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We consider the fractional analogue of the Ornstein-Uhlenbeck process i.e. the solution of a one-dimensional homogeneous linear stochastic differential equation driven by a fractional Brownian motion in place of the usual Brownian motion. The statistical problem of estimation of the drift and variance parameters is investigated on the basis of a… (More)

We investigate the optimal filtering problem in the simplest Gaussian linear system driven by fractional Brownian motions. At first we extend to this setting the Kalman-Bucy filtering equations which are well-known in the specific case of usual Brownian motions. Closed form Volterra type integral equations are derived both for the mean of the optimal filter… (More)

In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous time. For a completely observable controlled linear system driven by a fractional Brownian motion, we describe explicitely the optimal control policy which minimizes a quadratic performance criterion. Mathematics Subject… (More)

- Marina L. Kleptsyna, Alain Le Breton, Michel Viot
- SIAM J. Control and Optimization
- 2008

In 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… (More)

Recently, stochastic models appropriate for long-range dependent phenomena have attracted a great deal of interest and numerous theoretical results and successful applications have been already reported. In particular, several contributions in the literature have been devoted to the extension of the classical theory of continuoustime stochastic systems… (More)

The optimal filtering problem for multidimensional continuous possibly non-Markovian, Gaussian processes, observed through a linear channel driven by a Brownian motion, is revisited. Explicit Volterra type filtering equations involving the covariance function of the filtered process are derived both for the conditional mean and for the covariance of the… (More)

Various methods to derive new formulas for the Laplace transforms of some quadratic forms of Gaussian sequences are discussed. In the general setting, an approach based on the resolution of an appropriate auxiliary filtering problem is developed; it leads to a formula in terms of the solutions of Voterra type recursions describing characteristics of the… (More)

- M. L. Kleptsyna, M. Viot
- 2004

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… (More)

In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous-time. For a completely observable controlled linear system driven by a fractional Brownian motion, we describe explicitely the optimal control policy which minimizes a quadratic performance criterion.