1 Introduction
2 Preliminaries
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\(|\cdot |:\) the Euclidean norm in \(\mathbb {R},\) \(\mathbb {R}^n\) and \(\mathbb {R}^{n\times d};\)
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\(\mathcal {F}_t^{s,x}:\) \(\sigma \)-algebra generated by the diffusion process \(\{X_r, s\le r\le t, X_s=x\};\)
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\({\mathbb {E}}_{t}^{s,x}[\cdot ]:\) conditional expectation under \(\mathcal {F}_t^{s,x},\) i.e., \({\mathbb {E}}_{t}^{s,x}[\cdot |\mathcal {F}_t^{s,x}];\)
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\(C_b^k:\) the set of continuous functions with uniformly bounded derivatives up to order k;
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\(C^{k_1,k_2}:\) the set of functions with continuous partial derivatives \(\frac{\partial }{\partial t}\) and \(\frac{\partial }{\partial x}\) up to \(k_1\) and \(k_2,\) respectively;
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\(C_L:\) the set of uniformly Lipschitz continuous function with respect to the spatial variables;
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\(C^{\frac{1}{2}}_L:\) the subset of \(C_L\) such that its element is Hölder-\(\frac{1}{2}\) continuous with respect to time, with uniformly bounded Lipschitz and Hölder constants.
3 Calculation of the Weights in the FDM for Approximating Derivative
3.1 Combination of Two Temporal Points
\(\alpha _{k,i}\varDelta t\) | \(i=0\) | \(i=1\) | \(i=2\) | \(i=3\) | \(i=4\) | \(i=5\) | \(i=6\) | \(i=7\) |
---|---|---|---|---|---|---|---|---|
\(k=1\) | \(-\frac{1}{2}\) | \(\frac{1}{2}\) | ||||||
\(k=2\) | \(-1\) | \(\frac{3}{2}\) | \(-\frac{1}{2}\) | |||||
\(k=3\) | \(-\frac{17}{12}\) | \(\frac{11}{4}\) | \(-\frac{7}{4}\) | \(\frac{5}{12}\) | ||||
\(k=4\) | \(-\frac{7}{4}\) | \(\frac{49}{12}\) | \(-\frac{15}{4}\) | \(\frac{7}{4}\) | \(-\frac{1}{3}\) | |||
\(k=5\) | \(-\frac{121}{60}\) | \(\frac{65}{12}\) | \(-\frac{77}{12}\) | \(\frac{53}{12}\) | \(-\frac{5}{3}\) | \(\frac{4}{15}\) | ||
\(k=6\) | \(-\frac{67}{30}\) | \(\frac{403}{60}\) | \(-\frac{29}{3}\) | \(\frac{35}{4}\) | \(-\frac{59}{12}\) | \(\frac{47}{30}\) | \(-\frac{13}{60}\) | |
\(k=7\) | \(-\frac{2027}{840}\) | \(\frac{319}{40}\) | \(-\frac{1613}{120}\) | \(\frac{361}{24}\) | \(-\frac{269}{24}\) | \(\frac{641}{120}\) | \(-\frac{59}{40}\) | \(\frac{151}{840}\) |
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\(|\lambda _{k,j}|\le 1,\)
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\(P^{'}(\lambda _{k,j})\ne 0\) if \(|\lambda _{k,j}|=1\) (simple roots).
k | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|
\(\max \left( \left| \lambda _{k,j}\right| \right) \) | 0.5000 | 0.5424 | 0.6344 | 0.7438 | 0.8636 | 0.9915 | 1.1264 |
3.2 Combination of Four Temporal Points
\(\alpha _{k,i}\varDelta t\) | \(i=0\) | \(i=1\) | \(i=2\) | \(i=3\) | \(i=4\) | \(i=5\) | \(i=6\) | \(i=7\) | \(i=8\) | \(i=9\) |
---|---|---|---|---|---|---|---|---|---|---|
\(k=1\) | \(-\frac{1}{4}\) | \(\frac{1}{4}\) | ||||||||
\(k=2\) | \(-\frac{3}{4}\) | \(\frac{5}{4}\) | \(-\frac{1}{2}\) | |||||||
\(k=3\) | \(-\frac{4}{3}\) | 3 | \(-\frac{9}{4}\) | \(\frac{7}{12}\) | ||||||
\(k=4\) | \(-\frac{11}{6}\) | 5 | \(-\frac{21}{4}\) | \(\frac{31}{12}\) | \(-\frac{1}{2}\) | |||||
\(k=5\) | \(-\frac{87}{40}\) | \(\frac{161}{24}\) | \(-\frac{26}{3}\) | 6 | \(-\frac{53}{24}\) | \(\frac{41}{120}\) | ||||
\(k=6\) | \(-\frac{19}{8}\) | \(\frac{949}{120}\) | \(-\frac{35}{3}\) | 10 | \(-\frac{125}{24}\) | \(\frac{37}{24}\) | \(-\frac{1}{5}\) | |||
\(k=7\) | \(-\frac{419}{168}\) | \(\frac{1049}{120}\) | \(-\frac{85}{6}\) | \(\frac{85}{6}\) | \(-\frac{75}{8}\) | \(\frac{97}{24}\) | \(-\frac{31}{30}\) | \(\frac{5}{42}\) | ||
\(k=8\) | \(-\frac{145}{56}\) | \(\frac{2661}{280}\) | \(-\frac{101}{6}\) | \(\frac{39}{2}\) | \(-\frac{385}{24}\) | \(\frac{75}{8}\) | \(-\frac{37}{10}\) | \(\frac{37}{42}\) | -\(\frac{2}{21}\) | |
\(k=9\) | \(-\frac{6781}{2520}\) | \(\frac{2917}{280}\) | \(-\frac{4303}{210}\) | \(\frac{841}{30}\) | \(-\frac{3461}{120}\) | \(\frac{887}{40}\) | \(-\frac{367}{30}\) | \(\frac{953}{210}\) | -\(\frac{106}{105}\) | \(\frac{32}{315}\) |
k | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|
\(\max \left( \left| \lambda _{k,j}\right| \right) \) | 0.6667 | 0.6614 | 0.6875 | 0.7104 | 0.7224 | 0.7376 | 0.8134 | 0.9931 | 1.2286 |
4 The Semi-discrete Multi-step Scheme for Decoupled FBSDEs
4.1 Two Reference ODEs
4.2 The Semi-discrete Scheme
5 The Fully Discrete Multi-step Scheme for Decoupled FBSDEs
6 Numerical Schemes for Coupled FBSDEs
7 Numerical Experiments
Scheme 2 |
\(N_T=16\)
|
\(N_T=20\)
|
\(N_T=24\)
|
\(N_T=28\)
|
\(N_T=32\)
| CR | |
---|---|---|---|---|---|---|---|
