Stochastic differential equation of the optimal non-linear filtering of the conditional Gaussian process

2005 ◽  
pp. 152-161
Author(s):  
O. A. Glonti
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Gaussian Process ◽  
Linear Filtering ◽  
Non Linear

scholarly journals Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion

2018 ◽  
Vol 21 (5) ◽  
pp. 1420-1435 ◽  
Author(s):  
Mirko D’Ovidio ◽  
Silvia Vitali ◽  
Vittoria Sposini ◽  
Oleksii Sliusarenko ◽  
Paolo Paradisi ◽  
...  
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Gaussian Process ◽  
Relaxation Times ◽  
Random Variable ◽  
Fractional Diffusion ◽  
Time Dependent ◽  
Centre Of Mass ◽  
Time Scaling

Abstract We consider an ensemble of Ornstein–Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable corresponding to this ensemble is statistically equivalent to a process driven by a non-autonomous stochastic differential equation with time-dependent drift and a white noise. In particular, the time scaling and the density function of such variable are driven by the population of timescales and of noise amplitudes, respectively. Moreover, we show that this variable is equivalent in distribution to a randomly-scaled Gaussian process, i.e., a process built by the product of a Gaussian process times a non-negative independent random variable. This last result establishes a connection with the so-called generalized grey Brownian motion and suggests application to model fractional anomalous diffusion in biological systems.

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