Bilinear stochastic systems with fractional Brownian motion input. (English) Zbl 0948.60050

Iglói, E.; Terdik, Gy.

Ann. Appl. Probab. 9, No. 1, 46-77 (1999).

A meaning is done to an integration with respect to a fractional Brownian motion \[ w_t^h= \frac{1} {\Gamma(1+h)} \Biggl[ \int_{-\infty}^0 [(t-s)^h- (-s)^h] dw_s+ \int_0^t (t-s)^h dw_s \Biggr], \] \(t\in \mathbb{R}\), \(h\in ]0, \frac 12[\), as following. If \(f\in L^2 (\mathbb{R})\) such that \(\int|e^{itx} f(t) dt|^2 |x|^{-2h} dx< +\infty\), then \[ \int f(t) dw_t^h:= \iint e^{itx} f(t) (ix)^{-h} dt W(dx) \] where \(W(dx)\) is a complex Gaussian white noise spectral measure. The main tool is the technique of spectral domain representation of square integrable stationary functions of the fractional Brownian notion, using the spectral domain chaotic representation form and its transfer functions. Then, a stochastic integration is defined for processes satisfying a set of assumptions (namely \(\zeta\)), using their transfer functions, as a spectral domain chaotic representation. Such of stochastic integrals as \(dy_t= (\xi_t dt+ \eta_t dw_t^h)\) are stationary processes satisfying \(\partial_t y_t= \xi_t\) and \(\partial_{w_t^h} y_t= \eta_t\). Stochastic differential equation \(dy_t= (\alpha y_t+ \mu) dt+ i\gamma y_t dw_t^h\) is solved in the set of processes \(\zeta\), moreover satisfying \(\partial_t y_t= \alpha y_t+ \mu\) and \(\partial_{w_t^h} y_t= i\gamma y_t\). This solution is expressed once again with the spectral decomposition chaotic representation. Finally, a stationary Stratonovich solution of \(dy_t= (\mu+ \alpha y_t) dt+ \gamma y_t dw_t\) is obtained as a byproduct of the previous stochastic differential equations when \(h\) goes to 0.

Reviewer: Monique Pontier (Toulouse)

MSC:

60H05	Stochastic integrals
62M15	Inference from stochastic processes and spectral analysis
93E03	Stochastic systems in control theory (general)

Keywords:

bilinear systems; long-range dependence; fractional Brownian motion; stationarity

Software:

longmemo

References:

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