Hamzi - Eurandom

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$Continuous-time random walks, fractional calculus and ... - Eurandom$

Experiment Consider the SDE dX = −5X 5 + 10X 3 + √ 2dW . 1.4 1.2 1 0.8 0.6 0.4 0.2 true RKHS empirical 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 48 / 55
Parameter Estimation for SDEs • It is apparent from Lρ∞(x) = 0 that both drift and diffusion coefficients of an SDE are directly related to the stationary solution ρ∞ of the Fokker-Planck equation. • This relation can be employed to derive estimators for the unknown coefficients. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 49 / 55
Page 1 and 2: Preliminary Results on Control and
Page 3 and 4: Outline • Review of Some Concepts
Page 5 and 6: Review of Some Concepts for Linear
Page 11 and 12: Controllability and Observability E
Page 13 and 14: Balancing of Linear Control Systems
Page 15 and 16: Balancing of Nonlinear Systems •
Page 17 and 18: Empirical Estimates of the Gramians
Page 19 and 20: Reproducing Kernel Hilbert Spaces
Page 21 and 22: Reproducing Kernel Hilbert Spaces T
Page 33 and 34: Balanced Reduction of Nonlinear Con
Page 35 and 36: Balancing in RKHS • “Balancing
Page 37 and 38: Experiment: Inputs ◮ Excite with
Page 39 and 40: Experiment diag(TKcT T ) 10 −1 10
Page 45 and 46: Nonlinear SDEs in RKHSes • Assump
Page 47: Nonlinear SDEs in RKHSes • The st
Page 51 and 52: Parameter Estimation for SDEs • I
Page 53 and 54: Example (A Fokker-Planck Integrable
Page 55: Open Questions • Derivation of da

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