Abstract
In this paper, we study general polynomial discretizations in backward and forward looking, and the preservation of stability properties. We apply these results to the Ramsey model [4]. Its discrete-time version is a hybrid discretizations of a backward-looking budget constraint and a forward-looking Euler equation. Saddle-path stability is a robust property under discretization.
Highlights
Continuous-time systems can be approximated by discretetime systems
Its discrete-time version is a hybrid discretizations of a backward-looking budget constraint and a forward-looking Euler equation
1) Let the steady state be a sink in continuous time (Figure 1). 1.1) If T02 < 4D0, the steady state is a sink in discrete time if h < hH1 and a source if hH1 < h
Summary
Continuous-time systems can be approximated by discretetime systems. In the spirit of Krivine, Lesne and Treiner [2], we bridge continuous and discrete-time dynamics through general polynomial discretizations. We want to show that the steady state is invariant to the step, the order and the direction of discretization and its continuous-time stability properties (sink, saddle, source) are preserved under a sufficiently small discretization step in any case (backward, forward or hybrid). Let us discretize the continuous-time dynamical system x = f x with f C1 by second-order Taylor polynomials, that is approximate the ith component of xn 1 with a quadratic form. If f is an analytic function, infinite-order backward or forward discretizations converge exactly to xn 1 xn and (1) and (2) hold with equality: xin 1 xin = In this case, the Taylor polynomials become a convergent series and the discretized dynamics represent exactly the continuous-time system whatever the step h. A discretization is a closer approximation of a continuous-time system when the step h is smaller or the order of discretization q higher. The dynamic properties of a continuous-time system can be preserved lowering h or increasing q
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