First Passage Problems and Duality Relations

Abstract

The model of a system {Zt, ut, k} (see Sections 1.1, 1.5 and 1.5), where ut is a stochastic variable, can be regarded as a random walk of the process {Zt}. Suppose a particle traverses a straight line path; at the beginning of each time interval (t, t+1) or (τt, τt+1), where t =..., -1,0,1,2,..., the particle is at position Z (as measured from the origin 0), and immediately after has a displacement u (-∞ < ut < ∞); then if it does not have any further displacement, the position of the particle at the end of the time period is given by $${Z_{{t + 1}}} = \min {\{ {Z_{t}} + {u_{t}},k\} ^{ + }}$$ (2.1) The above implies that Zt+1 cannot be negative, at the same time if Zt + ut exceeds k, the particle continues to be at k; thus we have two impenetrable barriers at 0 and k. If k = ∞, then we have a walk on a line of infinite length with a single impenetrable barrier at 0. Similarly we can conceive of a walk Zt which has only one barrier at k but the line runs to infinity in the negative zone; here we get $${Z_{{t + 1}}} = \min \{ {Z_{t}} + {u_{t}},k\} , - \infty < {Z_{t}} \leqslant $$ (2.2) the model corresponds to an inventory system in which backlogging of orders is permitted.