About the linear-quadratic regulator problem under a fractional brownian perturbation

M. L. Kleptsyna; Alain Le Breton; M. Viot

ESAIM: Probability and Statistics (2003)

  • Volume: 7, page 161-170
  • ISSN: 1292-8100

Abstract

top
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.

How to cite

top

Kleptsyna, M. L., Breton, Alain Le, and Viot, M.. "About the linear-quadratic regulator problem under a fractional brownian perturbation." ESAIM: Probability and Statistics 7 (2003): 161-170. <http://eudml.org/doc/245312>.

@article{Kleptsyna2003,
abstract = {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.},
author = {Kleptsyna, M. L., Breton, Alain Le, Viot, M.},
journal = {ESAIM: Probability and Statistics},
keywords = {fractional brownian motion; linear system; optimal control; quadratic payoff; fractional Brownian motion; linear-quadratic Gaussian regulator},
language = {eng},
pages = {161-170},
publisher = {EDP-Sciences},
title = {About the linear-quadratic regulator problem under a fractional brownian perturbation},
url = {http://eudml.org/doc/245312},
volume = {7},
year = {2003},
}

TY - JOUR
AU - Kleptsyna, M. L.
AU - Breton, Alain Le
AU - Viot, M.
TI - About the linear-quadratic regulator problem under a fractional brownian perturbation
JO - ESAIM: Probability and Statistics
PY - 2003
PB - EDP-Sciences
VL - 7
SP - 161
EP - 170
AB - 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.
LA - eng
KW - fractional brownian motion; linear system; optimal control; quadratic payoff; fractional Brownian motion; linear-quadratic Gaussian regulator
UR - http://eudml.org/doc/245312
ER -

References

top
  1. [1] M.H.A. Davis, Linear Estimation and Stochastic Control. Chapman and Hall (1977). Zbl0437.60001MR476099
  2. [2] L. Decreusefond and A.S. Üstünel, Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999) 177-214. Zbl0924.60034MR1677455
  3. [3] T.E. Duncan, Y. Hu and B. Pasik–Duncan, Stochastic calculus for fractional Brownian motion I. Theory. SIAM J. Control Optim. 38 (2000) 582-612. Zbl0947.60061
  4. [4] G. Gripenberg and I. Norros, On the prediction of fractional Brownian motion. J. Appl. Probab. 33 (1997) 400-410. Zbl0861.60049MR1385349
  5. [5] Y. Hu, B. Øksendal and A. Sulem, A stochastic maximum principle for processes driven by fractional Brownian motion, Preprint 24. Pure Math. Dep. Oslo University (2000). 
  6. [6] M.L. Kleptsyna and A. Le Breton, Statistical analysis of the fractional Ornstein–Uhlenbeck type process. Statist. Inference Stochastic Process. (to appear). Zbl1021.62061
  7. [7] M.L. Kleptsyna and A. Le Breton, Extension of the Kalman–Bucy filter to elementary linear systems with fractional Brownian noises. Statist. Inference Stochastic Process. (to appear). Zbl1011.60018
  8. [8] M.L. Kleptsyna, A. Le Breton and M.-C. Roubaud, General approach to filtering with fractional Brownian noises – Application to linear systems. Stochastics and Stochastics Rep. 71 (2000) 119-140. Zbl0979.93117
  9. [9] M.L. Kleptsyna, A. Le Breton and M. Viot, Solution of some linear-quadratic regulator problem under a fractional Brownian perturbation and complete observation, in Prob. Theory and Math. Stat., Proc. of the 8th Vilnius Conference, edited by B. Grigelionis et al., VSP/TEV (to appear). Zbl1030.93059MR2148966
  10. [10] R.S. Liptser and A.N. Shiryaev, Statistics of Random Processes. Springer-Verlag (1978). Zbl1008.62072
  11. [11] I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5 (1999) 571-587. Zbl0955.60034MR1704556
  12. [12] C.J. Nuzman and H.V. Poor, Linear estimation of self-similar processes via Lamperti’s transformation. J. Appl. Probab. 37 (2000) 429-452. Zbl0963.60034

Citations in EuDML Documents

top
  1. Marina L. Kleptsyna, Alain Le Breton, Michel Viot, On the infinite time horizon linear-quadratic regulator problem under a fractional brownian perturbation
  2. Marina L. Kleptsyna, Alain Le Breton, Michel Viot, On the infinite time horizon linear-quadratic regulator problem under a fractional Brownian perturbation
  3. Marina L. Kleptsyna, Alain Le Breton, Michel Viot, Separation principle in the fractional Gaussian linear-quadratic regulator problem with partial observation

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.