Stochastic processes and moving boundaries

• Applicant: Professor Dr. Vitali Wachtel
• Subject Area: Mathematics
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 429723931

• Final Report Year: 2025

Final Report Abstract

I this project we have considered two classes of problems for conditioned random processes. First, we have studied first-passage problems over square-root boundaries for the Brownian motion and for discrete time random walks. We have found tails of firstpassage times over asymptotically square-root boundaries for the Brownian motion. Furthermore, we have constructed space-time harmonic functions for discrete time random walks killed at the boundary of the form c√t + b−a with b ≥ 0 and a > c√b. We have also studied tails of the corresponding exit times. Second, we have derived asymptotic expansions for local probabilities of random walks conditioned to stay positive. The ’coefficients’ in our expansions for lower deviation probabilities are polyharmonic functions, which, in general, are not polynomials. This is a quite new type of expansions. We have also proven a Berry-Esseen-type inequality for random walks conditioned to stay positive, which gives an optimal rate of convergence towards the Rayleigh distribution.

Publications

• First-Passage Times for Random Walks in the Triangular Array Setting. Progress in Probability, 181-203. Springer International Publishing.
  Denisov, Denis; Sakhanenko, Alexander & Wachtel, Vitali
  
• Crossing an Asymptotically Square-Root Boundary by the Brownian Motion. Proceedings of the Steklov Institute of Mathematics, 316(1), 105-120.
  Denisov, Denis E.; Hinrichs, Günter; Sakhanenko, Alexander I. & Wachtel, Vitali I.
  
• Persistence of autoregressive sequences with logarithmic tails. Electronic Journal of Probability, 27(none).
  Denisov, Denis; Hinrichs, Günter; Kolb, Martin & Wachtel, Vitali
  
• Berry-Esseen inequality for random walks conditioned to stay positive
  Denisov, D., Tarasov, A. & Wachtel, V.
  
• Expansions for random walks conditioned to stay positive
  Denisov, D., Tarasov, A. & Wachtel, V.
  
• Random walks in cones revisited. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 60(1).
  Denisov, Denis & Wachtel, Vitali
  
• Random walks with square-root boundaries: the case of exact boundaries g(t) = c √t + b − a
  Denisov, D., Sakhanenko, A., Terveer, S. & Wachtel, V.
  
• Asymptotic expansions for random walks conditioned to stay positive. Electronic Journal of Probability, 31(none).
  Denisov, Denis; Tarasov, Alexander & Wachtel, Vitali