A General Method for Construction of Bivariate Stochastic Processes Given two Marginal Processes

Given two arbitrary univariate stochastic processes {Yt}, {Zt}, assumed to only share the same time t. When considered as describing (time dependent) random quantities that are physically separated (the baseline case), the processes are independent for every time epoch t. From this trivial case we move to the case where physical interactions between the quantities make them, at any moment t, stochastically dependent. For each time epoch t, we impose stochastic dependence on two “initially independent” random variables Yt, Zt by multiplying the product of their survival functions by a proper ‘dependence factor’  φt (yt,zt) , obtaining in this way a universal (“canonical”) form of any (!) bivariate distribution (in some known cases, however, this form may become complicated thou it always exists). This factor, basically, may have the form φt(y,z) = exp[ – ∫0y ∫0z  ψt(s;u) dsdu ] whenever such a function ψt(s;u) exists, for each t.  That representation of stochastic dependence by the functions ψt (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 (1972) and, especially Aalen  (1989) for single pairs (or sets) of, time independent, random variables. That is why, until Section 4, we would rather consider single random vectors (Y, Z)’ joint survival functions, mostly as a preparation to the theory of bivariate stochastic processes {(Yt, Zt)} constructions as initiated in Section 4. 

The bivariate constructions are illustrated by examples of some applications in biomedical and econometric areas. Reliability applications, associated with the considered “micro shock 🡪 microdamage” paradigm, obviously may follow.  

The authors declare no conflict of interest.

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The datasets used in this study are openly available at [repository link] and the source code is available on GitHub at [GitHub link].

This work did not receive any external funding.