Linear filtering with fractional Brownian motion in the signal and observation processes

Integral equations for the mean-square estimate are obtained for the linear filtering problem, in which the noise generating the signal is a fractional Brownian motion with Hurst index h∈(3/4,1) and the noise in the observation process includes a fractional Brownian motion as well as a Wiener process. AMS subject classifications: 93E11, 60G20, 60G35

Asymptotically optimal filtering in linear systems with fractional Brownian noises

In this paper, the filtering problem is revisited in the basic Gaussian homogeneous linear system driven by fractional Brownian motions. We exhibit a simple approximate filter which is asymptotically optimal in the sense that, when the observation time tends to infinity, the variance of the corresponding filtering error converges to the same limit as for the exact optimal filter

Risk Sensitive and LEG filtering problems are not equivalent

International audienceFiltering problems with general exponential quadratic criteria are investigated for Gauss-Markov processes. In this setting, the linear exponential Gaussian and risk sensitive filtering problems are solved and it is shown that they may have different solutions

Separation principle in the fractional Gaussian linear-quadratic regulator problem with partial observation

International audienceIn this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous-time with partial observation. For a controlled linear system where both the state and observation processes are driven by fractional Brownian motions, we describe explicitly the optimal control policy which minimizes a quadratic performance criterion. Actually, we show that a separation principle holds, i.e., the optimal control separates into two stages based on optimal filtering of the unobservable state and optimal control of the filtered state. Both finite and infinite time horizon problems are investigated

Filtering with exponential criteria via linear observation channels

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On the linear-exponential filtering problem for general Gaussian processes

International audienceThe explicit solution of the filtering problem with exponential criteria for a general Gaussian signal is obtained through an approach which is based on a conditional Cameron-Martin type formula. This key formula is derived for conditional expectations of exponentials of some quadratic functionals of a general continuous Gaussian process. The formula involves conditional expectations and conditional covariances in some auxiliary optimal risk-neutral filtering problem

On the linear-exponential filtering problem for general Gaussian processes

International audienceThe explicit solution of the filtering problem with exponential criteria for a general Gaussian signal is obtained through an approach which is based on a conditional Cameron-Martin-type formula. This key formula is derived for conditional expectations of exponentials of some quadratic functionals of a general continuous Gaussian process. The formula involves conditional expectations and conditional covariances in some auxiliary optimal risk-neutral filtering problem which is used in the proof. Closed form equations of the Itô-Volterra- and Riccati-Volterra-types for these ingredients are provided. Particular cases for which the results can be further elaborated are investigated