Certain results concerning filtering and control of diffusion in small white noise

We examine in detail the binary hypothesis decision and/or estimation problem using a risk-sensitive cost criterion, when the state and observation processes are diffusion signals. We demonstrate the role played by a Feynman-Kac version of the Duncan-Mortensen-Zakai stochastic partial differential equation in making decisions. The question of performance is addressed by relating Chernoff bounds to this Feynman-Kac stochastic equation. We also examine in detail the behaviour of our calculations, in the limit as the covariances of the random inputs tend to zero. The procedure employs large deviations techniques. This approach enables us to establish relations between stochastic and deterministic methods in tackling the binary decision problem. The latter reveals a natural formulation of the binary decision problem in terms of an H^∞-disturbance attenuation framework