Fractional generalizations of filtering problems and their associated fractional Zakai equations

Umarov, Sabir; Daum, Fred; Nelson, Kenric P.

doi:10.2478/s13540-014-0197-x

“…The fractional, or time-changed version of the Zakai equation in the general case of time-changed Lévy processes is obtained in the paper [UDN14] for filtering problems driven by Lévy processes. The stochastic integral in equation (7.133) is well defined in the sense of Itô's integral.…”

Section: Filtering Problem: Fractional Zakai Equationmentioning

confidence: 99%

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

¹

2015

Developments in Mathematics

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Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments -distributed and variable fractional order differential operators.Fractional order derivatives interpolate integer order derivatives to real (not necessarily fractional) or complex order derivatives. There are different types of fractional derivatives not always equivalent. The first attempt to develop the fractional calculus systematically was taken by Liouville (1832) and Riemann (1847) in the first half of the nineteenth century, even though discussions on non-integer order derivatives had been started long ago. 1 In the 1870s Letnikov and Grünwald independently used an approach for the definition of the fractional order derivative and integral different from that of Riemann and Liouville. The Cauchy problem for fractional order differential equations with the Riemann-Liouville derivative is not well posed (Section 3.3), that is the Cauchy problem in this case is unphysical. In the 1960s Caputo and Djrbashian introduced independently, so-called, a regularization of the Riemann-Liouville fractional derivative, which was later named a fractional derivative in the sense of Caputo-Djrbashian (Section 3.5). The usefulness of the Caputo-Djrbashian derivative is that the Cauchy problem for fractional order differential equations with the Caputo-Djrbashian derivative is well posed. In the operators language one can write the latter in the form DJ = I, where I is the identity operator, which means that the operator D is a left inverse to the operator J. One can easily check that D is not a right inverse to J, since, according to the same fundamental theorem of calculus, for any differentiable function f the equality JD f (t) = f (t) − f (0) holds. The similar relations are true by induction for operators D n and J n , where D n = d n dt n , "n-th derivative," and J n is the n-fold integration operator. Namely,andThus D n is the left inverse to J n , and is the right inverse to J n in the class of functions satisfying additional conditions: f (k) (0) = 0, k = 0,..., n − 1. These relations between "differentiation" and "integration" operators valid for n = 1, 2, . . . , form the basis for the definitions of fractional derivatives in the sense of RiemannLiouville and Caputo-Djrbashian, as soon as the fractional order integration operator is defined. Fractional order integration operator 123 Fractional order integration operatorIn this section we introduce the fractional order integration operator of order α > 0 (ℜ(α) > 0). One can verify (by changing order of integration) that the n-fold integration operatorTaking into account the relationship Γ (n) = (n − 1)!, whereis the Euler's ...

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“…(ii) The fractional versions of the classical stochastic filtering (see [2] for the basics) has been actively studied recently, see e.g. [44]. (iii) The quantum mean-field games as developed by the author in [25] can now be extended to the theory of fractional quantum mean-field games.…”

Section: Introductionmentioning

confidence: 99%

CTRW modeling of quantum measurement and fractional equations of quantum stochastic filtering and control

Kolokoltsov

¹

2022

Fract Calc Appl Anal

1

0

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Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.

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“…(ii) The fractional versions of the classical stochastic filtering (see [2] for the basics) has been actively studied recently, see e.g. [44].…”

Section: Introductionmentioning

confidence: 99%

Continuous time random walks modeling of quantum measurement and fractional equations of quantum stochastic filtering and control

Kolokoltsov

2020

Preprint

0

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Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.

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Fractional generalizations of filtering problems and their associated fractional Zakai equations

Cited by 7 publications

References 18 publications

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

CTRW modeling of quantum measurement and fractional equations of quantum stochastic filtering and control

Continuous time random walks modeling of quantum measurement and fractional equations of quantum stochastic filtering and control

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