Approximation of random dynamical systems with discrete time by stochastic differential equations: I. Theory

2007 ◽  
Vol 15 (3) ◽  
pp. 205-222
Author(s):  
Yuri A. Godin ◽  
Stanislav Molchanov
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Random Dynamical Systems

Random dynamical systems generated by coalescing stochastic flows on ℝ

2018 ◽  
Vol 18 (04) ◽  
pp. 1850031 ◽  
Author(s):  
Georgii V. Riabov
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
State Space ◽  
Random Dynamical Systems ◽  
Stochastic Flows

Existence of random dynamical systems for a class of coalescing stochastic flows on [Formula: see text] is proved. A new state space for coalescing flows is built. As particular cases coalescing flows of solutions to stochastic differential equations and coalescing Harris flows are considered.

scholarly journals Almost periodicity and periodicity for nonautonomous random dynamical systems

2020 ◽  
pp. 2150034
Author(s):  
Paul Raynaud de Fitte
Keyword(s):  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Random Processes ◽  
Random Dynamical Systems ◽  
Almost Periodicity ◽  
Almost Periodic

We present a notion of almost periodicity which can be applied to random dynamical systems as well as almost periodic stochastic differential equations in Hilbert spaces (abstract stochastic partial differential equations). This concept allows for improvements of known results of almost periodicity in distribution, for general random processes and for solutions to stochastic differential equations.

On effects of discretization on estimators of drift parameters for diffusion processes

1996 ◽  
Vol 33 (04) ◽  
pp. 1061-1076 ◽  
Author(s):  
P. E. Kloeden ◽  
E. Platen ◽  
H. Schurz ◽  
M. Sørensen
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Diffusion Processes ◽  
Maximum Likelihood Estimators ◽  
Statistical Properties ◽  
Stochastic Integrals

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

Continuous Autoregressive Moving Average Models

2021 ◽  
pp. 118-141
Author(s):  
Yakup Ari
Keyword(s):  
Time Series ◽  
Parameter Estimation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Moving Average ◽  
Likelihood Estimation ◽  
Real Data ◽  
Autoregressive Moving Average ◽  
The Difference

The financial time series have a high frequency and the difference between their observations is not regular. Therefore, continuous models can be used instead of discrete-time series models. The purpose of this chapter is to define Lévy-driven continuous autoregressive moving average (CARMA) models and their applications. The CARMA model is an explicit solution to stochastic differential equations, and also, it is analogue to the discrete ARMA models. In order to form a basis for CARMA processes, the structures of discrete-time processes models are examined. Then stochastic differential equations, Lévy processes, compound Poisson processes, and variance gamma processes are defined. Finally, the parameter estimation of CARMA(2,1) is discussed as an example. The most common method for the parameter estimation of the CARMA process is the pseudo maximum likelihood estimation (PMLE) method by mapping the ARMA coefficients to the corresponding estimates of the CARMA coefficients. Furthermore, a simulation study and a real data application are given as examples.

scholarly journals Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Jinshu Chen
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Functional Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Convergence Of Operators

We aim to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales M. We first apply the 2D-Fock transform for operators from the testing functional space S(M) to the generalized functional space S⁎(M) and obtain a necessary and sufficient condition for such operators sequences to be strongly convergent. We then discuss the integration of these operator-valued functions. Finally, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations in terms of operators acting on functionals of discrete-time normal martingales M. And also we prove the continuity and continuous dependence on initial values of the solution.

scholarly journals Asymptotic Dynamics of Young Differential Equations

2021 ◽  
Author(s):  
Luu Hoang Duc ◽  
Phan Thanh Hong
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Random Dynamical Systems ◽  
Analytic Approach ◽  
Pullback Attractors ◽  
Random Attractors ◽  
Asymptotic Dynamics

AbstractWe provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.

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