scholarly journals The log log law for multidimensional stochastic integrals and diffusion processes

1971 ◽  
Vol 5 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Ludwig Arnold
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Wiener Process ◽  
Diffusion Processes ◽  
Stochastic Integrals ◽  
Dimensional Diffusion ◽  
Vector Process ◽  
Log Law ◽  
Image Position ◽  
And Diffusion

Let for t ∈ [a, b] ⊂ [0, ∞) where Ws is an n-dimensional Wiener process, f(s) an n-vector process and G(s) an n × m matrix process. f and G are nonanticipating and sample continuous. Then the set of limit points of the net in Rn is equal, almost surely, to the random ellipsoid Et = G(t)Sm, Sm = {x ∈ Rm: |x| ≤ 1}. The analogue of Lévy's law is also given. The results apply to n-dimensional diffusion processes which are solutions of stochastic differential equations, thus extending the versions of Hinčin's and Lévy's laws proved by H.P. McKean, Jr, and W.J. Anderson.

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.

Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Numerical Simulation of One-Dimensional Diffusion Process

1996 ◽  
Vol 61 (4) ◽  
pp. 512-535 ◽  
Author(s):  
Pavel Hasal ◽  
Vladimír Kudrna
Keyword(s):  
Numerical Simulation ◽  
Diffusion Coefficient ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Stochastic Integral ◽  
Random Motion ◽  
One Dimensional ◽  
Dimensional Diffusion

Some problems are analyzed arising when a numerical simulation of a random motion of a large ensemble of diffusing particles is used to approximate the solution of a one-dimensional diffusion equation. The particle motion is described by means of a stochastic differential equation. The problems emerging especially when the diffusion coefficient is a function of spatial coordinate are discussed. The possibility of simulation of various kinds of stochastic integral is demonstrated. It is shown that the application of standard numerical procedures commonly adopted for ordinary differential equations may lead to erroneous results when used for solution of stochastic differential equations. General conclusions are verified by numerical solution of three stochastic differential equations with different forms of the diffusion coefficient.

scholarly journals Diffusion Processes Satisfying a Conservation Law Constraint

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
J. Bakosi ◽  
J. R. Ristorcelli
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Conservation Law ◽  
Evolution Equations ◽  
Diffusion Processes ◽  
Random Variables ◽  
Planck Equation ◽  
Equation Model ◽  
First Four Moments ◽  
And Diffusion

We investigate coupled stochastic differential equations governing N nonnegative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires a set of fluctuating variables to be nonnegative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the nonnegativity and the unit-sum conservation law constraints are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes.

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

1996 ◽  
Vol 33 (4) ◽  
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.

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