Convergence of the derivative martingale for the branching random walk in time-inhomogeneous random environment

2023/06/27 by Wenming Hong, Hong, Wenming, Shengli Liang +1
Mathematics · Economics, Econometrics and Finance · #Stochastic processes and statistical mechanics #Stochastic processes and financial applications #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2306.15204

Abstract

Consider a branching random walk on the real line with a random environment in time (BRWRE). A necessary and sufficient condition for the non-triviality of the limit of the derivative martingale is formulated. To this end, we investigate the random walk in time-inhomogeneous random environment (RWRE), which related the BRWRE by the many-to-one formula. The key step is to figure out Tanaka's decomposition for the RWRE conditioned to stay non-negative (or above a line), which is interesting itself as well.

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