We study the optional projection of a standard Brownian motion on the natural filtration of certain kinds of observation processes. The observation process, $Y$, is defined as a solution of a stochastic differential equation such that it reveals some (possibly…

We present a new proof of the extended arc-sine law related to Walsh's Brownian motion, known also as Brownian spider. The main argument mimics the scaling property used previously, in particular by D. Williams, in the 1- dimensional Brownian case, which can…

Let $X_{1},X_{2},...$ denote independent, identically distributed random variables with common distribution $F$, and $S$ the corresponding random walk with $\rho :=\lim_{n\rightarrow \infty }P(S_{n}>0)$ and $\tau :=\inf \{n\geq 1:S_{n}\leq 0\}$. We assume that $X$ is in the domain of attraction of…