\`x^2+y_1+z_12^34\`
Article Contents
Article Contents

High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion

  • * Corresponding author

    * Corresponding author

This research is partially supported by the NSF of China (No. 12001539), the NSF of Hunan Province (No. 2020JJ5647) and China Postdoctoral Science Foundation (No. 2019TQ0073)

Abstract / Introduction Full Text(HTML) Figure(2) / Table(8) Related Papers Cited by
  • In this work, by combining the Feynman-Kac formula with an Itô-Taylor expansion, we propose a class of high order one-step schemes for backward stochastic differential equations, which can achieve at most six order rate of convergence and only need the terminal conditions on the last one step. Numerical experiments are carried out to show the efficiency and high order accuracy of the proposed schemes.

    Mathematics Subject Classification: Primary: 65C20, 65C30; Secondary: 60H35.

    Citation:

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  • Figure 1.  The plots of $ \log_2(|Y_0-Y^0|) $ and $ \log_2(|Z_0-Z^0|) $ w.r.t. $ \log_2(\triangle t) $ for Sch. 3.1.

    Figure 2.  The plots of $ \log_2(|Y_0-Y^0|) $ and $ \log_2(|Z_0-Z^0|) $ w.r.t. $ \log_2(\triangle t) $ for Sch. 3.2.

    Table 1.  Errors and convergence rates of Scheme 3.1

    $ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
    $ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
    $ 30 $ 1.900E-10 4.870E-10 2.571E-10 1.352E-09 4.061E-10 1.640E-09
    $ 40 $ 3.904E-11 1.136E-10 8.297E-11 3.120E-10 1.236E-10 3.781E-10
    $ 50 $ 1.115E-11 3.660E-11 3.300E-11 9.929E-11 4.771E-11 1.202E-10
    $ 60 $ 4.883E-12 1.478E-11 1.163E-11 4.090E-11 1.713E-11 4.962E-11
    $ 70 $ 2.558E-12 6.903E-12 4.084E-12 1.961E-11 6.298E-12 2.384E-11
    CR 5.128 5.028 4.810 5.008 4.850 5.006
    $\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
    $N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
    $30$ 5.551E-10 1.929E-09 8.532E-10 2.505E-09 1.300E-09 3.370E-09
    $40$ 1.643E-10 4.442E-10 2.457E-10 5.765E-10 3.677E-10 7.748E-10
    $50$ 6.243E-11 1.411E-10 9.186E-11 1.829E-10 1.360E-10 2.456E-10
    $60$ 2.264E-11 5.833E-11 3.365E-11 7.575E-11 5.016E-11 1.019E-10
    $70$ 8.513E-12 2.808E-11 1.294E-11 3.654E-11 1.958E-11 4.925E-11
    CR 4.870 5.005 4.889 5.003 4.901 5.001
     | Show Table
    DownLoad: CSV

    Table 2.  Errors and convergence rates of Scheme 3.2

    $ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
    $ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
    $ 30 $ 6.977E-12 1.462E-11 1.584E-11 3.048E-11 1.880E-11 3.576E-11
    $ 40 $ 1.238E-12 2.621E-12 2.920E-12 5.410E-12 3.481E-12 6.340E-12
    $ 50 $ 2.914E-13 7.099E-13 7.284E-13 1.448E-12 8.737E-13 1.690E-12
    $ 60 $ 7.283E-14 2.430E-13 2.172E-13 4.964E-13 2.661E-13 5.821E-13
    $ 70 $ 2.698E-14 2.542E-14 7.605E-14 2.017E-13 8.982E-14 2.398E-13
    CR 6.612 7.045 6.298 5.921 6.286 5.910
    $\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
    $N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
    $30$ 2.176E-11 4.106E-11 2.767E-11 5.163E-11 3.653E-11 6.749E-11
    $40$ 4.042E-12 7.276E-12 5.163E-12 9.138E-12 6.846E-12 1.191E-11
    $50$ 1.019E-12 1.942E-12 1.309E-12 2.429E-12 1.746E-12 3.170E-12
    $60$ 3.140E-13 6.706E-13 4.107E-13 8.249E-13 5.551E-13 1.078E-12
    $70$ 1.161E-13 2.445E-13 1.448E-13 3.592E-13 2.083E-13 4.128E-13
    CR 6.183 6.008 6.187 5.888 6.103 5.996
     | Show Table
    DownLoad: CSV

