The implicit function theorem asserts that there exists a ball of nonzero radius within which one can express a certain subset of variables, in a system of equations, as functions of the remaining variables. We derive a lower bound for the radius of this ball in the case of Lipschitz maps. Under a sign-preserving condition, we prove that an implicit function exists in the case of a set of inequalities. Also in this case, we state an estimate for the size of the domain. An application to the local Lipschitz behavior of solution maps is discussed.

We describe the methods employed for the generation of possible scenarios for term structure evolution. The problem originated as a request from the Italian Ministry of Economy and Finance to find an optimal strategy for the issuance of Public Debt securities. The basic idea is to split the evolution of each rate into two parts. The first component is driven by the evolution of the official rate (the European Central Bank official rate in the present case). The second component of each rate is represented by the fluctuations having null correlation with the ECB rate.

In this paper we extend the stability results of cascaded closed-loop predictors for deterministic systems to stochastic linear systems with additive noise and arbitrarily large time-varying input delay, under suitable hypothesis on the delay function. We propose an output-feedback controller design based on a set of differential delay equations with size depending on the delay bound and we show that it ensures exponential stability in mean and mean square stability of the controlled system. The result is extended to multiple delays on separate input signals and to systems with arbitrarily large output delays.