On representations and simulation of conditioned random walks on integer lattices

We consider a random walk on a multidimensional integer lattice with random bounds on local times. We introduce a family of auxiliary “accompanying” processes that have regenerative structures and play key roles in our analysis. We obtain a number of representations for the distribution of the random walk in terms of similar distributions of the “accompanying” processes. Based on that, we obtain representations for the conditional distribution of the random walk, conditioned on the event that it hits a high level before its death. Under more restrictive assumptions a representation of such type has been obtained earlier by the same authors in a recent paper published in the Springer series on Progress in Probability, 77 (2021), where a certain “limiting” process was used in place of “accompanying” processes of the present paper.