scholarly journals Dynamical Behavior of the Stochastic Delay Mutualism System

2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Peiyan Xia ◽  
Daqing Jiang ◽  
Xiaoyue Li
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Positive Solution ◽  
Asymptotic Property ◽  
Dynamical Behavior ◽  
Asymptotic Properties ◽  
Unique Positive Solution ◽  
Stochastic Delay ◽  
Mutualism System

We discuss the dynamical behavior of the stochastic delay three-specie mutualism system. We develop the technique for stochastic differential equations to deal with the asymptotic property. Using it we obtain the existence of the unique positive solution, the asymptotic properties, and the nonpersistence. Finally, we give the numerical examinations to illustrate our results.

scholarly journals Estimation of the Hurst index of the solutions of fractional SDE with locally Lipschitz drift

2020 ◽  
Vol 25 (6) ◽  
pp. 1059-1078
Author(s):  
Kęstutis Kubilius
Keyword(s):  
Differential Equations ◽  
Positive Solution ◽  
Euler Scheme ◽  
Hurst Index ◽  
Unique Positive Solution ◽  
Backward Euler Scheme ◽  
Asymptotically Normal ◽  
Backward Euler ◽  
Locally Lipschitz ◽  
Strongly Consistent

Strongly consistent and asymptotically normal estimates of the Hurst index H are obtained for stochastic differential equations (SDEs) that have a unique positive solution. A strongly convergent approximation of the considered SDE solution is constructed using the backward Euler scheme. Moreover, it is proved that the Hurst estimator preserves its properties, if we replace the solution with its approximation.

scholarly journals Parameter Estimation for Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Na Song ◽  
Zaiming Liu
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Asymptotic Properties ◽  
Regularity Conditions ◽  
Minimum Distance Estimator ◽  
Mixed Fractional Brownian Motion ◽  
Distance Estimator ◽  
Drift Parameter

We study the asymptotic properties of minimum distance estimator of drift parameter for a class of nonlinear scalar stochastic differential equations driven by mixed fractional Brownian motion. The consistency and limit distribution of this estimator are established as the diffusion coefficient tends to zero under some regularity conditions.

scholarly journals Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations

2011 ◽  
Vol 43 (02) ◽  
pp. 572-596 ◽  
Author(s):  
Bernt Øksendal ◽  
Agnès Sulem ◽  
Tusheng Zhang
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Backward Stochastic Differential Equation ◽  
Maximum Principles ◽  
Optimal Consumption ◽  
Control Problems ◽  
Stochastic Delay ◽  
Stochastic Delay Equations

We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

scholarly journals Nonparametric estimation of trend function for stochastic differential equations driven by a bifractional Brownian motion

2020 ◽  
Vol 12 (1) ◽  
pp. 128-145
Author(s):  
Abdelmalik Keddi ◽  
Fethi Madani ◽  
Amina Angelika Bouchentouf
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Nonparametric Estimation ◽  
Unknown Function ◽  
Asymptotic Properties ◽  
Kernel Estimator ◽  
Trend Function ◽  
Bifractional Brownian Motion ◽  
Numerical Example

AbstractThe main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential equations of the type {\rm{d}}{{\rm{X}}_{\rm{t}}} = {\rm{S}}\left( {{{\rm{X}}_{\rm{t}}}} \right){\rm{dt + }}\varepsilon {\rm{dB}}_{\rm{t}}^{{\rm{H,K}}},\,{{\rm{X}}_{\rm{0}}} = {{\rm{x}}_{\rm{0}}},\,0 \le {\rm{t}} \le {\rm{T,}}where {{\rm{B}}_{\rm{t}}^{{\rm{H,K}}},{\rm{t}} \ge {\rm{0}}} is a bifractional Brownian motion with known parameters H ∈ (0, 1), K ∈ (0, 1] and HK ∈ (1/2, 1). We estimate the unknown function S(xt) by a kernel estimator ̂St and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.

scholarly journals Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations

2011 ◽  
Vol 43 (2) ◽  
pp. 572-596 ◽  
Author(s):  
Bernt Øksendal ◽  
Agnès Sulem ◽  
Tusheng Zhang
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Backward Stochastic Differential Equation ◽  
Maximum Principles ◽  
Optimal Consumption ◽  
Control Problems ◽  
Stochastic Delay ◽  
Stochastic Delay Equations

We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

scholarly journals Statistical Inference for Stochastic Differential Equations with Small Noises

