Quadratic functionals and small ball probabilities for the \(m\)-fold integrated Brownian motion
Let \(W(t)\), \(t\geq 0\), be a standard Brownian motion starting at \(0\). Denote by \(X_0(t)= W(t)\), \(X_m(t)= \int^t_0 X_{m-1}(s) ds\), \(t\geq 0\), \(m\geq 1\), the \(m\)-fold integrated Brownian motion for positive integer \(m\). This paper is concerned with the quadratic functionals for the real process \(X_m(t)\) as well as the asymptotic behavior of the small ball probability. A method for estimating the small probability such as \(P(\sup_{0\leq i\leq 1}|X_m(t)|\leq\varepsilon)\), as \(\varepsilon\to 0\), is developed via the appropriate quadratic functionals.
scientific article; zbMATH DE number 1933040
scientific article