Hamzi - Eurandom

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

Parameter Estimation for SDEs • Consider the scalar-valued SDE • Equation Lρ∞(x) = 0 reduces to dXt = f(Xt) dt + g(Xt) dWt . f(x)ρ∞(x) = 1 2 (g(x)2 ρ∞(x)) ′ . The methodology described before yields an estimator ˆρ∞ of ρ∞, but this is obviously not sufficient to estimate both unknown functions f and g directly using the above relation. If we, however, have knowledge of either one, then a natural nonparametric estimator for the other one is given via or ˆf(x) = g(x)g ′ (x) + g(x)2 ˆρ ′ ∞(x) 2ˆρ∞(x) ˆg(x) 2 = 2 ˆρ∞(x) ∫ x 0 f(u)ˆρ∞(u) du , . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 50 / 55
Parameter Estimation for SDEs • It is reasonable to assume that the diffusion coefficient is known up to a multiplicative factor g(x; θ) = θg(x) with θ being an unknown parameter and g is known here. In this case we estimate θ via the quadratic variation of the path estimator ˆ θ ˆθ = ∑n−1 i=0 (Xti+1 − Xti )2 ∫ t 0 g(Xs) ds . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 51 / 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 and 48: Nonlinear SDEs in RKHSes • The st
Page 49: Parameter Estimation for SDEs • I
Page 53 and 54: Example (A Fokker-Planck Integrable
Page 55: Open Questions • Derivation of da

Hamzi - Eurandom

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