1,661 research outputs found
Maximum Principle for General Controlled Systems Driven by Fractional Brownian Motions
We obtain a maximum principle for stochastic control problem of general
controlled stochastic differential systems driven by fractional Brownian
motions (of Hurst parameter ). This maximum principle specifies a system
of equations that the optimal control must satisfy (necessary condition for the
optimal control). This system of equations consists of a backward stochastic
differential equation driven by both fractional Brownian motion and the
corresponding underlying standard Brownian motion. In addition to this backward
equation, the maximum principle also involves the Malliavin derivatives. Our
approach is to use conditioning and Malliavin calculus. To arrive at our
maximum principle we need to develop some new results of stochastic analysis of
the controlled systems driven by fractional Brownian motions via fractional
calculus. Our approach of conditioning and Malliavin calculus is also applied
to classical system driven by standard Brownian motion while the controller has
only partial information. As a straightforward consequence, the classical
maximum principle is also deduced in this more natural and simpler way.Comment: 44 page
Fractional stochastic differential equations satisfying fluctuation-dissipation theorem
We propose in this work a fractional stochastic differential equation (FSDE)
model consistent with the over-damped limit of the generalized Langevin
equation model. As a result of the `fluctuation-dissipation theorem', the
differential equations driven by fractional Brownian noise to model memory
effects should be paired with Caputo derivatives, and this FSDE model should be
understood in an integral form. We establish the existence of strong solutions
for such equations and discuss the ergodicity and convergence to Gibbs measure.
In the linear forcing regime, we show rigorously the algebraic convergence to
Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this
verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the
correct physical behavior. We further discuss possible approaches to analyze
the ergodicity and convergence to Gibbs measure in the nonlinear forcing
regime, while leave the rigorous analysis for future works. The FSDE model
proposed is suitable for systems in contact with heat bath with power-law
kernel and subdiffusion behaviors
The Global Maximum Principle for Optimal Control of Partially Observed Stochastic Systems Driven by Fractional Brownian Motion
In this paper we study the stochastic control problem of partially observed
(multi-dimensional) stochastic system driven by both Brownian motions and
fractional Brownian motions. In the absence of the powerful tool of Girsanov
transformation, we introduce and study new stochastic processes which are used
to transform the original problem to a "classical one". The adjoint backward
stochastic differential equations and the necessary condition satisfied by the
optimal control (maximum principle) are obtained.Comment: 30 page
Fractional generalizations of filtering problems and their associated fractional Zakai equations
In this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process
Multifractal analysis of discretized X-ray CT images for the characterization of soil macropore structures
A correct statistical model of soil pore structure can be critical for understanding flow and transport processes in soils, and creating synthetic soil pore spaces for hypothetical and model testing, and evaluating similarity of pore spaces of different soils. Advanced visualization techniques such as X-ray computed tomography (CT) offer new opportunities of exploring heterogeneity of soil properties at horizon or aggregate scales. Simple fractal models such as fractional Brownian motion that have been proposed to capture the complex behavior of soil spatial variation at field scale rarely simulate irregularity patterns displayed by spatial series of soil properties. The objective of this work was to use CT data to test the hypothesis that soil pore structure at the horizon scale may be represented by multifractal models. X-ray CT scans of twelve, water-saturated, 20-cm long soil columns with diameters of 7.5 cm were analyzed. A reconstruction algorithm was applied to convert the X-ray CT data into a stack of 1480 grayscale digital images with a voxel resolution of 110 microns and a cross-sectional size of 690 Ă— 690 pixels. The images were binarized and the spatial series of the percentage of void space vs. depth was analyzed to evaluate the applicability of the multifractal model. The series of depth-dependent macroporosity values exhibited a well-defined multifractal structure that was revealed by singularity and RĂ©nyi spectra. The long-range dependencies in these series were parameterized by the Hurst exponent. Values of the Hurst exponent close to one were observed indicating the strong persistence in variations of porosity with depth. The multifractal modeling of soil macropore structure can be an efficient method for parameterizing and simulating the vertical spatial heterogeneity of soil pore space
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