Algebras Generated by the Bergman Projection and Operators of Multiplication by Piecewise Continuous Functions

The “function” δ(x - xo) is known as the Dirac Delta function and may be defined as zero everywhere except at xo, where it is infinite in such a way that1having property that for every continuous function φ(x) on (a, b)2It is well known [2] δ(x-xo) can be approximated as a limit of a sequence of piecewise continuous functions, and there is an abundance of such sequences.

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Some explicit results on one kind of sticky diffusion

Journal of Applied Probability ◽

10.1017/jpr.2019.22 ◽

2019 ◽

Vol 56 (2) ◽

pp. 398-415

Author(s):

Yiming Jiang ◽

Shiyu Song ◽

Yongjin Wang

Keyword(s):

Diffusion Process ◽

Continuous Functions ◽

Price Dynamics ◽

Call Option ◽

Distributional Properties ◽

Financial Application ◽

Piecewise Continuous

AbstractIn this paper we derive several explicit results on one special sticky diffusion process which is constructed as a time-changed version of a diffusion with no sticky points. A theorem concerning the process-related Green operators defined on some nonnegative piecewise continuous functions is provided. Then, based on this theorem, we explore the distributional properties of the sticky diffusion. A financial application is presented where we compute the value of the European vanilla call option written on the underlying with sticky price dynamics.

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Synthesis of intelligent fuzzy control in the space of bounded piecewise-continuous functions based on the combined maximum principle

MATEC Web of Conferences ◽

10.1051/matecconf/201822604031 ◽

2018 ◽

Vol 226 ◽

pp. 04031 ◽

Cited By ~ 1

Author(s):

Andrey A. Kostoglotov ◽

Sergey V. Lazarnko ◽

Igor A. Nikitin

Keyword(s):

Control Method ◽

Control Object ◽

Continuous Functions ◽

Terminal State ◽

Control Synthesis ◽

Continuous Control ◽

Control Laws ◽

Relay Control ◽

Piecewise Continuous ◽

Two Stages

It is shown that the solution of the problem of control synthesis using the Hamilton-Ostrogradskii principle leads to a variational inequality, from which the conditions for the maximum of the the generalized power function in the space of bounded piecewise continuous functions follow. It allows to find a feedback structure up to a synthesis function. Using the methods of the structure construction the nonlinear structures of the relay and continuous control laws are obtained. The proposed control method allows to avoid the mode with frequented switching. It consists of the following two stages. At the first stage, the control object is brought into the vicinity of the terminal state using the relay control law. In the second stage, the quasi-optimal continuous control is used. The uncertainty of the transition area size is resolved using fuzzy logic. The efficiency of the intelligent controls is demonstrated by the example of mathematical modeling of the system dynamics.

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