Estimation of first crossing time distribution for N-parameter Brownian motion processes relative to upper class boundaries
Let \(X_ t\), \(t=(t_ 1,...,t_ N)\), be an N-parameter Brownian sheet, \(\phi\) (u) a positive function with suitable growth conditions, \(g(u)=u^{1/2}\phi (u)\). Denote \(\delta (t)=t_ 1...t_ N\), \(I=[0,1]^ N\), \(T_ g=\inf \{\partial (t):\) \(t\in I\), \(X_ t\geq g(\delta (t))\}\), \(P_{\tau}=P(T_ g\leq \tau)\). The main result is the following: For each \(\phi\) (\(\cdot)\) from the upper class for \(X_ t\) at the origin \[ 0\leq a_ N\leq \underline{\lim}_{\tau \to 0}P_{\tau}/I_ g(\tau,N)\leq \overline{\lim}_{\tau \to 0}P_{\tau}/I_ g(\tau,N)\leq 2^ Ne^ N(1-N^{-1})^{N-1}/N!, \] where \[ I_ g(\tau,N)=(2\pi)^{-1/2}\int_{0}^{\tau}H_ g(u,N)du,\quad H_ g(u,N)=\exp \{-\phi^ 2(u)\}\phi^{2N-1}(u)| \log u^{- 1}|^{N-1}u^{\quad -1}. \] To prove this fact the authors used methods analogous to their earlier work for standard Brownian motion [Estimation of first crossing time distributions for Brownian motion processes relative to upper class boundaries. to appear]. Some numerical results of evaluating \(a_ N\) are given.
scientific article; zbMATH DE number 3923816