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This dissertation contains two parts. The first part provides the first algorithm that, under minimal assumptions, allows to simulate the stationary waiting-time sequence of a single-server queue backwards in time, jointly with the input processes of the queue (inter-arrival and service times). The single-server queue is useful in applications of DCFTP (Dominated Coupling From The Past), which is a well known protocol for simulation without bias from steady-state distributions. Our algorithm terminates in finite time assuming only finite mean of the inter-arrival and service times. In order to simulate the single-server queue in stationarity until the first idle period in finite expected termination time we require the existence of finite variance. This requirement is also necessary for such idle time (which is a natural coalescence time in DCFTP applications) to have finite mean. Thus, in this sense, our algorithm is applicable under minimal assumptions. The second part studies the behavior of diffusion processes in a random environment. We consider an adversary that moves in a given domain and our goal is to produce an optimal strategy to detect and neutralize him by a given deadline. We assume that the target's dynamics follows a diffusion process whose parameters are informed by available intelligence information. We will dedicate one chapter to the rigorous formulation of the detection problem, an introduction of several frameworks that can be considered when applying our methods, and a discussion on the challenges of finding the analytical optimal solution. Then, in the following chapter, we will present our main result, a large deviation behavior of the adversary's survival probability under a given strategy. This result will be later give rise to asymptotically efficient Monte Carlo algorithms.