The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate
Abstract The Euler scheme is one of the standard schemes to obtain numerical approximations of solutions of stochastic differential equations (SDEs). Its convergence properties are well known in the case of globally Lipschitz continuous coefficients. However, in many situations, relevant systems do...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-10-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-019-2361-4 |
| _version_ | 1829114755627876352 |
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| author | S. Göttlich K. Lux A. Neuenkirch |
| author_facet | S. Göttlich K. Lux A. Neuenkirch |
| author_sort | S. Göttlich |
| collection | DOAJ |
| description | Abstract The Euler scheme is one of the standard schemes to obtain numerical approximations of solutions of stochastic differential equations (SDEs). Its convergence properties are well known in the case of globally Lipschitz continuous coefficients. However, in many situations, relevant systems do not show a smooth behavior, which results in SDE models with discontinuous drift coefficient. In this work, we analyze the long time properties of the Euler scheme applied to SDEs with a piecewise constant drift and a constant diffusion coefficient and carry out intensive numerical tests for its convergence properties. We emphasize numerical convergence rates and analyze how they depend on the properties of the drift coefficient and the initial value. We also give theoretical interpretations of some of the arising phenomena. For application purposes, we study a rank-based stock market model describing the evolution of the capital distribution within the market and provide theoretical as well as numerical results on the long time ranking behavior. |
| first_indexed | 2024-12-12T16:02:49Z |
| format | Article |
| id | doaj.art-115dc6dfc21647d998d232a2095c0b43 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-12T16:02:49Z |
| publishDate | 2019-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-115dc6dfc21647d998d232a2095c0b432022-12-22T00:19:23ZengSpringerOpenAdvances in Difference Equations1687-18472019-10-012019112110.1186/s13662-019-2361-4The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rateS. Göttlich0K. Lux1A. Neuenkirch2Department of Mathematics, University of MannheimDepartment of Mathematics, University of MannheimDepartment of Mathematics, University of MannheimAbstract The Euler scheme is one of the standard schemes to obtain numerical approximations of solutions of stochastic differential equations (SDEs). Its convergence properties are well known in the case of globally Lipschitz continuous coefficients. However, in many situations, relevant systems do not show a smooth behavior, which results in SDE models with discontinuous drift coefficient. In this work, we analyze the long time properties of the Euler scheme applied to SDEs with a piecewise constant drift and a constant diffusion coefficient and carry out intensive numerical tests for its convergence properties. We emphasize numerical convergence rates and analyze how they depend on the properties of the drift coefficient and the initial value. We also give theoretical interpretations of some of the arising phenomena. For application purposes, we study a rank-based stock market model describing the evolution of the capital distribution within the market and provide theoretical as well as numerical results on the long time ranking behavior.http://link.springer.com/article/10.1186/s13662-019-2361-4Discontinuous driftNumerical schemesConvergence ratesExperimental study |
| spellingShingle | S. Göttlich K. Lux A. Neuenkirch The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate Advances in Difference Equations Discontinuous drift Numerical schemes Convergence rates Experimental study |
| title | The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate |
| title_full | The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate |
| title_fullStr | The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate |
| title_full_unstemmed | The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate |
| title_short | The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate |
| title_sort | euler scheme for stochastic differential equations with discontinuous drift coefficient a numerical study of the convergence rate |
| topic | Discontinuous drift Numerical schemes Convergence rates Experimental study |
| url | http://link.springer.com/article/10.1186/s13662-019-2361-4 |
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