• Title/Summary/Keyword: Stochastic differential equations

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AN EXISTENCE OF THE SOLUTION TO NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS UNDER SPECIAL CONDITIONS

  • KIM, YOUNG-HO
    • Journal of applied mathematics & informatics
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    • v.37 no.1_2
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    • pp.53-63
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    • 2019
  • In this paper, we show the existence of solution of the neutral stochastic functional differential equations under non-Lipschitz condition, a weakened linear growth condition and a contractive condition. Furthermore, in order to obtain the existence of solution to the equation we used the Picard sequence.

WEAKLY STOCHASTIC RUNGE-KUTTA METHOD WITH ORDER 2

  • Soheili, Ali R.;Kazemi, Zahra
    • Journal of applied mathematics & informatics
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    • v.26 no.1_2
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    • pp.135-149
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    • 2008
  • Many deterministic systems are described by Ordinary differential equations and can often be improved by including stochastic effects, but numerical methods for solving stochastic differential equations(SDEs) are required, and work in this area is far less advanced than for deterministic differential equations. In this paper,first we follow [7] to describe Runge-Kutta methods with order 2 from Taylor approximations in the weak sense and present two well known Runge-Kutta methods, RK2-TO and RK2-PL. Then we obtain a new 3-stage explicit Runge-Kutta with order 2 in weak sense and compare the numerical results among these three methods.

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BERRY-ESSEEN BOUND FOR MLE FOR LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION

  • RAO B.L.S. PRAKASA
    • Journal of the Korean Statistical Society
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    • v.34 no.4
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    • pp.281-295
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    • 2005
  • We investigate the rate of convergence of the distribution of the maximum likelihood estimator (MLE) of an unknown parameter in the drift coefficient of a stochastic process described by a linear stochastic differential equation driven by a fractional Brownian motion (fBm). As a special case, we obtain the rate of convergence for the case of the fractional Ornstein- Uhlenbeck type process studied recently by Kleptsyna and Le Breton (2002).

ON MARTINGALE PROPERTY OF THE STOCHASTIC INTEGRAL EQUATIONS

  • KIM, WEONBAE
    • Korean Journal of Mathematics
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    • v.23 no.3
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    • pp.491-502
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    • 2015
  • A martingale is a mathematical model for a fair wager and the modern theory of martingales plays a very important and useful role in the study of the stochastic fields. This paper is devoted to investigate a martingale and a non-martingale on the several stochastic integral or differential equations. Specially, we show that whether the stochastic integral equation involving a standard Wiener process with the associated filtration is or not a martingale.

REFLECTED BSDE DRIVEN BY A L$\acute{E}$VY PROCESS WITH STOCHASTIC LIPSCHITZ COEFFICIENT

  • Lu, Wen
    • Journal of applied mathematics & informatics
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    • v.28 no.5_6
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    • pp.1305-1314
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    • 2010
  • In this paper, we deal with a class of one-dimensional reflected backward stochastic differential equations driven by a Brownian motion and the martingales of Teugels associated with an independent L$\acute{e}$vy process having a stochastic Lipschitz coefficient. We derive the existence and uniqueness of solutions for these equations via Snell envelope and the fixed point theorem.

Stochastic vibration response of a sandwich beam with nonlinear adjustable visco-elastomer core and supported mass

  • Ying, Z.G.;Ni, Y.Q.;Duan, Y.F.
    • Structural Engineering and Mechanics
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    • v.64 no.2
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    • pp.259-270
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    • 2017
  • The stochastic vibration response of the sandwich beam with the nonlinear adjustable visco-elastomer core and supported mass under stochastic support motion excitations is studied. The nonlinear dynamic properties of the visco-elastomer core are considered. The nonlinear partial differential equations for the horizontal and vertical coupling motions of the sandwich beam are derived. An analytical solution method for the stochastic vibration response of the nonlinear sandwich beam is developed. The nonlinear partial differential equations are converted into the nonlinear ordinary differential equations representing the nonlinear stochastic multi-degree-of-freedom system by using the Galerkin method. The nonlinear stochastic system is converted further into the equivalent quasi-linear system by using the statistic linearization method. The frequency-response function, response spectral density and mean square response expressions of the nonlinear sandwich beam are obtained. Numerical results are given to illustrate new stochastic vibration response characteristics and response reduction capability of the sandwich beam with the nonlinear visco-elastomer core and supported mass under stochastic support motion excitations. The influences of geometric and physical parameters on the stochastic response of the nonlinear sandwich beam are discussed, and the numerical results of the nonlinear sandwich beam are compared with those of the sandwich beam with linear visco-elastomer core.

EXISTENCE UNIQUENESS AND STABILITY OF NONLOCAL NEUTRAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH RANDOM IMPULSES AND POISSON JUMPS

  • CHALISHAJAR, DIMPLEKUMAR;RAMKUMAR, K.;RAVIKUMAR, K.;COX, EOFF
    • Journal of Applied and Pure Mathematics
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    • v.4 no.3_4
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    • pp.107-122
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    • 2022
  • This manuscript aims to investigate the existence, uniqueness, and stability of non-local random impulsive neutral stochastic differential time delay equations (NRINSDEs) with Poisson jumps. First, we prove the existence of mild solutions to this equation using the Banach fixed point theorem. Next, we demonstrate the stability via continuous dependence initial value. Our study extends the work of Wang, and Wu [16] where the time delay is addressed by the prescribed phase space 𝓑 (defined in Section 3). To illustrate the theory, we also provide an example of our methods. Using our results, one could investigate the controllability of random impulsive neutral stochastic differential equations with finite/infinite states. Moreover, one could extend this study to analyze the controllability of fractional-order of NRINSDEs with Poisson jumps as well.

DIFFERENTIABILITY OF NEUTRAL STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY G-BROWNIAN MOTION WITH RESPECT TO THE INITIAL DATA

  • Zakaria Boumezbeur;Hacene Boutabia
    • Honam Mathematical Journal
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    • v.45 no.3
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    • pp.433-456
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    • 2023
  • This paper deals with differentiability of solutions of neutral stochastic differential equations with respect to the initial data in the G-framework. Since the initial data belongs to the space BC ([-r, 0] ; ℝn) of bounded continuous ℝn-valued functions defined on [-r, 0] (r > 0), the derivative belongs to the Banach space 𝓛BC (ℝn) of linear bounded operators from BC ([-r, 0] ; ℝn) to ℝn. We give the neutral stochastic differential equation of the derivative. In addition, we exhibit two examples confirming the accuracy of the obtained results.

ON FUZZY STOCHASTIC DIFFERENTIAL EQUATIONS

  • KIM JAI HEUI
    • Journal of the Korean Mathematical Society
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    • v.42 no.1
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    • pp.153-169
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    • 2005
  • A fuzzy stochastic differential equation contains a fuzzy valued diffusion term which is defined by stochastic integral of a fuzzy process with respect to 1-dimensional Brownian motion. We prove the existence and uniqueness of the solution for fuzzy stochastic differential equation under suitable Lipschitz condition. To do this we prove and use the maximal inequality for fuzzy stochastic integrals. The results are illustrated by an example.