scholarly journals Governing equations for Probability densities of stochastic differential equations with discrete time delays

2017 ◽  
Vol 22 (9) ◽  
pp. 3615-3628 ◽  
Author(s):  
Yayun Zheng ◽  
◽  
Xu Sun
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Time Delays ◽  
Governing Equations ◽  
Probability Densities

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 Nonparametric inference of stochastic differential equations based on the relative entropy rate

2021 ◽  
Author(s):  
Min Dai ◽  
Jinqiao Duan ◽  
jianyu Hu ◽  
Xiangjun Wang
Keyword(s):  
Differential Equations ◽  
Complex Systems ◽  
Stochastic Differential Equations ◽  
Relative Entropy ◽  
Nonparametric Inference ◽  
Entropy Rate ◽  
Governing Equations ◽  
Nonparametric Approach ◽  
Information Detection ◽  
Kernel Theory

The information detection of complex systems from data is currently undergoing a revolution,driven by the emergence of big data and machine learning methodology. Discovering governingequations and quantifying dynamical properties of complex systems are among central challenges. Inthis work, we devise a nonparametric approach to learn the relative entropy rate from observationsof stochastic differential equations with different drift functions. The estimator corresponding tothe relative entropy rate then is presented via the Gaussian process kernel theory. Meanwhile, thisapproach enables to extract the governing equations. We illustrate our approach in several examples.Numerical experiments show the proposed approach performs well for rational drift functions, notonly polynomial drift functions.

On Optimum Filtering for a Class of Linear Distributed-Parameter Systems

1969 ◽  
Vol 91 (2) ◽  
pp. 173-178 ◽  
Author(s):  
F. E. Thau
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Process ◽  
Discrete Time ◽  
Continuous Time ◽  
Time Delays ◽  
Distributed Parameter ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Variance Equation

Filtering equations are derived for processes described by linear partial differential equations with known homogeneous boundary conditions. Both discrete-time and continuous-time measurements are treated. As in the case of linear systems with time delays, the filtering and variance equations become partial differential equations for processes with continuous measurements. A numerical solution to the nonlinear variance equation is obtained for a particular diffusion process.

The passage from random walk to diffusion in quantum probability

1988 ◽  
Vol 25 (A) ◽  
pp. 151-166 ◽  
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Random Walk ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Continuous Time ◽  
Quantum Probability ◽  
Diffusion Limit ◽  
Quantum Random Walk

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.

scholarly journals A discrete-time approximation for doubly reflected BSDEs

2009 ◽  
Vol 41 (01) ◽  
pp. 101-130 ◽  
Author(s):  
Jean-François Chassagneux
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Euler Scheme ◽  
Time Approximation ◽  
Type Approximation ◽  
Convergence Results ◽  
Discrete Time Approximation ◽  
Regularity Results ◽  
Reflected Bsdes

We study the discrete-time approximation of doubly reflected backward stochastic differential equations (BSDEs) in a multidimensional setting. As in Ma and Zhang (2005) or Bouchard and Chassagneux (2008), we introduce the discretely reflected counterpart of these equations. We then provide representation formulae which allow us to obtain new regularity results. We also propose an Euler scheme type approximation and give new convergence results for both discretely and continuously reflected BSDEs.

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