qmcintdim20

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=20


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

N: 511        SOB: -9.46    NX: 1.45      HAl: -32.51   UNF: -0.14
N: 1023       SOB: -3.99    NX: 3.24      HAl: -22.87   UNF: 0.42
N: 2047       SOB: -2.36    NX: 3.77      HAl: -14.32   UNF: -1.51
N: 4095       SOB: 1.1      NX: 4.1       HAl: -6.9     UNF: 4.78
N: 8191       SOB: 0.96     NX: 3.09      HAl: -4.55    UNF: 1.83
N: 16383      SOB: 0.31     NX: 3.36      HAl: -3.24    UNF: 2.01
N: 32767      SOB: 0.29     NX: 3.63      HAl: -1.49    UNF: 0.93



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

N: 511        SOB: 0.32     NX: -14.81    HAl: -23.8    UNF: 3.07
N: 1023       SOB: -6.77    NX: -2.61     HAl: -13.56   UNF: 0.29
N: 2047       SOB: -0.87    NX: -1.73     HAl: -6.48    UNF: 6.54
N: 4095       SOB: -0.81    NX: 0.27      HAl: -4.56    UNF: 1.39
N: 8191       SOB: 0.15     NX: 0.07      HAl: -3.31    UNF: -0.34
N: 16383      SOB: 0.81     NX: 0.35      HAl: -1.6     UNF: 0.69
N: 32767      SOB: -0.19    NX: 0.33      HAl: -1.45    UNF: 0.11



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

N: 511        SOB: -6.99    NX: 1.07      HAl: -18.82   UNF: -11.38
N: 1023       SOB: -7.01    NX: 1.92      HAl: -10.95   UNF: -10.21
N: 2047       SOB: -1.64    NX: 0.15      HAl: -2.71    UNF: -5.9
N: 4095       SOB: -1.62    NX: -0.05     HAl: -2.48    UNF: -0.46
N: 8191       SOB: -0.87    NX: 0.75      HAl: -0.95    UNF: -0.87
N: 16383      SOB: -1.12    NX: 0.21      HAl: -0.58    UNF: -1.7
N: 32767      SOB: -1.19    NX: 0.57      HAl: -0.02    UNF: -0.06



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

N: 511        SOB: -94.53   NX: -47.59    HAl: -99.54   UNF: -94.1
N: 1023       SOB: -87.68   NX: -54.73    HAl: -97.51   UNF: -96.36
N: 2047       SOB: -77.38   NX: -70.64    HAl: -97.12   UNF: -94.72
N: 4095       SOB: -76.53   NX: -71.08    HAl: -93.76   UNF: -90.21
N: 8191       SOB: -68.67   NX: -78.29    HAl: -92.98   UNF: -79.48
N: 16383      SOB: -77.78   NX: -36.06    HAl: -88.11   UNF: -79.52
N: 32767      SOB: -75.46   NX: -29.75    HAl: -83.93   UNF: -75.32



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

N: 511        SOB: -92.57   NX: -92.99    HAl: -100.0   UNF: -86.03
N: 1023       SOB: -93.26   NX: -72.22    HAl: -100.0   UNF: -46.16
N: 2047       SOB: -93.57   NX: -79.06    HAl: -100.0   UNF: -61.77
N: 4095       SOB: -52.82   NX: -81.17    HAl: -100.0   UNF: -73.13
N: 8191       SOB: -59.67   NX: -84.13    HAl: -100.0   UNF: 144.89
N: 16383      SOB: 123.63   NX: -29.77    HAl: -100.0   UNF: 30.97
N: 32767      SOB: 19.86    NX: -37.28    HAl: -100.0   UNF: -16.88



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

N: 511        SOB: -92.69   NX: -95.44    HAl: -100.0   UNF: -84.37
N: 1023       SOB: -94.09   NX: -56.21    HAl: -100.0   UNF: -90.91
N: 2047       SOB: -83.28   NX: -69.93    HAl: -100.0   UNF: -66.35
N: 4095       SOB: -86.11   NX: -74.2     HAl: -100.0   UNF: -72.81
N: 8191       SOB: -78.88   NX: -59.43    HAl: -100.0   UNF: 462.52
N: 16383      SOB: -81.27   NX: -50.97    HAl: -100.0   UNF: 195.27
N: 32767      SOB: -76.18   NX: -58.6     HAl: -100.0   UNF: 88.53



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

N: 511        SOB: 410941.79NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 205420.9 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 102660.45NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 51280.22 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 25065.82 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 12482.91 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 6191.46  NX: -100.0    HAl: -100.0   UNF: -100.0



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

N: 511        SOB: 821983.58NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 409893.22NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 205420.9 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 102660.45NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 51280.22 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 25065.82 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 12482.91 NX: -100.0    HAl: -100.0   UNF: -100.0



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

N: 511        SOB: 410941.79NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 205420.9 NX: -100.0    HAl: -100.0   UNF: 948.58
N: 2047       SOB: 102660.45NX: -100.0    HAl: -100.0   UNF: 948.58
N: 4095       SOB: 51280.22 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 25065.82 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 12482.91 NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 6191.46  NX: -100.0    HAl: -100.0   UNF: -100.0



