Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling points t(i) is an element of [t(0), t(J)] for i = 0, 1, . . . , J of the solution. Two examples are provided.

The role of the term spread in an augmented Taylor rule: An empirical investigation

Using US data for the period 1967:5-2002:4, this paper empirically investigates the performance of an augmented version of the Taylor rule (ATR) that (i) allows for the presence of switching regimes, (ii) considers the long-short term spread in addition to the typical variables, (iii) uses an alternative monthly indicator of general economic activity suggested by Stock and Watson (1999), and (iv) considers interest rate smoothing. The estimation results show the existence of switching regimes, one characterized by low volatility and the other by high volatility. Moreover, the scale of the responses of the Federal funds rate to movements in the term spread, inflation and the economic activity index depend on the regime. The estimation results also show robust empirical evidence that the ATR has been more stable during the term of office of Chairman Greenspan than in the pre-Greenspan period. However, a closer look at the Greenspan period shows the existence of two alternative regimes and that the response of the Fed funds rate to inflation has not been significant during this period once the term spread is considered.

Valuing Expansions of the Electricity Transmission Network under Uncertainty: The Binodal Case

Transmission investments are currently needed to meet an increasing electricity demand, to address security of supply concerns, and to reach carbon-emissions targets. A key issue when assessing the benefits from an expanded grid concerns the valuation of the uncertain cash flows that result from the expansion. We propose a valuation model that accommodates both physical and economic uncertainties following the Real Options approach. It combines optimization techniques with Monte Carlo simulation. We illustrate the use of our model in a simplified, two-node grid and assess the decision whether to invest or not in a particular upgrade. The generation mix includes coal-and natural gas-fired stations that operate under carbon constraints. The underlying parameters are estimated from observed market data.