\(k=3\)
|
\(|Y^0-Y_0|\)
| 4.717e-06 | 2.613e-06 | 1.569e-06 | 1.015e-06 | 6.905e-07 | 2.78 |
\(|Z^0-Z_0|\)
| 2.547e-05 | 1.552e-05 | 1.009e-05 | 6.889e-06 | 4.903e-06 | 2.39 | |
RT | 0.37 | 0.49 | 0.65 | 0.91 | 1.19 | ||
\(k=4\)
|
\(|Y^0-Y_0|\)
| 6.871e-07 | 3.152e-07 | 1.629e-07 | 9.240e-08 | 5.618e-08 | 3.61 |
\(|Z^0-Z_0|\)
| 6.879e-06 | 3.097e-06 | 1.595e-06 | 9.027e-07 | 5.488e-07 | 3.65 | |
RT | 0.39 | 0.61 | 0.85 | 1.12 | 1.41 | ||
\(k=5\)
|
\(|Y^0-Y_0|\)
| 5.623e-08 | 2.077e-08 | 9.011e-09 | 4.355e-09 | 2.343e-09 | 4.59 |
\(|Z^0-Z_0|\)
| 6.522e-07 | 2.427e-07 | 1.047e-07 | 5.016e-08 | 2.704e-08 | 4.60 | |
RT | 0.46 | 0.69 | 0.95 | 1.25 | 1.58 | ||
\(k=6\)
|
\(|Y^0-Y_0|\)
| 3.549e-09 | 1.073e-09 | 3.929e-10 | 1.623e-10 | 7.549e-11 | 5.56 |
\(|Z^0-Z_0|\)
| 5.658e-08 | 1.632e-08 | 6.000e-09 | 2.519e-09 | 1.168e-09 | 5.59 | |
RT | 0.52 | 0.85 | 1.26 | 1.74 | 2.10 | ||
\(k=7\)
|
\(|Y^0-Y_0|\)
| 2.156e-10 | 4.809e-11 | 1.457e-11 | 5.075e-12 | 2.019e-12 | 6.73 |
\(|Z^0-Z_0|\)
| 6.349e-09 | 1.556e-09 | 4.796e-10 | 1.749e-10 | 7.147e-11 | 6.47 | |
RT | 0.60 | 1.10 | 1.65 | 2.15 | 2.79 | ||
\(k=8\)
|
\(|Y^0-Y_0|\)
| 6.025e-11 | 8.573e-12 | 3.292e-12 | 7.027e-13 | 4.868e-13 | 7.10 |
\(|Z^0-Z_0|\)
| 1.029e-09 | 1.934e-10 | 5.811e-11 | 1.459e-11 | 6.696e-12 | 7.35 | |
RT | 0.62 | 1.17 | 1.73 | 2.37 | 3.11 | ||
\(k=9\)
|
\(|Y^0-Y_0|\)
| 2.315e-11 | 4.131e-12 | 9.073e-13 | 2.169e-13 | 2.398e-14 | 9.55 |
\(|Z^0-Z_0|\)
| 3.672e-10 | 5.073e-11 | 1.184e-11 | 2.528e-12 | 5.760e-13 | 9.19 | |
RT | 0.69 | 1.32 | 2.04 | 2.83 | 3.74 |
Scheme 3 |
\(N_T=13\)
|
\(N_T=15\)
|
\(N_T=17\)
|
\(N_T=19\)
|
\(N_T=21\)
| CR | |
---|---|---|---|---|---|---|---|
\(k=3\)
|
\(|Y^0-Y_0|\)
| 2.269e-04 | 1.398e-04 | 9.186e-05 | 6.025e-05 | 4.336e-05 | 3.47 |
\(|Z^0-Z_0|\)
| 1.562e-04 | 1.143e-04 | 6.555e-05 | 4.431e-05 | 3.359e-05 | 3.36 | |
RT | 3.89 | 4.79 | 5.78 | 6.53 | 7.84 | ||
\(k=4\)
|
\(|Y^0-Y_0|\)
| 9.569e-06 | 7.654e-06 | 4.799e-06 | 2.734e-06 | 1.571e-06 | 3.83 |
\(|Z^0-Z_0|\)
| 1.447e-04 | 1.126e-04 | 6.268e-05 | 3.009e-05 | 1.223e-05 | 5.14 | |
RT | 4.51 | 5.73 | 6.79 | 8.12 | 10.43 | ||
