This paper proposes a universal framework for constructing bivariate stochastic processes,going beyond the limitations of copulas and offering a potentially simpler alternative.The achieved generality of the construct...This paper proposes a universal framework for constructing bivariate stochastic processes,going beyond the limitations of copulas and offering a potentially simpler alternative.The achieved generality of the construction methods extends its applicability to diverse stochastic processes also including discrete as well as continuous time cases.The initially given two arbitrary univariate stochastic processes{Y_(t)},{Z_(t)},are only assumed to share the same time t.When considered as describing(time dependent)random quantities that are physically separated(the baseline case),the processes are independent.From this trivial case we move to the case when physical interactions between the quantities make them stochastically dependent random variables at any moment t.For each time epoch t,we impose stochastic dependence on two“initially independent”random variables Y_(t),Z_(t) by multiplying the product of their survival functions by a proper“dependence factor”φ_(t)(y_(t), z_(t)),obtaining in this way a universal(“canonical”)form valid for any(!)bivariate distribution.In some known cases,however,this form may become complicated thou it always exists and is unique.The dependence factor,basically,but not always,has the form φ_(t)(y, z)=exp[-∫^(y)_(0)∫^(z)_(0)Ψ_(t)(s ,u )dsdu]whenever such a continuous function Ψ_(t)(s ,u ) exists,for each t.That representation of stochastic dependence by the functions Ψ_(t)(s ,u ) leads,in turn,to the phenomenon of change of the original(baseline)hazard rates of the marginals,similar to those analyzed by Cox and,especially Aalen for single pairs(or sets)of,time independent,random variables.That is why,until Section 4,we consider only single random vectors(Y,Z)'joint survival functions,mostly as a preparation to the theory of bivariate stochastic processes{(Y_(t),Z_(t))}constructions as initiated in Section 4.The bivariate constructions are illustrated by examples of some applications in biomedical and econometric areas.展开更多
文摘This paper proposes a universal framework for constructing bivariate stochastic processes,going beyond the limitations of copulas and offering a potentially simpler alternative.The achieved generality of the construction methods extends its applicability to diverse stochastic processes also including discrete as well as continuous time cases.The initially given two arbitrary univariate stochastic processes{Y_(t)},{Z_(t)},are only assumed to share the same time t.When considered as describing(time dependent)random quantities that are physically separated(the baseline case),the processes are independent.From this trivial case we move to the case when physical interactions between the quantities make them stochastically dependent random variables at any moment t.For each time epoch t,we impose stochastic dependence on two“initially independent”random variables Y_(t),Z_(t) by multiplying the product of their survival functions by a proper“dependence factor”φ_(t)(y_(t), z_(t)),obtaining in this way a universal(“canonical”)form valid for any(!)bivariate distribution.In some known cases,however,this form may become complicated thou it always exists and is unique.The dependence factor,basically,but not always,has the form φ_(t)(y, z)=exp[-∫^(y)_(0)∫^(z)_(0)Ψ_(t)(s ,u )dsdu]whenever such a continuous function Ψ_(t)(s ,u ) exists,for each t.That representation of stochastic dependence by the functions Ψ_(t)(s ,u ) leads,in turn,to the phenomenon of change of the original(baseline)hazard rates of the marginals,similar to those analyzed by Cox and,especially Aalen for single pairs(or sets)of,time independent,random variables.That is why,until Section 4,we consider only single random vectors(Y,Z)'joint survival functions,mostly as a preparation to the theory of bivariate stochastic processes{(Y_(t),Z_(t))}constructions as initiated in Section 4.The bivariate constructions are illustrated by examples of some applications in biomedical and econometric areas.
