Ikeda, N., S. Watanabe: Stochastic Differential Equations and Diffusion Processes. North Holland Publ. Co., Amsterdam — Oxford — New York 1981, 480 S., Dfl. 175

1986 ◽  
Vol 28 (4) ◽  
pp. 510-510
Author(s):  
K. Matthes
Keyword(s):  
New York ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Diffusion Processes ◽  
North Holland Publ ◽  
And Diffusion

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.

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.

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