In this paper we describe a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence. Modified equations are derived for a variety of numerical methods, such as the Euler or the Milstein method. Existence of higher order modified equations is also discussed. In the case of linear SDEs, using the Gaussianity of the underlying...

The purpose of this paper is to measure the strong and weak order of convergence of both the Euler and Milstein schemes using a stochastic volatility model and an N−dimensional Exponential Brownian Motion Process (EBM). An exact solution is normally required to calculate the order of convergence, however there are none available for this volatility process. We propose a method to solve this pro...

This article studies an infinite dimensional analog of Milstein’s scheme for finite dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity type condition. This article introduces the infinite dimensional analog of this commutativity type condition and observes that a certain class ...

Jelena Stajic, J. N. Milstein, Qijin Chen, M. L. Chiofalo, M. J. Holland, and K. Levin James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309, USA Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA Clas...

Recently we reviewed the work of Nlels K. Jerne, C&ar Milstein, and Georges J.F. Kohler, who shared the 1984 Nobel Prize in medicine. 1 We also examined the work of Carlo Rubbia and Simon van der Meer, joint recipients of the physics prize, and that of R. Bruce Merrifield, who was awarded the chemistry prize.z In this essay we review the achievements of the prizewinners in economics and literat...

Robert A. Hahn, PhD, MPH, John A. Knopf, MPH, Sandra Jo Wilson, PhD, Benedict I. Truman, MD, MPH, Bobby Milstein, PhD, MPH, Robert L. Johnson, MD, Jonathan E. Fielding, MD, MPH, MBA, Carles J.M. Muntaner, MD, PhD, Camara Phyllis Jones, MD, PhD, MPH, Mindy T. Fullilove, MD, MS, Regina Davis Moss, PhD, MPH, MCHES, Erin Ueffing, BHSc, MHSc, Pete C. Hunt, MPH, and the Community Preventive Services ...

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDEs). The scheme has convergence order 1. In order to establish the scheme, we prove an infinitedimensional Itô formula for “tame” functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires...

(whom he had met at student political meetings in Buenos Aires) was not similarly treated and had to tag along in a third class cabin. Despite this inauspicious start, the Milsteins maintained a devoted partnership that lasted till César's death with the two occasionally collaborating on aspects of immunoglobulin biochemistry. Milstein's PhD with Dixon (his second doctorate) was on the activati...