q-Deformed Brownian Motion

Brownian motion may be embedded in the Fock space of bosonic free field in one dimension.Extending this correspondence to a family of creation and annihilation operators satisfying a q-deformed algebra, the notion of q-deformation is carried from the algebra to the domain of stochastic processes.The properties of q-deformed Brownian motion, in particular its non-Gaussian nature and cumulant structure,are established.Comment: 6 page

Endogenous Cycles in Optimal Monetary Policy with a Nonlinear Phillips Curve

There is by now a large consensus in modern monetary policy. This consensus has been built upon a dynamic general equilibrium model of optimal monetary policy with sticky prices a la Calvo and forward looking behavior. In this paper we extend this standard model by introducing nonlinearity into the Phillips curve. As the linear Phillips curve may be questioned on theoretical grounds and seems not to be favoured by empirical evidence, a similar procedure has already been undertaken in a series papers over the last few years, e.g., Schaling (1999), Semmler and Zhang (2004), Nobay and Peel (2000), Tambakis (1999), and Dolado et al. (2004). However, these papers were mainly concerned with the analysis of the problem of inflation bias, by deriving an interest rate rule which is nonlinear, leaving the issues of stability and the possible existence of endogenous cycles in such a framework mostly overlooked. Under the specific form of nonlinearity proposed in our paper (which allows for both convexity and concavity and secures closed form solutions), we show that the introduction of a nonlinear Phillips curve into a fully deterministic structure of the standard model produces significant changes to the major conclusions regarding stability and the efficiency of monetary policy in the standard model. We should emphasize the following main results: (i) instead of a unique fixed point we end up with multiple equilibria; (ii) instead of saddle--path stability, for different sets of parameter values we may have saddle stability, totally unstable and chaotic fixed points (endogenous cycles); (iii) for certain degrees of convexity and/or concavity of the Phillips curve, where endogenous fluctuations arise, one is able to encounter various results that seem interesting. Firstly, when the Central Bank pays attention essentially to inflation targeting, the inflation rate may have a lower mean and is certainly less volatile; secondly, for changes in the degree of price stickiness the results are not are clear cut as in the previous case, however, we can also observe that when such stickiness is high the inflation rate tends to display a somewhat larger mean and also higher volatility; and thirdly, it shows that the target values for inflation and the output gap (π^,x^), both crucially affect the dynamics of the economy in terms of average values and volatility of the endogenous variables --- e.g., the higher the target value of the output gap chosen by the Central Bank, the higher is the inflation rate and its volatility --- while in the linear case only the π^ does so (obviously, only affecting in this case the level of the endogenous variables). Moreover, the existence of endogenous cycles due to chaotic motion may raise serious questions about whether the old dictum of monetary policy (that the Central Bank should conduct policy with discretion instead of commitment) is not still very much in the business of monetary policy.Optimal monetary policy, Interest Rate Rules, Nonlinear Phillips Curve, Endogenous Fluctuations and Stabilization

Symbolic Sequences and Tsallis Entropy

We address this work to investigate symbolic sequences with long-range correlations by using computational simulation. We analyze sequences with two, three and four symbols that could be repeated $l$ times, with the probability distribution $p(l)\propto 1/ l^{\mu}$. For these sequences, we verified that the usual entropy increases more slowly when the symbols are correlated and the Tsallis entropy exhibits, for a suitable choice of $q$, a linear behavior. We also study the chain as a random walk-like process and observe a nonusual diffusive behavior depending on the values of the parameter $\mu$.Comment: Published in the Brazilian Journal of Physic