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Liang Shen ◽  
Qingsong Xu
Keyword(s):  
Differential Equations ◽  
Normal Distribution ◽  
Least Squares ◽  
Stochastic Differential Equations ◽  
Asymptotic Distribution ◽  
Asymptotic Property ◽  
Stable Distribution ◽  
Least Squares Method ◽  
Regularity Conditions ◽  
Drift Parameter

This paper proposes the least squares method to estimate the drift parameter for the stochastic differential equations driven by small noises, which is more general than pure jumpα-stable noises. The asymptotic property of this least squares estimator is studied under some regularity conditions. The asymptotic distribution of the estimator is shown to be the convolution of a stable distribution and a normal distribution, which is completely different from the classical cases.

scholarly journals Asymptotic properties of coupled forward–backward stochastic differential equations

2014 ◽  
Vol 14 (03) ◽  
pp. 1450004 ◽  
Author(s):  
Ana Bela Cruzeiro ◽  
André de Oliveira Gomes ◽  
Liangquan Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Asymptotic Properties ◽  
Backward Stochastic Differential Equations ◽  
Type Formula ◽  
Deviation Principle

In this paper, we consider coupled forward–backward stochastic differential equations (FBSDEs in short) with parameter ε > 0, of the following type [Formula: see text] We study the asymptotic behavior of its solutions and establish a large deviation principle for the corresponding processes.

Reconstruction of Complex Dynamical Systems Using Stochastic Differential Equations

2020 ◽  
Author(s):  
Forough Hassanibesheli ◽  
Niklas Boers ◽  
Jürgen Kurths
Keyword(s):  
Time Series ◽  
Differential Equations ◽  
Complex Systems ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Degrees Of Freedom ◽  
Evolution Equations ◽  
Ice Core ◽  
Dynamical Behavior ◽  
Generalized Langevin Equation

<p>A complex system is a system composed of highly interconnected components in which the collective property of an underlying system cannot be described by dynamical behavior of the individual parts. Typically, complex systems are governed by nonlinear interactions and intricate fluctuations, thus to retrieve dynamics of a system, it is required to characterize and asses interactions between deterministic tendencies and random fluctuations. </p><p>For systems with large numbers of degrees of freedom, interacting across various time scales, deriving time-evolution equations from data is computationally expensive. A possible way to circumvent this problem is to isolate a small number of relatively slow degrees of freedom that may suffice to characterize the underlying dynamics and solve the governing motion equation for the reduced-dimension system in the framework of stochastic differential equations(SDEs).  For some specific example settings, we have studied the performance of three stochastic dimension-reduction methods (Langevin equation(LE), generalized Langevin Equation(GLE) and Empirical Model Reduction(EMR)) to model various synthetic and real-world time series. In this study corresponding numerical simulations of all models have been examined by probability distribution function(PDF) and Autocorrelation function(ACF) of the average simulated time series as statistical benchmarks for assessing the differnt models' performance. </p><p>First we reconstruct the Niño-3 monthly sea surface temperature (SST) indices averages across (5°N–5°S, 150°–90°W) from 1891 to 2015 using the three aforementioned stochastic models. We demonstrate that all these considered models can reproduce the same skewed and heavy-tailed distributions of Niño-3 SST, comparing ACFs, GLE exhibits a tendency towards achieving a higher accuracy than LE and EMR. A particular challenge for deriving the underlying dynamics of complex systems from data is given by situations of abrupt transitions between alternative states. We show how the Kramers-Moyal approach to derive drift and diffusion terms for LEs can help in such situations. A prominent example of such 'Tipping Events' is given by the Dansgaard-Oeschger events during previous glacial intervals. We attempt to obtain the statistical properties of high-resolution, 20yr average, δ<sup>18</sup>O and Ca<sup>+</sup><sup>2</sup> collected from the same ice core from the NGRIP on the GICC05 time scale. Through extensive analyses of various systems, our results signify that stochastic differential equation models considering memory effects are comparatively better approaches for understanding  complex systems.</p><p> </p>

scholarly journals Uniqueness of Successive Positive Solution for Nonlocal Singular Higher-Order Fractional Differential Equations Involving Arbitrary Derivatives

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Qiuyan Zhong ◽  
Xingqiu Zhang ◽  
Xinyi Lu ◽  
Zhengqing Fu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Positive Solution ◽  
Fractional Differential Equations ◽  
Higher Order ◽  
Unique Positive Solution ◽  
Mixed Monotone Operator ◽  
Iterative Schemes ◽  
Fractional Differential

In this article, by means of fixed point theorem on mixed monotone operator, we establish the uniqueness of positive solution for some nonlocal singular higher-order fractional differential equations involving arbitrary derivatives. We also give iterative schemes for approximating this unique positive solution.

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