Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients
Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. However, the important case of superlinearly growing coefficients...
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| Published in | Foundations of computational mathematics Vol. 11; no. 6; pp. 657 - 706 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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New York
Springer-Verlag
01.12.2011
Springer Springer Nature B.V |
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| ISSN | 1615-3375 1615-3383 |
| DOI | 10.1007/s10208-011-9101-9 |
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| Abstract | Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. However, the important case of superlinearly growing coefficients has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth. |
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| AbstractList | Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. However, the important case of superlinearly growing coefficients has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth. Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. However, the important case of superlinearly growing coefficients has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth.[PUBLICATION ABSTRACT] |
| Audience | Academic |
| Author | Hutzenthaler, Martin Jentzen, Arnulf |
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| Cites_doi | 10.1214/aop/1029867124 10.1007/978-1-84800-048-3 10.1098/rspa.2010.0348 10.1007/978-3-642-57913-4 10.1007/s00032-002-0006-6 10.1137/S0036142901389530 10.1007/BF01303802 10.1214/aoap/1177004598 10.1007/BFb0093180 10.1137/S0036144500378302 10.1016/S0304-4149(02)00150-3 10.1137/040612026 10.1007/978-94-015-8455-5 10.1007/978-3-642-88264-7 10.1112/S1461157000001388 10.1007/978-3-662-12616-5 |
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| Keywords | Stochastic differential equations Monte Carlo Euler method 65C30 Local Lipschitz condition 65C05 Euler scheme 60H35 |
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| SubjectTerms | Applications of Mathematics Computational mathematics Computer Science Computer simulation Convergence Differential equations Economics Eulers equations Foundations Linear and Multilinear Algebras Math Applications in Computer Science Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Matrix Theory Monte Carlo methods Numerical Analysis Polynomials Stochastic models Stochasticity |
| Title | Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients |
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