This paper develops a two-stage stochastic transport model for the detection and tracking of a moving target governed by a multidimensional random walk. In the first stage, the target's initial location is estimated through an optimal search strategy over independent regions, where a truncated bivariate distribution is used to describe prior uncertainty. Explicit analytical expressions for the optimal search distances minimizing the expected detection time are derived. In the second stage, the tracking problem is formulated along intersecting trajectories, where coordinated searchers aim to intercept a stochastically moving target. Closed-form expressions for the first-encounter (first-passage) time and the corresponding tracking distances are obtained. Moreover, sufficient conditions ensuring the finiteness of the expected first-encounter time are established for a bounded controlled tracking framework, where unbiased random walk motion is coupled with coordinated recurrent coverage of the admissible tracking trajectories and a uniform positive encounter probability. The proposed model provides a unified analytical framework for first-passage phenomena in multidimensional stochastic transport systems.

We developed a multivariate discrete range distribution derived from the Wiener process to model high-low price dynamics of multiple assets observed at discrete times and subject to market imposed bounds. The model provides closed-form expressions for the joint PMF, CDF, survival and hazard functions, reversed and second order failure rates, moments, stress-strength reliability, and a full system of multivariate order statistics. A truncated version of the distribution was also established to account for realistic price limit regimes, showing how probability mass redistributes within constrained domains. These theoretical properties were supplemented by a numerical study based on real high-low data and confirmed that the model can capture clustered volatility, attenuation of tail risk, and joint range behavior more precisely than unconstrained formulations. The proposed framework offers a mathematically coherent and computationally practical tool for the analysis of range-based behavior in constrained financial markets.