Some Blow-Up Problems for a Semilinear Parabolic Equation with a Potential

The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=\Delta u+V(x) |u|^{p-1}u$ in $\Omega \times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=M\vf (x) as $M$ goes to infinity, which have been found in \cite{cer}, are improved under some reasonable and weaker conditions compared with \cite{cer}.Comment: 29 page

Forecasting Value-at-Risk Using the Markov-Switching ARCH Model

This paper analyzes the application of the Markov-switching ARCH model (Hamilton and Susmel, 1994) in improving value-at-risk (VaR) forecast. By considering a mixture of normal distributions with varying variances over different time and regimes, we find that the “spurious high persistence†found in the GARCH model is adjusted. Under relative performance and hypothesis-testing evaluations, the VaR forecasts derived from the Markov-switching ARCH model are preferred to alternative parametric and nonparametric VaR models that only consider time-varying volatility. JEL classification: C22, C52, G28. Keywords: Value-at-Risk, Switching-regime ARCH models.Value-at-Risk, Switching-regime ARCH models