Sampling the Ornstein Uhlenbeck Process for Remote Estimation over an Unreliable Channel

We study a sampling problem, where the sampler chooses a time to take a sample of the Ornstein Uhlenbeck (OU) process and transmits the sample to a remote estimator over an unreliable channel. The goal is to design these sampling times to minimize the long-term average mean square estimation error (MSE). Our problem strictly generalizes the previous study from the Wiener process to the OU process and poses significant challenges. First, the OU process includes an additional drift term, complicating the reward function. Then, under an unreliable channel, computing the action value functions needs iterations. These make it hard to compute an optimal value function via value iteration. Moreover, the drift term makes the third derivative of the reward function nonlinear, making it complicated to validate the optimality. Despite the challenges, through thorough calculations and a convexity assumption, we are still able to obtain an optimal sampling policy that remains a simple threshold structure with low complexity. In addition, we add a sampling rate constraint that is not considered in the previous study, and provide an exact optimal sampling policy. A counterintuitive and different result is that the zero-wait policy is optimal when the channel is highly unreliable.