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

N: 511        SOB: -99.55   NX: 10.81     HAl: -99.45   UNF: -72.46
N: 1023       SOB: -89.12   NX: -33.46    HAl: -86.66   UNF: -41.96
N: 2047       SOB: -90.91   NX: -58.68    HAl: -91.33   UNF: -52.51
N: 4095       SOB: -87.78   NX: -67.56    HAl: -84.58   UNF: -19.87
N: 8191       SOB: -87.17   NX: -74.71    HAl: -86.76   UNF: -51.53
N: 16383      SOB: -54.63   NX: 0.58      HAl: -62.14   UNF: -40.31
N: 32767      SOB: -44.03   NX: -27.44    HAl: -47.69   UNF: -23.42



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

N: 511        SOB: -100.0   NX: -100.0    HAl: -99.99   UNF: -99.73
N: 1023       SOB: -100.0   NX: -83.29    HAl: -99.85   UNF: -99.67
N: 2047       SOB: -99.29   NX: -91.54    HAl: -98.02   UNF: -99.43
N: 4095       SOB: -98.37   NX: -95.27    HAl: -98.91   UNF: -99.46
N: 8191       SOB: -98.83   NX: -96.41    HAl: -99.0    UNF: -84.35
N: 16383      SOB: -89.01   NX: -93.26    HAl: -96.12   UNF: -87.61
N: 32767      SOB: -91.86   NX: -73.35    HAl: -96.39   UNF: -89.99



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

N: 511        SOB: -99.99   NX: -99.99    HAl: -100.0   UNF: -100.0
N: 1023       SOB: -99.99   NX: -99.98    HAl: -100.0   UNF: -100.0
N: 2047       SOB: -99.98   NX: -99.96    HAl: -99.99   UNF: -100.0
N: 4095       SOB: -99.87   NX: -99.98    HAl: -99.84   UNF: -99.65
N: 8191       SOB: -99.59   NX: -99.88    HAl: -99.9    UNF: -99.82
N: 16383      SOB: -99.71   NX: -99.87    HAl: -99.93   UNF: -97.77
N: 32767      SOB: -99.79   NX: -99.58    HAl: -78.67   UNF: -92.47



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

N: 511        SOB: -630.98  NX: -956.48   HAl: -145.63  UNF: -385.27
N: 1023       SOB: -305.17  NX: -637.8    HAl: -550.59  UNF: 1293.66
N: 2047       SOB: -86.82   NX: -188.22   HAl: -125.91  UNF: 772.01
N: 4095       SOB: -48.38   NX: 276.15    HAl: 449.65   UNF: -76.94
N: 8191       SOB: 195.54   NX: 474.84    HAl: 6.58     UNF: 438.57
N: 16383      SOB: -35.82   NX: 57.52     HAl: 76.33    UNF: 382.18
N: 32767      SOB: -198.8   NX: -24.97    HAl: -96.32   UNF: 202.99



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

N: 511        SOB: 411195.62NX: 432.62    HAl: -306.76  UNF: -562.27
N: 1023       SOB: 205199.29NX: -114.49   HAl: -224.72  UNF: -489.86
N: 2047       SOB: 102228.08NX: -820.28   HAl: 210.85   UNF: -68.88
N: 4095       SOB: 50490.94 NX: -803.28   HAl: -325.96  UNF: -125.52
N: 8191       SOB: 24644.65 NX: -608.17   HAl: -55.77   UNF: -156.69
N: 16383      SOB: 12542.52 NX: -561.82   HAl: -178.89  UNF: -246.21
N: 32767      SOB: 6154.52  NX: -244.81   HAl: -198.54  UNF: -211.26



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

N: 511        SOB: -956.54  NX: 499.17    HAl: -2395.0  UNF: 411.84
N: 1023       SOB: -131.24  NX: -168.95   HAl: -1152.76 UNF: 349.01
N: 2047       SOB: 405.44   NX: -305.55   HAl: 239.08   UNF: -14.45
N: 4095       SOB: 209.55   NX: -399.27   HAl: -22.52   UNF: 248.39
N: 8191       SOB: 227.05   NX: -294.59   HAl: 50.02    UNF: 17.1
N: 16383      SOB: -31.78   NX: -154.39   HAl: -484.38  UNF: 103.79
N: 32767      SOB: 15.08    NX: -81.73    HAl: -357.99  UNF: 283.07



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: 1.36736633176E9NX: -100.0    HAl: -100.0   UNF: -100.0
N: 1023       SOB: 6.8234518395E8NX: -100.0    HAl: -100.0   UNF: -100.0
N: 2047       SOB: 3.4083903988E8NX: -100.0    HAl: -100.0   UNF: -100.0
N: 4095       SOB: 1.7033621661E8NX: -100.0    HAl: -100.0   UNF: -100.0
N: 8191       SOB: 8.514726022E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 16383      SOB: 4.256838249E7NX: -100.0    HAl: -100.0   UNF: -100.0
N: 32767      SOB: 2.128284208E7NX: -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