    Table 3.  Errors and convergence rates of the Euler, C-N and multi-step schemes

    Euler Crank-Nicolson 3-step
    $ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
    $ 30 $ 5.533E-02 4.186E-02 1.714E-04 2.541E-04 3.355E-05 3.992E-05
    $ 40 $ 4.176E-02 3.130E-02 9.883E-05 1.348E-04 1.452E-05 1.662E-05
    $ 50 $ 3.356E-02 2.448E-02 6.315E-05 8.619E-05 7.549E-06 8.439E-06
    $ 60 $ 2.799E-02 2.092E-02 4.513E-05 5.539E-05 4.411E-06 4.855E-06
    $ 70 $ 2.403E-02 1.796E-02 3.356E-05 3.969E-05 2.797E-06 3.044E-06
    CR 0.985 1.000 1.927 2.188 2.932 3.037
    4-step 5-step 6-step
    $N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
    $30$ 2.764E-07 5.529E-07 3.497E-08 7.795E-08 1.683E-09 1.863E-09
    $40$ 8.634E-08 2.089E-07 9.029E-09 1.918E-08 3.175E-10 3.188E-10
    $50$ 3.494E-08 9.405E-08 3.107E-09 6.418E-09 8.611E-11 8.138E-11
    $60$ 1.668E-08 4.811E-08 1.289E-09 2.614E-09 2.943E-11 2.671E-11
    $70$ 8.927E-09 2.709E-08 6.099E-10 1.221E-09 1.179E-11 1.033E-11
    CR 4.052 3.563 4.779 4.906 5.853 6.128
     | Show Table
    DownLoad: CSV

    Table 4.  Errors and convergence rates of Scheme 3.1

    $ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
    $ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
    $ 20 $ 1.826E-10 1.007E-09 1.912E-10 1.391E-09 1.914E-10 1.578E-09
    $ 30 $ 2.753E-11 1.342E-10 2.860E-11 1.937E-10 2.871E-11 2.186E-10
    $ 40 $ 6.742E-12 3.115E-11 7.027E-12 4.862E-11 7.076E-12 5.457E-11
    $ 50 $ 2.003E-12 8.484E-12 2.109E-12 1.795E-11 2.132E-12 1.985E-11
    $ 60 $ 7.782E-13 3.367E-12 8.229E-13 7.359E-12 8.341E-13 8.109E-12
    CR 4.975 5.212 4.964 4.756 4.953 4.784
    $\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
    $N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
    $20$ 1.917E-10 1.765E-09 1.921E-10 2.139E-09 1.928E-10 2.701E-09
    $30$ 2.882E-11 2.436E-10 2.904E-11 2.935E-10 2.937E-11 3.683E-10
    $40$ 7.125E-12 6.052E-11 7.223E-12 7.243E-11 7.371E-12 9.028E-11
    $50$ 2.155E-12 2.175E-11 2.200E-12 2.554E-11 2.269E-12 3.124E-11
    $60$ 8.448E-13 8.866E-12 8.664E-13 1.037E-11 8.988E-13 1.265E-11
    CR 4.942 4.807 4.921 4.841 4.890 4.874
     | Show Table
    DownLoad: CSV

    Table 5.  Errors and convergence rates of Scheme 3.2

    $ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
    $ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
    $ 20 $ 2.785E-11 1.610E-10 2.690E-11 1.632E-10 2.659E-11 1.639E-10
    $ 30 $ 4.607E-12 1.151E-11 4.522E-12 1.170E-11 4.494E-12 1.176E-11
    $ 40 $ 8.240E-13 2.929E-12 8.087E-13 2.953E-12 8.034E-13 2.968E-12
    $ 50 $ 2.065E-13 9.485E-13 2.025E-13 9.570E-13 2.011E-13 9.599E-13
    $ 60 $ 5.729E-14 3.326E-13 5.607E-14 3.395E-13 5.573E-14 3.423E-13
    CR 5.635 5.526 5.624 5.524 5.620 5.523
    $\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
    $N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
    $20$ 2.628E-11 1.646E-10 2.565E-11 1.660E-10 2.471E-11 1.681E-10
    $30$ 4.465E-12 1.182E-11 4.408E-12 1.194E-11 4.322E-12 1.213E-11
    $40$ 7.985E-13 2.973E-12 7.884E-13 2.997E-12 7.730E-13 3.029E-12
    $50$ 1.998E-13 9.618E-13 1.971E-13 9.737E-13 1.933E-13 9.786E-13
    $60$ 5.485E-14 3.362E-13 5.396E-14 3.354E-13 5.251E-14 3.469E-13
    CR 5.621 5.538 5.615 5.544 5.605 5.535
     | Show Table
    DownLoad: CSV

