A weak trapezoidal method for a class of stochastic differential equations
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2011-03-01
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We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally in the study of population processes and chemical reaction kinetics. We show that the method constructs paths that are second order accurate in the weak sense. The method is simpler than many second order methods in that it neither requires the construction of iterated It̂o integrals nor the evaluation of any derivatives. The method consists of two steps. In the first an explicit Euler step is used to take a fractional step. The resulting fractional point is then combined with the initial point to obtain a higher order, trapezoidal like, approximation. The higher order of accuracy stems from the fact that both the drift and the quadratic variation of the underlying SDE are approximated to second order. © 2011 International Press.
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Jonathan Christopher Mattingly
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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