Abstract
The objective of the thesis is to study some properties and applications of stochastic equations driven by a fractional Brownian motion with Hurst parameter H. I n particular, we study the continuity with respect to H of the heat and wave multiplicative and additive stochastic partial differential equations driven by a noise which is white in the time variable and behaves like a fractional Brownian motion in the space variable. Morevoer, we study an analogous problem for a class of one-dimensional stochastic differential equations driven by a fractional noise, in the setting of rough paths theory. On the side of applications, we define and evaluate a stochastic model with the objective of forecasting the future electricity prices in the italian market. This model includes as the main stochastic component an equation driven by a fractional Brownian motion, plus a jump component which shows self-exciting properties, namely a Hawkes process.
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