scholarly journals STATISTICAL CAUSALITY AND STABLE SUBSPACES OF 

2012 ◽  
Vol 88 (1) ◽  
pp. 17-25 ◽  
Author(s):  
LJILJANA PETROVIĆ ◽  
DRAGANA VALJAREVIĆ
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Weak Solutions ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Statistical Concept ◽  
Extremal Measures ◽  
Probability Spaces ◽  
Necessary And Sufficient

AbstractIn this paper we consider the statistical concept of causality in continuous time between filtered probability spaces, based on Granger’s definitions of causality. Then we consider some stable subspaces of $H^p$ which contain right continuous modifications of martingales $P(A \mid {\mathcal {G}}_t)$. We give necessary and sufficient conditions, in terms of statistical causality, for these spaces to coincide with $H^p$. These results can be applied to extremal measures and regular weak solutions of stochastic differential equations.

scholarly journals ON THE UNITARITY OF STOCHASTIC EVOLUTIONS DRIVEN BY THE SQUARE OF WHITE NOISE

2001 ◽  
Vol 04 (04) ◽  
pp. 579-588 ◽  
Author(s):  
LUIGI ACCARDI ◽  
ANDREAS BOUKAS ◽  
HUI-HSUNG KUO
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Using the closed Itô's table for the renormalized square of white noise, recently obtained by Accardi, Hida, and Kuo in Ref. 4, we consider the problem of providing necessary and sufficient conditions for the unitarity of the solutions of a certain type of quantum stochastic differential equations.

Solutions of Multiparameter Systems of Elliptic Differential Equations

2005 ◽  
Vol 5 (2) ◽  
Author(s):  
Joanna Gawrycka ◽  
Slawomir Rybicki
Keyword(s):  
Differential Equations ◽  
Weak Solutions ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Necessary And Sufficient Conditions ◽  
Variational System ◽  
Bifurcation Points ◽  
Elliptic Differential Equations ◽  
Necessary And Sufficient

AbstractIn this article we study bifurcations of weak solutions of the following multiparameter variational system of elliptic differential equations:We formulate necessary and sufficient conditions for the existence of bifurcation points and global bifurcation points of weak solutions of this system.

scholarly journals Quantum stochastic operator cocycles via associated semigroups

2007 ◽  
Vol 142 (3) ◽  
pp. 535-556 ◽  
Author(s):  
J. MARTIN LINDSAY ◽  
STEPHEN J. WILLS
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
General Principle ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Necessary And Sufficient Conditions ◽  
Contraction Operator ◽  
Stochastic Operator ◽  
Necessary And Sufficient

AbstractA recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation.

STATISTICAL CAUSALITY AND MARTINGALE REPRESENTATION PROPERTY WITH APPLICATION TO STOCHASTIC DIFFERENTIAL EQUATIONS

2014 ◽  
Vol 90 (2) ◽  
pp. 327-338 ◽  
Author(s):  
LJILJANA PETROVIĆ ◽  
DRAGANA VALJAREVIĆ
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Time ◽  
Change Of Measure ◽  
Statistical Concept ◽  
Martingale Representation ◽  
Continuous Semimartingale ◽  
Probability Spaces ◽  
Definition Of ◽  
Securities Trading

AbstractThe paper considers a statistical concept of causality in continuous time between filtered probability spaces, based on Granger’s definition of causality. This causality concept is connected with the preservation of the martingale representation property when the filtration is getting smaller. We also give conditions, in terms of causality, for every martingale to be a continuous semimartingale, and we consider the equivalence between the concept of causality and the preservation of the martingale representation property under change of measure. In addition, we apply these results to weak solutions of stochastic differential equations. The results can be applied to the economics of securities trading.

The λ-classification of continuous-time birth-and-death processes

2003 ◽  
Vol 35 (04) ◽  
pp. 1111-1130 ◽  
Author(s):  
Andrew G. Hart ◽  
Servet Martínez ◽  
Jaime San Martín
Keyword(s):  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Birth And Death Processes ◽  
Positive Recurrent ◽  
Necessary And Sufficient

We study the λ-classification of absorbing birth-and-death processes, giving necessary and sufficient conditions for such processes to be λ-transient, λ-null recurrent and λ-positive recurrent.

Positive fractional continuous-time linear systems with singular pencils

2012 ◽  
Vol 60 (1) ◽  
pp. 9-12 ◽  
Author(s):  
T. Kaczorek
Keyword(s):  
Linear Systems ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Descriptor Systems ◽  
State Variables ◽  
Algebraic Constraints ◽  
Necessary And Sufficient ◽  
Time Linear

Positive fractional continuous-time linear systems with singular pencils A method for checking the positivity and finding the solution to the positive fractional descriptor continuous-time linear systems with singular pencils is proposed. The method is based on elementary row and column operations of the fractional descriptor systems to equivalent standard systems with some algebraic constraints on state variables and inputs. Necessary and sufficient conditions for the positivity of the fractional descriptor systems are established.

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