The correlated random walk with boundaries: A combinatorial solution

Abstract

The transition functions for the correlated random walk with two absorbing boundaries are derived by means of a combinatorial construction which is based on Krattenthaler's theorem for counting lattice paths with turns. Results for walks with one boundary and for unrestricted walks are presented as special cases. Finally we give an asymptotic formula, which proves to be useful for computational purposes.

Limit Theorems and Absorption Problems for One-Dimensional Correlated Random Walks

“…We remark that p(l, m), q(l, m), r(l, m), s(l, m) in Ξ(l, m) correspond to the types of paths described above in terms of P and Q, respectively. A similar argument can be seen in the proof of Theorem 2.1 of Böhm [7]. On the other hand, for the quantum walk case, this proof appears in Konno [15].…”

“…(19) and (20) in Corollary 2.6 of [7], respectively. By using Lemma 2, we have an expression for the characteristic function of X ϕ n .…”

“…The correlated random walk has been studied by various authors. Recent work relating to the present paper includes [2,3,7,8,9,10,20,22,23,26]. More detailed information on their results will be stated in the next section.…”

Analytical Solution to Transport in Brownian Ratchets via the Gambler’s Ruin Model

“…Extensions to the classic Gambler's Ruin or other MC problems, e.g. inclusion of correlations [18] or multiple currencies [19], may yield further insights into other areas of stochastic dynamics, e.g. turbulence or high-dimensional thermally activated dynamics.…”

Optimal Buy/Sell Rules for Correlated Random Walks

“…Gillis' conjecture was proved in Iossif (1986) and then for more general correlated random walks in Chen and Renshaw (1994). Other results on correlated random walks and boundary problems include Proudfoot and Lampard (1972), Jain (1973), Mukherjea and Steele (1987), Zhang (1992), and Böhm (2000).…”