On the error exponents for detecting randomly sampled noisy diffusion processes

This paper deals with the detection of a continuous random process described by an Ornstein-Uhlenbeck (O-U) stochastic differential equation. Randomly spaced sensors or equivalently a random time sampler which deliver noisy samples of the process are used for this detection. Two types of tests are considered: either H0 refers to the presence of the noisy O-U process or H0 refers to the sole presence of noise. For any fixed false alarm probability, it is shown that the Type II error probability decreases to zero exponentially in the number of samples. The exponents, which do not depend on the false alarm probability, are characterized. This work completes former contributions that consider noiseless O-U process with a random sampling or noisy O-U processes with a regular sampling.