In this paper we present a numerical scheme for stochastic differential equations based upon the Wiener chaos expansion. The approximation of a square integrable stochastic differential equation is obtained by cutting off the infinite chaos expansion in chaos order and in number of basis elements. We derive an explicit upper bound for the

We consider a one-dimensional continuous stochastic process

In this paper we take a different route and propose to use the Wiener chaos expansion (also called polynomial chaos in the literature) to approximate the solution of the SDE (

The paper is structured as follows. In Section

In this section we introduce some basic concepts of Malliavin calculus. The interested readers are referred to [

Now, we introduce multiple stochastic integrals of order

Next, we introduce the notion of Malliavin derivative and its adjoint operator. We define the set of smooth random variables via

The operator

We start with the analysis of the Wiener chaos expansion introduced in (

Using the SDE (

We remark that the propagator system (

(i) (

(ii) (

For a general specification of the drift coefficient

The strong approximation error associated with the truncation (

Let us give some remarks about the statement (

Let us explicitly compute the last terms of the upper bound (

(i) (

(ii) (

Throughout the proofs

We show the assertion of Proposition

Now, we proceed with the proof of the main result of Theorem

Absolute errors of the variance as calculated with different polynomial chaos expansions for the Geometric Brownian Motion,

To get an idea of the performance of the polynomial chaos expansion for different bases and truncations, we apply the trigonometric basis from Example

Figure

Table

The information contained in the coefficient functions

Hence, we can define

Let

Let

An even more rigorous reduction of the number of coefficients included in the propagator system can be achieved by using a

Let

Considering again the setting of the previous example with

Table

Computational times in seconds and error estimates for trigonometric and Haar bases from Example

