Mean-square dissipative methods for stochastic age-dependent capital system with fractional Brownian motion and jumps

The aim of this paper is to study a generalization of fractional Airy differential equations whose input data (coefficient and initial conditions) are random variables. Under appropriate hypotheses assumed upon the input data, we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense. Afterwards, we provide reliable explicit approximations for the main statistical information of the solution process (mean, variance and covariance). Further, we show a set of numerical examples where our obtained theory is illustrated. More precisely, we show that our results for the random fractional Airy equation are in full agreement with the corresponding to classical random Airy differential equation available in the extant literature. Finally, we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional Airy differential equation by combining the knowledge of the mean and the variance and the Principle of Maximum Entropy.

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“…So, if we choose 7), it is guarantee that Y(t) defined by (3) will satisfy the random fractional differential equa-…”

Section: Constructing the Solution Stochastic Processmentioning

confidence: 99%

“…The success of FDEs to deal with this type of problems lies in their ability to account for memory effects [1,2,3]. Some recent papers dealing with analytic and numerical techniques to study interesting problems about FDEs can be found in [4,5,6,7,8,9], for example.…”

Section: Introductionmentioning

confidence: 99%

Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus

Burgos

López

Debbouche

et al. 2019

Applied Mathematics and Computation

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“…However, the randomness of the external environment is not always well modeled by the standard Brownian motion because of the long-range dependence of price of the financial products. In recent years, some researchers adopted fractional Brownian motion (fBm) to describe this long-range dependence of price in the financial market [5][6][7][8][9]. Therefore, it is a very interesting topic to take Poisson jumps and fBm into account for the stochastic capital systems, and we will consider the following system in this paper:…”

Section: Introductionmentioning

confidence: 99%

Strong convergence of the split-step θ-method for stochastic age-dependent capital system with Poisson jumps and fractional Brownian motion

Kang

Zhang

2018

Adv Differ Equ

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Most stochastic age-dependent capital systems cannot be solved explicitly, so it is necessary to develop numerical methods and study the properties of numerical solutions. In this paper, we consider a class of stochastic age-dependent capital systems with Poisson jumps and fractional Brownian motion (fBm) and investigate the convergence of the split-step θ-method (SSθ) for this system. It is proved that the numerical approximation solutions converge to the analytic solutions for the equations, and the order of approximation is also provided. Finally, a numerical experiment is simulated to illustrate that the SSθ method has better accuracy than the Euler method.

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“…This property helps researchers to propose more accurate models of various phenomena [34,6,3,5,16,7,21]. Thus, many problems in physics, economics and other sciences have been modeled as fractional stochastic differential equations [38,44,1,4]. In [14] the global existence and uniqueness of solutions for the FSDEs in the Caputo sense are described.…”

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confidence: 99%

A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay

Banihashemi¹,

Jafaria²,

Babaei³

2022

DCDS-S

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In present work, a step-by-step Legendre collocation method is employed to solve a class of nonlinear fractional stochastic delay differential equations (FSDDEs). The step-by-step method converts the nonlinear FSDDE into a non-delay nonlinear fractional stochastic differential equation (FSDE). Then, a Legendre collocation approach is considered to obtain the numerical solution in each step. By using a collocation scheme, the non-delay nonlinear FSDE is reduced to a nonlinear system. Moreover, the error analysis of this numerical approach is investigated and convergence rate is examined. The accuracy and reliability of this method is shown on three test examples and the effect of different noise measures is investigated. Finally, as an useful application, the proposed scheme is applied to obtain the numerical solution of a stochastic SIRS model.

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Mean-square dissipative methods for stochastic age-dependent capital system with fractional Brownian motion and jumps

Cited by 7 publications

References 22 publications

Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus

Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus

Strong convergence of the split-step θ-method for stochastic age-dependent capital system with Poisson jumps and fractional Brownian motion

A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay

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