scholarly journals Generalized Memory: Fractional Calculus Approach

2018 ◽  
Vol 2 (4) ◽  
pp. 23 ◽  
Author(s):  
Vasily E. Tarasov

The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

2016 ◽  
Vol 30 (21) ◽  
pp. 1650274 ◽  
Author(s):  
Hyun-Joo Kim

Effects of scale-free avalanche walks on anomalous diffusions have been studied by introducing simple non-Markovian walk models. The scale-free avalanche walk is realized as a walker goes to one direction consistently in a time interval, the distribution of which follows a power-law. And it is applied to the memory models, in which the entire history of a walk process is memorized or the memory for the latest step is enhanced with time. The power-law avalanche walk with memory effects strengthens the persistence between steps and thus makes the Hurst exponent be larger than the cases without avalanche walks, while does not affect the anti-persistent nature.


Author(s):  
Zengjing Chen ◽  
Bo Wang

AbstractIn this paper, we first give a sufficient condition on the coefficients of a class of infinite time interval backward stochastic differential equations (BSDEs) under which the infinite time interval BSDEs have a unique solution for any given square integrable terminal value, and then, using the infinite time interval BSDEs, we study the convergence of g-martingales introduced by Peng via a kind of BSDEs. Finally, we study the applications of g-expectations and g-martingales in both finance and economics.


Author(s):  
N. V. Vareh ◽  
O. Y. Volfson ◽  
O. A. Padalka

In this paper systems of differential equations with deviation of an argument with nonlinearity of general form in each equation are considered. The asymptotic properties of solutions of systems with a pair and odd number of equations on an infinite time interval are studied


2010 ◽  
Vol 7 (4) ◽  
pp. 1458-1461
Author(s):  
Baghdad Science Journal

In this paper, a sufficient condition for stability of a system of nonlinear multi-fractional order differential equations on a finite time interval with an illustrative example, has been presented to demonstrate our result. Also, an idea to extend our result on such system on an infinite time interval is suggested.


Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1857-1868 ◽  
Author(s):  
Zhaojun Zong ◽  
Feng Hu

In this paper, we study the existence and uniqueness theorem for Lp (1 < p < 2) solutions to a class of infinite time interval backward doubly stochastic differential equations (BDSDEs). Furthermore, we obtain the comparison theorem for 1-dimensional infinite time interval BDSDEs in Lp.


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