qmcintdim15

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=15


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

N: 511        SOB: -5.95    NX: 3.13      HAl: -20.55   UNF: 2.7
N: 1023       SOB: -1.54    NX: 3.27      HAl: -10.69   UNF: 3.82
N: 2047       SOB: -0.69    NX: 3.63      HAl: -7.58    UNF: 1.09
N: 4095       SOB: 0.81     NX: 1.59      HAl: -4.44    UNF: 4.06
N: 8191       SOB: 0.44     NX: 0.13      HAl: -2.15    UNF: 1.67
N: 16383      SOB: 0.13     NX: 0.15      HAl: -1.18    UNF: 1.08
N: 32767      SOB: 0.13     NX: 0.14      HAl: -0.83    UNF: -0.33



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

N: 511        SOB: -3.66    NX: -4.4      HAl: -11.85   UNF: -7.49
N: 1023       SOB: -7.19    NX: -2.4      HAl: -8.29    UNF: 0.28
N: 2047       SOB: -1.68    NX: 0.08      HAl: -4.11    UNF: -1.31
N: 4095       SOB: -0.45    NX: -0.45     HAl: -2.02    UNF: 1.73
N: 8191       SOB: 0.44     NX: -0.04     HAl: -0.75    UNF: 3.34
N: 16383      SOB: 0.2      NX: -0.31     HAl: -0.66    UNF: 2.6
N: 32767      SOB: 0.28     NX: -0.01     HAl: -0.67    UNF: 0.89



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

N: 511        SOB: -5.62    NX: 2.23      HAl: -11.33   UNF: 1.79
N: 1023       SOB: -4.81    NX: -0.21     HAl: -5.76    UNF: 2.03
N: 2047       SOB: -2.16    NX: -2.44     HAl: -0.31    UNF: -0.5
N: 4095       SOB: -2.35    NX: -0.44     HAl: -1.11    UNF: -0.13
N: 8191       SOB: -1.15    NX: -0.42     HAl: -0.65    UNF: -0.02
N: 16383      SOB: -1.12    NX: 0.0       HAl: -0.05    UNF: 0.19
N: 32767      SOB: -0.59    NX: 0.27      HAl: 0.1      UNF: -0.32



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

N: 511        SOB: -91.45   NX: -82.32    HAl: -97.21   UNF: -91.11
N: 1023       SOB: -74.24   NX: -50.17    HAl: -86.7    UNF: -88.86
N: 2047       SOB: -74.33   NX: 182.4     HAl: -85.64   UNF: -86.97
N: 4095       SOB: 166.38   NX: 83.89     HAl: -79.62   UNF: -57.33
N: 8191       SOB: 73.13    NX: 4.12      HAl: -62.71   UNF: 67.96
N: 16383      SOB: 69.2     NX: 6.62      HAl: -63.43   UNF: 19.44
N: 32767      SOB: 49.45    NX: -11.27    HAl: -62.52   UNF: -12.3



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

N: 511        SOB: -87.64   NX: -87.28    HAl: -100.0   UNF: -14.64
N: 1023       SOB: -71.17   NX: -60.75    HAl: -100.0   UNF: -47.63
N: 2047       SOB: -55.72   NX: -63.05    HAl: -100.0   UNF: -59.36
N: 4095       SOB: -62.74   NX: -28.64    HAl: -100.0   UNF: -69.0
N: 8191       SOB: -61.35   NX: 16.38     HAl: -100.0   UNF: -67.0
N: 16383      SOB: -56.35   NX: -19.27    HAl: -100.0   UNF: -58.3
N: 32767      SOB: -52.93   NX: -9.23     HAl: -100.0   UNF: -54.16



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

N: 511        SOB: -90.83   NX: -76.65    HAl: -100.0   UNF: -76.54
N: 1023       SOB: -87.16   NX: -79.73    HAl: -100.0   UNF: -69.02
N: 2047       SOB: -63.32   NX: -66.43    HAl: -100.0   UNF: -67.58
N: 4095       SOB: -68.51   NX: -9.08     HAl: -100.0   UNF: -57.04
N: 8191       SOB: -53.51   NX: -13.39    HAl: -100.0   UNF: -31.34
N: 16383      SOB: -46.45   NX: -34.21    HAl: -100.0   UNF: -49.32
N: 32767      SOB: -49.45   NX: -13.42    HAl: -100.0   UNF: -14.9



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

N: 511        SOB: 12777.82 NX: 96.61     HAl: -67.23   UNF: 63.84
N: 1023       SOB: 6322.53  NX: 63.84     HAl: -67.23   UNF: 31.07
N: 2047       SOB: 3340.64  NX: 31.07     HAl: -34.46   UNF: -1.7
N: 4095       SOB: 1669.47  NX: -1.7      HAl: -1.7     UNF: 31.07
N: 8191       SOB: 883.04   NX: -1.7      HAl: 31.07    UNF: -1.7
N: 16383      SOB: 424.29   NX: -1.7      HAl: -1.7     UNF: -1.7
N: 32767      SOB: 227.68   NX: -1.7      HAl: -1.7     UNF: -1.7



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

N: 511        SOB: 25622.88 NX: -67.23    HAl: -67.23   UNF: 31.07
N: 1023       SOB: 12745.06 NX: -67.23    HAl: -67.23   UNF: -1.7
N: 2047       SOB: 6355.3   NX: -1.7      HAl: -34.46   UNF: -1.7
N: 4095       SOB: 3144.03  NX: -34.46    HAl: -34.46   UNF: -1.7
N: 8191       SOB: 1538.4   NX: -34.46    HAl: -1.7     UNF: -1.7
N: 16383      SOB: 751.97   NX: -34.46    HAl: -34.46   UNF: -1.7
N: 32767      SOB: 358.75   NX: -1.7      HAl: -1.7     UNF: -1.7