Trigonometric basis | Haar basis | ||||||||

type | time | time | |||||||

2 | 1 | 3 | full | 0.04 | 6.04 | 6.04 | 0.03 | 5.31 | 5.31 |

4 | 1 | 5 | full | 0.02 | 5.68 | 5.68 | 0.02 | 5.31 | 5.31 |

8 | 1 | 9 | full | 0.22 | 5.49 | 5.49 | 0.04 | 5.30 | 5.30 |

16 | 1 | 17 | full | 0.05 | 5.40 | 5.40 | 0.19 | 5.31 | 5.31 |

32 | 1 | 33 | full | 0.17 | 5.35 | 5.35 | 1.31 | 5.30 | 5.30 |

64 | 1 | 65 | full | 1.28 | 5.33 | 5.33 | 10.12 | 5.31 | 5.31 |

2 | 2 | 6 | full | 0.02 | 3.04 | 3.04 | 0.02 | 1.61 | 1.83 |

4 | 2 | 15 | full | 0.03 | 2.35 | 2.35 | 0.05 | 1.61 | 1.76 |

8 | 2 | 45 | full | 0.20 | 1.98 | 1.98 | 0.82 | 1.61 | 1.69 |

8 | 2 | 41 | sp^{1} |
0.31 | 1.99 | 1.99 | 0.24 | 1.61 | 1.69 |

8 | 2 | 19 | sp^{2} |
0.04 | 2.16 | 2.16 | 0.07 | 1.61 | 1.72 |

16 | 2 | 153 | full | 3.85 | 1.80 | 1.80 | 18.61 | 1.61 | 1.65 |

16 | 2 | 141 | sp^{3} |
1.00 | 1.80 | 1.80 | 4.60 | 1.61 | 1.65 |

16 | 2 | 27 | sp^{4} |
0.05 | 2.07 | 2.07 | 0.19 | 1.61 | 1.67 |

32 | 2 | 561 | full | 86.83 | 1.71 | 1.71 | 554.61 | 1.61 | 1.63 |

32 | 2 | 537 | sp^{5} |
22.80 | 1.71 | 1.71 | 143.90 | 1.61 | 1.63 |

32 | 2 | 69 | sp^{6} |
0.31 | 1.84 | 1.84 | 3.22 | 1.61 | 1.64 |

64 | 2 | 2145 | full | 2189.70 | 1.66 | 1.66 | 17234.51 | 1.61 | 1.62 |

2 | 3 | 10 | full | 0.02 | 2.15 | 2.15 | 0.02 | 0.38 | 1.35 |

4 | 3 | 35 | full | 0.10 | 1.29 | 1.29 | 0.19 | 0.38 | 1.01 |

8 | 3 | 165 | full | 2.76 | 0.84 | 0.84 | 10.58 | 0.38 | 0.76 |

8 | 3 | 127 | sp^{7} |
0.53 | 0.85 | 0.85 | 1.77 | 0.37 | 0.76 |

8 | 3 | 37 | sp^{8} |
0.06 | 1.11 | 1.11 | 0.13 | 0.38 | 0.86 |

16 | 3 | 969 | full | 200.34 | 0.61 | 0.61 | 1070.17 | 0.38 | 0.58 |

16 | 3 | 763 | sp^{9} |
45.61 | 0.62 | 0.62 | 169.16 | 0.38 | 0.59 |

16 | 3 | 45 | sp^{10} |
0.14 | 1.02 | 1.02 | 0.39 | 0.38 | 0.76 |

2 | 4 | 15 | full | 0.02 | 1.94 | 1.94 | 0.02 | 0.07 | 1.32 |

4 | 4 | 70 | full | 0.30 | 1.04 | 1.04 | 0.79 | 0.07 | 0.87 |

8 | 4 | 495 | full | 26.99 | 0.57 | 0.57 | 117.55 | 0.07 | 0.56 |

8 | 4 | 303 | sp^{11} |
4.34 | 0.57 | 0.57 | 12.30 | 0.07 | 0.52 |

8 | 4 | 32 | sp^{12} |
0.05 | 0.96 | 0.96 | 0.12 | 0.07 | 0.75 |

16 | 4 | 4845 | full | 5986.44 | 0.32 | 0.32 | 30591.94 | 0.07 | 0.33 |

16 | 4 | 40 | sp^{13} |
0.08 | 0.87 | 0.87 | 0.36 | 0.07 | 0.67 |

32 | 4 | 92 | sp^{14} |
0.50 | 0.59 | 0.59 | 3.56 | 0.07 | 0.45 |

2 | 5 | 21 | full | 0.03 | 1.91 | 1.91 | 0.03 | 0.01 | 1.27 |

4 | 5 | 126 | full | 0.92 | 1.00 | 1.00 | 2.04 | 0.01 | 0.87 |

8 | 5 | 1287 | full | 192.98 | 0.51 | 0.51 | 855.07 | 0.01 | 0.53 |

8 | 5 | 599 | sp^{15} |
10.04 | 0.51 | 0.51 | 50.03 | 0.01 | 0.55 |

8 | 5 | 36 | sp^{16} |
0.03 | 0.92 | 0.92 | 0.11 | 0.01 | 0.74 |

16 | 5 | 20349 | full | 120469.96 | 0.26 | 0.26 | 600591.17 | 0.01 | 0.28 |

16 | 5 | 44 | sp^{17} |
0.06 | 0.83 | 0.83 | 0.28 | 0.01 | 0.65 |

32 | 5 | 98 | sp^{18} |
0.51 | 0.55 | 0.55 | 3.60 | 0.01 | 0.42 |

List of first and second order sparse indices used in Section

symbol | order | index |
|||

sp^{1} |
2 | 8 | 1 | 41 | |

sp^{2} |
2 | 8 | 2 | 19 | |

sp^{3} |
2 | 16 | 1 | 141 | |

sp^{4} |
2 | 16 | 2 | 27 | |

sp^{5} |
2 | 32 | 1 | 537 | |

sp^{6} |
2 | 32 | 2 | 69 | |

sp^{7} |
3 | 8 | 1 | 127 | |

sp^{8} |
3 | 8 | 2 | 37 | |

sp^{9} |
3 | 16 | 1 | 763 | |

sp^{10} |
3 | 16 | 2 | 45 | |

sp^{11} |
4 | 8 | 1 | 303 | |

sp^{12} |
4 | 8 | 2 | 32 | |

sp^{13} |
4 | 16 | 2 | 40 | |

sp^{14} |
4 | 32 | 2 | 92 | |

sp^{15} |
5 | 8 | 1 | 599 | |

sp^{16} |
5 | 8 | 2 | 36 | |

sp^{17} |
5 | 16 | 2 | 44 | |

sp^{18} |
5 | 32 | 2 | 98 | |