Volatility forecasting is crucial for option pricing, risk management and
portfolio management. Nowadays, volatility has become the subject of
trading. There are now exchange-traded contracts written on volatility.
Financial market volatility also has a wider impact on financial regulation,
monetary policy and macroeconomy.
This research monograph concerns the design and analysis of discrete-time
approximations for stochastic differential equations (SDEs) driven by Wiener
processes and Poisson processes or Poisson jump measures. In financial and
actuarial modeling and other areas of application, such jump diffusions are
often used to describe the dynamics of various state variables. In finance these
may represent, for instance, asset prices, credit ratings, stock indices, interest
rates, exchange rates or commodity prices.
Contents 9 Sample path properties of local times 9.1 Bounded discontinuities 9.2 A necessary condition for unboundedness 9.3 Suﬃcient conditions for continuity 9.4 Continuity and boundedness of local times 9.5 Moduli of continuity 9.6 Stable mixtures 9.7 Local times for certain Markov chains 9.8 Rate of growth of unbounded local times 9.9 Notes and references p-variation 10.1 Quadratic variation of Brownian motion 10.2 p-variation of Gaussian processes 10.3 Additional variational results for Gaussian processes 10.4 p-variation of local times 10.
In terms of the shape of the EKC, debates and further studies have shown other
variations from the inverted-U shape originally proposed: cubic function and L-shaped curves.
Torras and Boyce (1998) suggested that instead of a quadratic function, the EKC
actually follows a cubic one. This allows for the possibility that a downturn in pollution (at
the peak of the inverted U) can be followed by a later upturn, that is, a reversal of the
tendency for pollution levels to decline with further increases in per capita income.