNF: -37.85



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

N: 511        SOB: 3027.43  NX: -4.51     HAl: -59.24   UNF: 143.35
N: 1023       SOB: 1328.61  NX: 53.81     HAl: 35.29    UNF: 30.27
N: 2047       SOB: 927.74   NX: -4.79     HAl: 27.73    UNF: 44.92
N: 4095       SOB: 412.59   NX: -34.19    HAl: -64.52   UNF: 70.2
N: 8191       SOB: 254.41   NX: -28.83    HAl: 8.71     UNF: 113.5
N: 16383      SOB: 148.67   NX: -4.6      HAl: 4.38     UNF: 34.66
N: 32767      SOB: 121.0    NX: -7.19     HAl: 18.28    UNF: 32.36



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

N: 511        SOB: -11.68   NX: 205.29    HAl: 151.93   UNF: 73.22
N: 1023       SOB: -179.1   NX: 142.16    HAl: 62.39    UNF: -10.18
N: 2047       SOB: -16.55   NX: 82.02     HAl: 104.04   UNF: -82.89
N: 4095       SOB: -79.11   NX: 71.08     HAl: 55.38    UNF: -61.26
N: 8191       SOB: -52.45   NX: 9.34      HAl: -5.64    UNF: 0.42
N: 16383      SOB: -74.48   NX: 14.37     HAl: -14.93   UNF: 48.62
N: 32767      SOB: -60.32   NX: -14.63    HAl: -12.76   UNF: 30.11



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: 625124.71NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 311900.59NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 155747.8 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 77785.83 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 38833.41 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 19364.33 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 9631.57  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