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

N: 511        SOB: 12843.36 NX: -34.46    HAl: -67.23   UNF: 31.07
N: 1023       SOB: 6388.06  NX: -34.46    HAl: -34.46   UNF: -1.7
N: 2047       SOB: 3144.03  NX: -34.46    HAl: -34.46   UNF: -34.46
N: 4095       SOB: 1571.17  NX: -34.46    HAl: -1.7     UNF: -1.7
N: 8191       SOB: 784.74   NX: -1.7      HAl: 31.07    UNF: -1.7
N: 16383      SOB: 391.52   NX: -1.7      HAl: -1.7     UNF: -1.7
N: 32767      SOB: 194.91   NX: -1.7      HAl: -1.7     UNF: -1.7



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

N: 511        SOB: -81.24   NX: -42.38    HAl: -95.15   UNF: -78.48
N: 1023       SOB: 49.43    NX: -31.55    HAl: -3.18    UNF: -68.99
N: 2047       SOB: -3.7     NX: -9.49     HAl: 5.49     UNF: -52.0
N: 4095       SOB: 396.85   NX: -28.22    HAl: -5.35    UNF: -62.54
N: 8191       SOB: 196.91   NX: -49.08    HAl: 29.43    UNF: -39.12
N: 16383      SOB: 123.86   NX: 35.39     HAl: -7.21    UNF: 0.85
N: 32767      SOB: 150.96   NX: 15.06     HAl: 10.65    UNF: -18.36



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

N: 511        SOB: -99.19   NX: -98.94    HAl: -95.45   UNF: -68.35
N: 1023       SOB: -99.24   NX: -73.61    HAl: -95.35   UNF: -79.68
N: 2047       SOB: -72.49   NX: -74.94    HAl: -84.65   UNF: -89.73
N: 4095       SOB: -85.82   NX: -70.68    HAl: -82.07   UNF: -89.08
N: 8191       SOB: -89.08   NX: -62.55    HAl: -87.45   UNF: -91.39
N: 16383      SOB: -59.89   NX: -71.6     HAl: -86.0    UNF: -80.75
N: 32767      SOB: -64.06   NX: -55.46    HAl: -70.14   UNF: -76.44



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

N: 511        SOB: -99.33   NX: -99.37    HAl: -100.0   UNF: -99.9
N: 1023       SOB: -99.46   NX: -99.65    HAl: -97.28   UNF: -99.94
N: 2047       SOB: -96.19   NX: -99.7     HAl: -94.51   UNF: -99.86
N: 4095       SOB: -97.54   NX: -90.1     HAl: -97.2    UNF: -97.29
N: 8191       SOB: -92.98   NX: -83.52    HAl: -97.46   UNF: -97.74
N: 16383      SOB: -95.82   NX: -90.62    HAl: -96.92   UNF: -72.4
N: 32767      SOB: 77.48    NX: -82.03    HAl: -88.64   UNF: -63.05



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

N: 511        SOB: 184.25   NX: 465.27    HAl: 53.07    UNF: -47.68
N: 1023       SOB: 108.88   NX: 442.68    HAl: 90.24    UNF: -226.57
N: 2047       SOB: 79.08    NX: 321.93    HAl: -111.75  UNF: 26.56
N: 4095       SOB: 43.9     NX: 90.57     HAl: -68.55   UNF: 179.91
N: 8191       SOB: -17.45   NX: 102.6     HAl: -207.93  UNF: 72.69
N: 16383      SOB: -60.32   NX: 30.32     HAl: -46.52   UNF: 61.61
N: 32767      SOB: -11.09   NX: 8.77      HAl: -48.98   UNF: 34.5



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

N: 511        SOB: 12897.21 NX: 214.64    HAl: -68.98   UNF: 306.16
N: 1023       SOB: 6277.28  NX: 203.91    HAl: 35.64    UNF: 160.44
N: 2047       SOB: 2666.95  NX: -37.26    HAl: 74.75    UNF: 58.95
N: 4095       SOB: 1439.14  NX: -53.62    HAl: -112.87  UNF: 141.17
N: 8191       SOB: 719.7    NX: -50.02    HAl: 32.03    UNF: 18.51
N: 16383      SOB: 404.86   NX: 34.56     HAl: -2.38    UNF: -13.13
N: 32767      SOB: 284.73   NX: 0.81      HAl: -30.07   UNF: 23.26



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

N: 511        SOB: 24.82    NX: -268.74   HAl: 179.76   UNF: -68.95
N: 1023       SOB: -35.64   NX: -147.19   HAl: 95.48    UNF: -87.8
N: 2047       SOB: 252.01   NX: 8.1       HAl: 13.14    UNF: 68.69
N: 4095       SOB: 123.78   NX: 14.78     HAl: -43.58   UNF: 119.27
N: 8191       SOB: 40.39    NX: 113.15    HAl: 75.25    UNF: 10.48
N: 16383      SOB: 0.88     NX: 49.27     HAl: 74.24    UNF: -22.11
N: 32767      SOB: -60.86   NX: 52.55     HAl: 25.4     UNF: -10.33



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: 5626922.35NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 2807905.28NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 1402530.21NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 700872.5 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 350300.66NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 175078.94NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 87484.12 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