Convergence in fractional models and applications (Q1767559)

Let \(b_{\alpha}\) be fractional Brownian motion with Hurst parameter \(\alpha\in (0,1)\) and let \(X\) be the solution of the stochastic differential equation \(dX(t)=\sigma(X(t))\,db_{\alpha}(t)\) on \([0,\infty)\) with \(X(0)=c\). To regularize the processes consider a positive kernel with \(L^1\)-norm equal to one, define \(\varphi_{\varepsilon}(t)=\varphi(t/\varepsilon)/\varepsilon\) and set \(b_{\alpha}^{\varepsilon}=\varphi_{\varepsilon}\ast b_{\alpha}\) and \(X^{\varepsilon}=\varphi_{\varepsilon}\ast X\), respectively. The authors discuss for the process \(b_{\alpha}^{\varepsilon}\) and functionals of this process like \(X^{\varepsilon}\) the relation of level crossings to local time and the rate of convergence of the involved limit theorem. Moreover, they analyze the rate of convergence of a limit theorem concerning the increments of fractional Brownian motion. Finally, the authors propose probabilistic and statistical applications of their results.