Browsing by Subject "stability"
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Item Open Access An adaptive Euler-Maruyama scheme for SDEs: Convergence and stability(IMA Journal of Numerical Analysis, 2007-01-01) Lamba, H; Mattingly, JC; Stuart, AMThe 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.Item Open Access Interdependent Utility and Truthtelling in Two-Sided Matching(Economic Research Initiatives at Duke (ERID) Working Paper, 2013-07-18) Wang, XYMechanisms which implement stable matchings are often observed to work well in practice, even in environments where the stable outcome is not unique, information is complete, and the number of players is small. Why might individuals refrain from strategic manipulation, even when the complexity cost of manipulation is low? I study a two-sided, one-to-one matching problem with no side transfers, where utility is interdependent in the following intuitive sense: an individual's utility from a match depends not only on her preference ranking of her assigned partner, but also on that partner's ranking of her. I show that, in a world of complete information and linear interdependence, a unique stable matching emerges, and is attained by a modified Gale-Shapley deferred acceptance algorithm. As a result, a stable rule supports truth-telling as an equilibrium strategy. Hence, these results offer a new intuition for why stable matching mechanisms seem to work well in practice, despite their theoretic manipulability: individuals may value being liked.