arXiv subjects
Publications and source records attributed to Cameron Bruggeman.
Consider a reflected diffusion on the positive half-line. We approximate it by solutions of stochastic differential equations using the penalty method: We emulate the "hard barrier" of reflection by a "soft barrier" of a large drift coefficient, which compells the diffusion to return to the positive half-line. The main tool of the proof is convergence of scale functions.
Consider a finite system of competing Brownian particles on the real line. Each particle moves as a Brownian motion, with drift and diffusion coefficients depending only on its current rank relative to the other particles. We find a sufficient condition for a.s. absence of a total collision (when all particles collide) and of other types of collisions, say of the three lowest-ranked particles. This continues the work of Ichiba, Karatzas, Shkolnikov (2013) and Sarantsev (2015).
A one-dimensional, continuous, regular, and strong Markov process $X$ with state space $E$ hits any point $z \in E$ fast with positive probability. To wit, if $τ_z = \inf \{t \geq 0:X_{t} = z\}$, then $P_ξ({ τ}_z<\varepsilon)>0$ for all $ξ\in E$ and $\varepsilon>0$.