Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
C.5 Ornstein-Ulhenbeck Process
Let us now rewrite equation (C.1) as in [100], with the help of the positive constant
parameters β and σ:
dXt = −βXtdt + σ � 2βdBt.
Therfore, with X(0) = 0, the mean value of Xt is 0 and the term α2
2ρ
σ2 .
of the variance, becomes
In order to make a simulation of the Ornstein-Uhlenbeck process, we consider equation
(C.2) and replace:
− α by σ √ 2β,
− ρ by β.
The update equation is (with µ = e −β∆t )
X(t + ∆t) =X(t)µ + σ(1 − µ 2 )zn.
The actual simulation is done according to the diagram of Figure C.1, with X(0) = 0, µ ∈]0; 1[
and σ > 0. Remark that σ is the asymptotic standard deviation of the variables X(t), t > 0.
An important theorem due to J. L. Doob [48] ensures that such a process necessarily satisfies a linear stochastic
differential equation identical to the one used in the definition above.
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