The DOI matches its Crossref record; 6 of the 6 references it cites were found in a registry or an index.
Numerical Treatment of Stochastic Differential Equations
Abstract
We define general Runge–Kutta approximations for the solution of stochastic differential equations (sde). These approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift. The explicit form of the correction term is given. Concerning the order of convergence, we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than $O(h^3 )$. This order is attained, e.g., by the stochastic analogue of Heun’s method. In the n-dimensional case, the highest order of convergence is in general only $O(h^2 )$, attained by Euler’s method. The order $O(h^3 )$ can only be reached if $(\nabla _x \sigma ^r )\sigma ^s = (\nabla _x \sigma ^s )\sigma ^r $ for the diffusion matrix.
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Cites
On the Convergence of Ordinary Integrals to Stochastic Integrals
Wong, E., & Zakai, M. (1965) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177699916 · Cited by 686
Stochastic Differential Equations: Theory and Applications
Baxendale, P. H., & Lototsky, S. V. (2007) · Interdisciplinary mathematical sciences · doi:10.1142/6453 · Cited by 605
Continuous Markov processes and stochastic equations
Maruyama, G. (1955) · Rendiconti del Circolo Matematico di Palermo · doi:10.1007/bf02846028 · Cited by 540
The numerical solution of stochastic differential equations
Kloeden, P. E., & Pearson, R. A. (1977) · The Journal of the Australian Mathematical Society. Series B. Applied Mathematics · doi:10.1017/s0334270000001405 · Cited by 264
Stochastic Calculus and Stochastic Models
(1974) · Elsevier eBooks · doi:10.1016/c2013-0-07439-7 · Cited by 240
Riemann-Stieltjes approximations of stochastic integrals
Wong, E., & Zakai, M. (1969) · Probability Theory and Related Fields · doi:10.1007/bf00531642 · Cited by 96
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Langevin-dynamics study of the dynamical properties of small magnetic particles
García-Palacios, J. L., & Lázaro, F. J. (1998) · Physical Review B · doi:10.1103/physrevb.58.14937 · Cited by 675
Stochastic Parameterization: Toward a New View of Weather and Climate Models
Berner, J., Achatz, U., Batté, L., Bengtsson, L., Cámara, A. D. L., Christensen, H. M., Colangeli, M., Coleman, D. R. B., Crommelin, D., Dolaptchiev, S. I., et al. (2017) · Bulletin of the American Meteorological Society · doi:10.1175/bams-d-15-00268.1 · Cited by 411
Option Pricing when the Variance is Changing
Johnson, H., & Shanno, D. (1987) · The Journal of Financial and Quantitative Analysis · doi:10.2307/2330709 · Cited by 375
An introduction to numerical methods for stochastic differential equations
Platen, E. (1999) · Acta Numerica · doi:10.1017/s0962492900002920 · Cited by 273
Wavelet-Improved Score-Based Generative Model for Medical Imaging
Wu, W., Wang, Y., Liu, Q., Wang, G., & Zhang, J. (2024) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2023.3325824 · Cited by 105
Metadata
- Authors
- Rüemelin, W.
- Venue
- SIAM Journal on Numerical Analysis
- Published
- 1982-06
- Volume
- 19
- Issue
- 3
- Pages
- 604-613
- ISSN
- 0036-1429 (ISSN-L)
- 1095-7170
- OpenAlex
- W1991239884
- Cited by
- 395
- References
- 6
- FWCI
- 4.66
- Type
- journal article
- Language
- en
- Open access
- closed
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