\(k=5\)
|
\(|Y^0-Y_0|\)
| 4.773e-07 | 1.740e-07 | 3.835e-08 | 6.464e-08 | 2.325e-08 | 5.96 |
\(|Z^0-Z_0|\)
| 2.433e-06 | 6.129e-07 | 2.215e-08 | 1.988e-07 | 2.968e-07 | 4.89 | |
RT | 21.07 | 28.68 | 35.43 | 41.14 | 53.55 | ||
\(k=6\)
|
\(|Y^0-Y_0|\)
| 4.469e-08 | 2.509e-08 | 1.361e-08 | 6.572e-09 | 3.257e-09 | 5.47 |
\(|Z^0-Z_0|\)
| 4.257e-07 | 3.145e-07 | 1.629e-07 | 7.017e-08 | 2.121e-08 | 6.15 | |
RT | 33.49 | 48.76 | 63.83 | 80.93 | 99.07 | ||
\(k=7\)
|
\(|Y^0-Y_0|\)
| 6.510e-10 | 4.904e-10 | 1.536e-11 | 4.256e-11 | 3.250e-11 | 7.20 |
\(|Z^0-Z_0|\)
| 1.218e-08 | 1.207e-08 | 8.072e-10 | 3.704e-09 | 2.249e-10 | 7.56 | |
RT | 34.39 | 53.76 | 75.34 | 97.09 | 120.69 | ||
\(k=8\)
|
\(|Y^0-Y_0|\)
| 1.876e-10 | 8.477e-11 | 3.098e-11 | 1.028e-11 | 5.961e-12 | 7.51 |
\(|Z^0-Z_0|\)
| 8.841e-09 | 2.877e-09 | 1.727e-10 | 7.179e-11 | 5.141e-10 | 8.19 | |
RT | 35.76 | 90.17 | 149.87 | 172.70 | 247.68 | ||
\(k=9\)
|
\(|Y^0-Y_0|\)
| 1.926e-11 | 5.744e-12 | 5.828e-13 | 9.910e-13 | 3.098e-14 | 12.10 |
\(|Z^0-Z_0|\)
| 2.502e-10 | 1.526e-10 | 3.190e-11 | 3.039e-11 | 2.234e-12 | 9.07 | |
RT | 55.19 | 173.80 | 275.15 | 384.02 | 503.85 |
Scheme 2 |
\(N_T=16\)
|
\(N_T=20\)
|
\(N_T=24\)
|
\(N_T=28\)
|
\(N_T=32\)
| CR | |
---|---|---|---|---|---|---|---|
\(k=3\)
|
\(|Y^{0,1}-Y^1_0|\)
| 4.308e-03 | 2.447e-03 | 1.495e-03 | 9.726e-04 | 6.651e-04 | 2.70 |
\(|Y^{0,2}-Y^2_0|\)
| 3.896e-03 | 2.200e-03 | 1.340e-03 | 8.705e-04 | 5.949e-04 | 2.71 | |
\(|Z^{0,1}-Z^1_0|\)
| 3.887e-03 | 2.041e-03 | 1.201e-03 | 7.602e-04 | 5.110e-04 | 2.93 | |
\(|Z^{0,2}-Z^2_0|\)
| 2.980e-03 | 1.491e-03 | 8.465e-04 | 5.243e-04 | 3.451e-04 | 3.11 | |
RT | 34.65 | 53.68 | 74.87 | 92.42 | 130.14 | ||
\(k=4\)
|
\(|Y^{0,1}-Y^1_0|\)
| 8.829e-04 | 4.437e-04 | 2.417e-04 | 1.413e-04 | 8.759e-05 | 3.34 |
\(|Y^{0,2}-Y^2_0|\)
| 7.776e-04 | 3.845e-04 | 2.072e-04 | 1.202e-04 | 7.413e-05 | 3.39 | |
\(|Z^{0,1}-Z^1_0|\)
| 6.653e-04 | 2.556e-04 | 1.161e-04 | 5.899e-05 | 3.270e-05 | 4.35 | |
\(|Z^{0,2}-Z^2_0|\)
| 6.845e-04 | 2.659e-04 | 1.218e-04 | 6.268e-05 | 3.521e-05 | 4.29 | |
RT | 43.52 | 70.30 | 100.41 | 142.95 | 195.94 | ||
\(k=5\)
