Browsing by Subject "stochastic differential equations"
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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 Noise Driven Transitions between Stable Equilibria in Stochastic Dynamical Systems(2011-05-13) Daniels, KaitlinIn this paper we examine two specific models of dynamical systems in which noise plays a central role. The first is a stochastic differential equation (SDE) modeling a particle in a potential well; the second is a simplified version of the Morris-Lecar model of a neuron. In each case, we consider both the underlying deterministic dynamical system, which is a governed by an ordinary differential equation, as well as the randomly-perturbed dynamical system, whose solution is a stochastic process satisfying a stochastic integral equation. We investigate the how perturbations of the drift functions influence transitions between stable equilibria and what effects such perturbations may have on exit times. We compare the results from computer simulations to analytical derivations of expected exit times. These results contribute to the understanding of the forces driving transitions between stable equilibria in perturbed dynamical systems.Item Open Access Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations(Communications in Contemporary Mathematics, 2005-10-01) Bakhtin, Y; Mattingly, JCWe explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier-Stokes equation and stochastic Ginsburg-Landau equation. © World Scientific Publishing Company.Item Open Access Time-Scaled Stochastic Input to Biochemical Reaction Networks(2010) Thomas, Rachel LeeBiochemical reaction networks with a sufficiently large number of molecules may be represented as systems of differential equations. Many networks receive inputs that fluctuate continuously in time. These networks may never settle down to a static equilibrium and are of great interest both mathematically and biologically. Biological systems receive inputs that vary on multiple time scales. Hormonal and neural inputs vary on a scale of seconds or minutes; inputs from meals and circadian rhythms vary on a scale of hours or days; and long term environmental changes (such as diet, disease, and pollution) vary on a scale of years. In this thesis, we consider the limiting behavior of networks in which the input is on a different time scale compared to the reaction kinetics within the network.
We prove analytic results of how the variance of reaction rates within a system compares to the variance of the input when the input is on a different time scale than the reaction kinetics within the network. We consider the behavior of simple chains, single species complex networks, reversible chains, and certain classes of non-linear systems with time-scaled stochastic input, as the input speeds up and slows down. In all cases, as the input fluctuates more and more quickly, the variance of species within the system approaches to zero. As the input fluctuates more and more slowly, the variance of the species approaches the variance of the input, up to a normalization factor.