A New Approach to Compare the Strong Convergence of the Milstein Scheme with the Approximate Coupling Method
Abstract
:1. Introduction
2. Preliminaries
- (i)
- (with probability one).
- (ii)
- The random variable is given by increment , for is normally distributed with mean zero and variance . Equivalently, .
- (iii)
- The increments and for
- Measurability: Let : and : be jointly Borel measurable in T.
- Lipschitz condition: There is a positive constant such that , and , for all u T] and .
- Growth condition: There is a constant such that , and , for all u T] and .
2.1. Approximation Schemes
2.2. Strong Order of Convergence
3. Comparison between Milstein and Approximate Coupling Methods
- [Error for approximate coupling]
- S=[ 400, 800, 1600, 3200, 6400];
- Error1=zeros(1,length1(S));
- for i=1:length1(S)
- Error1(1,i)=
- log(approximat2022(’YA’,[1; 0],1,S(1,i)));
- end
- h=1./S;
- fad1=log(h)
- plot(log(h), Error1)
4. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Steps | Step-Size | Absolute Error | Elapsed Time (Hour) | |
---|---|---|---|---|
1 | 400 | 0.0025 | 0.0692 | 0.126 |
2 | 800 | 0.0013 | 0.0353 | 0.433 |
3 | 1600 | 0.0006 | 0.0176 | 21.839 |
4 | 3200 | 0.00030 | 0.0091 | 102.817 |
5 | 6400 | 0.000150 | 0.0046 | 261.888 |
Steps | Step-Size | Absolute Error | Elapsed Time (Hour) | |
---|---|---|---|---|
1 | 400 | 0.0025 | 0.0029 | 0.05805 |
2 | 800 | 0.0013 | 0.0015 | 0.01163 |
3 | 1600 | 0.0006 | 0.00075 | 0.2325 |
4 | 3200 | 0.0003 | 0.00036 | 0.4664 |
5 | 6400 | 0.00015 | 0.00018 | 0.9344 |
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Alnafisah, Y. A New Approach to Compare the Strong Convergence of the Milstein Scheme with the Approximate Coupling Method. Fractal Fract. 2022, 6, 339. https://doi.org/10.3390/fractalfract6060339
Alnafisah Y. A New Approach to Compare the Strong Convergence of the Milstein Scheme with the Approximate Coupling Method. Fractal and Fractional. 2022; 6(6):339. https://doi.org/10.3390/fractalfract6060339
Chicago/Turabian StyleAlnafisah, Yousef. 2022. "A New Approach to Compare the Strong Convergence of the Milstein Scheme with the Approximate Coupling Method" Fractal and Fractional 6, no. 6: 339. https://doi.org/10.3390/fractalfract6060339