On asymptotic stability of continuous-time risk-sensitive filters with respect to initial conditions

Tn this paper, we consider the problem of risk-sensitive filtering for cont inuous-time stochastic linear Gaussian time-invariant systems. In particula r, we address the problem of forgetting of initial conditions. Our results show that suboptimal risk-sensitive filters initialized with arbitrary Gaus sian initial conditions asymptotically approach the optimal risk-sensitive filter for a linear Gaussian system with Gaussian but unknown initial condi tions in the mean square sense at an exponential rate, provided the arbitra ry initial covariance matrix results in a stabilizing solution of the (H-in finity-like) Riccati equation associated with the risk-sensitive problem. M ore importantly, in the case of non-Gaussian initial conditions, a suboptim al risk-sensitive filter asymptotically approaches the optimal risk-sensiti ve filter in the mean square sense under a boundedness condition satisfied by the fourth order absolute moment of the initial non-Gaussian density and a slow growth condition satisfied by a certain Radon-Nikodym derivative. ( C) 2000 Elsevier Science B.V. All rights reserved.