    Table 6.  Errors and convergence rates of multi-step schemes

    5-step 6-step
    $ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
    $ 20 $ 2.50E-09 5.12E-07 7.75E-10 1.25E-07
    $ 30 $ 3.49E-10 8.50E-08 6.74E-11 1.56E-08
    $ 40 $ 7.99E-11 2.29E-08 1.18E-11 3.37E-09
    $ 50 $ 2.49E-11 8.10E-09 3.03E-12 9.99E-10
    $ 60 $ 1.06E-11 3.43E-09 1.01E-12 3.64E-10
    CR 5.009 4.555 6.051 5.313
     | Show Table
    DownLoad: CSV

    Table 7.  Errors and convergence rates of Scheme 3.1

    $ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
    $ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
    $ 30 $ 1.179E-04 2.343E-05 4.801E-04 6.426E-05 6.009E-04 9.333E-05
    $ 40 $ 2.545E-05 6.052E-06 1.289E-04 1.786E-05 1.634E-04 2.582E-05
    $ 50 $ 7.607E-06 2.115E-06 4.550E-05 6.408E-06 5.814E-05 9.247E-06
    $ 60 $ 2.816E-06 8.941E-07 1.921E-05 2.730E-06 2.468E-05 3.938E-06
    $ 70 $ 1.211E-06 4.310E-07 9.207E-06 1.316E-06 1.187E-05 1.898E-06
    CR 5.405 4.715 4.667 4.590 4.632 4.598
    $\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
    $N$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$ $|Y_0-Y^{0}|$ $|Z_0-Z^{0}|$
    $30$ 7.216E-04 1.223E-04 9.632E-04 1.801E-04 1.326E-03 2.661E-04
    $40$ 1.980E-04 3.377E-05 2.670E-04 4.966E-05 3.705E-04 7.344E-05
    $50$ 7.077E-05 1.209E-05 9.604E-05 1.776E-05 1.339E-04 2.627E-05
    $60$ 3.014E-05 5.148E-06 4.107E-05 7.562E-06 5.747E-05 1.118E-05
    $70$ 1.454E-05 2.481E-06 1.987E-05 3.644E-06 2.786E-05 5.390E-06
    CR 4.609 4.601 4.581 4.604 4.559 4.603
     | Show Table
    DownLoad: CSV

    Table 8.  Errors and convergence rates of Scheme 3.2

    $ \theta_2=0 $ $ \theta_2=\frac{3}{10} $ $ \theta_2=\frac{2}{5} $
    $ N $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $ $ |Y_0-Y^{0}| $ $ |Z_0-Z^{0}| $
    $ 30 $ 2.412E-05 5.723E-06 9.919E-05 1.822E-05 1.242E-04 2.238E-05
    $ 40 $ 3.982E-06 9.642E-07 2.045E-05 3.726E-06 2.594E-05 4.646E-06
    $ 50 $ 9.623E-07 2.367E-07 5.855E-06 1.059E-06 7.485E-06 1.334E-06
    $ 60 $ 2.981E-07 7.128E-08 2.078E-06 3.720E-07 2.672E-06 4.715E-07
    $ 70 $ 1.095E-07 2.891E-08 8.586E-07 1.581E-07 1.108E-06 2.001E-07
    CR 6.367 6.277 5.606 5.614 5.570 5.577
    $\theta_2=\frac{1}{2}$ $\theta_2=\frac{7}{10}$ $\theta_2=1$
    N |Y0-Y0| |Z0-Z0| |Y0-Y0| |Z0-Z0| |Y0-Y0| |Z0-Z0|
    30 1.492E-04 2.653E-05 1.993E-04 3.483E-05 2.744E-04 4.729E-05
    40 3.143E-05 5.566E-06 4.242E-05 7.413E-06 5.888E-05 1.016E-05
    50 9.116E-06 1.608E-06 1.238E-05 2.161E-06 1.727E-05 2.981E-06
    60 3.265E-06 5.727E-07 4.452E-06 7.756E-07 6.232E-06 1.073E-06
    70 1.358E-06 2.418E-07 1.857E-06 3.220E-07 2.606E-06 4.511E-07
    CR 5.547 5.552 5.519 5.527 5.496 5.494
     | Show Table
    DownLoad: CSV
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