An adaptive Euler-Maruyama scheme for SDEs: Convergence and stability

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2007-01-01

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Abstract

The understanding of adaptive algorithms for stochastic differential equations (SDEs) is an open area, where many issues related to both convergence and stability (long-time behaviour) of algorithms are unresolved. This paper considers a very simple adaptive algorithm, based on controlling only the drift component of a time step. Both convergence and stability are studied. The primary issue in the convergence analysis is that the adaptive method does not necessarily drive the time steps to zero with the user-input tolerance. This possibility must be quantified and shown to have low probability. The primary issue in the stability analysis is ergodicity. It is assumed that the noise is nondegenerate, so that the diffusion process is elliptic, and the drift is assumed to satisfy a coercivity condition. The SDE is then geometrically ergodic (averages converge to statistical equilibrium exponentially quickly). If the drift is not linearly bounded, then explicit fixed time step approximations, such as the Euler-Maruyama scheme, may fail to be ergodic. In this work, it is shown that the simple adaptive time-stepping strategy cures this problem. In addition to proving ergodicity, an exponential moment bound is also proved, generalizing a result known to hold for the SDE itself. © The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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10.1093/imanum/drl032

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Lamba, H, JC Mattingly and AM Stuart (2007). An adaptive Euler-Maruyama scheme for SDEs: Convergence and stability. IMA Journal of Numerical Analysis, 27(3). pp. 479–506. 10.1093/imanum/drl032 Retrieved from https://hdl.handle.net/10161/24760.

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Mattingly

Jonathan Christopher Mattingly

Kimberly J. Jenkins Distinguished University Professor of New Technologies

Jonathan Christopher  Mattingly grew up in Charlotte, NC where he attended Irwin Ave elementary and Charlotte Country Day.  He graduated from the NC School of Science and Mathematics and received a BS is Applied Mathematics with a concentration in physics from Yale University. After two years abroad with a year spent at ENS Lyon studying nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to attend Princeton University where he obtained a PhD in Applied and Computational Mathematics in 1998. After 4 years as a Szego assistant professor at Stanford University and a year as a member of the IAS in Princeton, he moved to Duke in 2003. He is currently a Professor of Mathematics and of Statistical Science.

His expertise is in the longtime behavior of stochastic system including randomly forced fluid dynamics, turbulence, stochastic algorithms used in molecular dynamics and Bayesian sampling, and stochasticity in biochemical networks.

Since 2013 he has also been working to understand and quantify gerrymandering and its interaction of a region's geopolitical landscape. This has lead him to testify in a number of court cases including in North Carolina, which led to the NC congressional and both NC legislative maps being deemed unconstitutional and replaced for the 2020 elections. 

He is the recipient of a Sloan Fellowship and a PECASE CAREER award.  He is also a fellow of the IMS and the AMS. He was awarded the Defender of Freedom award by  Common Cause for his work on Quantifying Gerrymandering.



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