|
\(|Y^{0,1}-Y^1_0|\)
| 5.686e-05 | 2.367e-05 | 1.084e-05 | 5.445e-06 | 2.952e-06 | 4.27 |
\(|Y^{0,2}-Y^2_0|\)
| 5.131e-05 | 2.112e-05 | 9.605e-06 | 4.801e-06 | 2.595e-06 | 4.31 | |
\(|Z^{0,1}-Z^1_0|\)
| 7.703e-05 | 2.621e-05 | 1.067e-05 | 4.923e-06 | 2.517e-06 | 4.94 | |
\(|Z^{0,2}-Z^2_0|\)
| 7.237e-05 | 2.402e-05 | 9.582e-06 | 4.363e-06 | 2.201e-06 | 5.04 | |
RT | 50.72 | 79.97 | 132.00 | 186.09 | 231.84 | ||
\(k=6\)
|
\(|Y^{0,1}-Y^1_0|\)
| 4.277e-06 | 1.706e-06 | 7.258e-07 | 3.354e-07 | 1.674e-07 | 4.68 |
\(|Y^{0,2}-Y^2_0|\)
| 3.857e-06 | 1.509e-06 | 6.336e-07 | 2.901e-07 | 1.437e-07 | 4.75 | |
\(|Z^{0,1}-Z^1_0|\)
| 5.569e-06 | 1.667e-06 | 5.803e-07 | 2.309e-07 | 1.016e-07 | 5.78 | |
\(|Z^{0,2}-Z^2_0|\)
| 5.531e-06 | 1.655e-06 | 5.791e-07 | 2.313e-07 | 1.026e-07 | 5.76 | |
RT | 64.97 | 110.97 | 173.50 | 246.96 | 337.71 | ||
\(k=7\)
|
\(|Y^{0,1}-Y^1_0|\)
| 5.753e-07 | 1.843e-07 | 6.513e-08 | 2.568e-08 | 1.115e-08 | 5.70 |
\(|Y^{0,2}-Y^2_0|\)
| 5.312e-07 | 1.669e-07 | 5.836e-08 | 2.283e-08 | 9.864e-09 | 5.76 | |
\(|Z^{0,1}-Z^1_0|\)
| 9.466e-07 | 2.291e-07 | 7.027e-08 | 2.528e-08 | 1.027e-08 | 6.52 | |
\(|Z^{0,2}-Z^2_0|\)
| 9.091e-07 | 2.157e-07 | 6.529e-08 | 2.326e-08 | 9.404e-09 | 6.59 | |
RT | 73.89 | 139.46 | 234.34 | 339.44 | 453.70 | ||
\(k=8\)
|
\(|Y^{0,1}-Y^1_0|\)
| 2.586e-08 | 1.080e-08 | 4.077e-09 | 1.592e-09 | 6.652e-10 | 5.30 |
\(|Y^{0,2}-Y^2_0|\)
| 2.384e-08 | 9.742e-09 | 3.623e-09 | 1.400e-09 | 5.802e-10 | 5.39 | |
\(|Z^{0,1}-Z^1_0|\)
| 6.238e-08 | 1.548e-08 | 4.186e-09 | 1.299e-09 | 4.791e-10 | 7.06 | |
\(|Z^{0,2}-Z^2_0|\)
| 6.290e-08 | 1.534e-08 | 4.077e-09 | 1.255e-09 | 4.579e-10 | 7.14 | |
RT | 84.65 | 177.96 | 299.48 | 436.73 | 592.02 | ||
\(k=9\)
|
\(|Y^{0,1}-Y^1_0|\)
| 8.294e-09 | 2.371e-09 | 6.964e-10 | 2.250e-10 | 8.045e-11 | 6.70 |
\(|Y^{0,2}-Y^2_0|\)
| 7.739e-09 | 2.153e-09 | 6.229e-10 | 1.991e-10 | 6.989e-11 | 6.80 | |
\(|Z^{0,1}-Z^1_0|\)
| 2.771e-08 | 5.329e-09 | 1.262e-09 | 3.491e-10 | 1.245e-10 | 7.84 | |
\(|Z^{0,2}-Z^2_0|\)
| 2.748e-08 | 5.202e-09 | 1.217e-09 | 3.350e-10 | 9.998e-11 | 8.08 | |
RT | 91.19 | 210.99 | 356.54 | 534.76 | 760.94 |