The problem of parametrization of the conditional survival function in the case of partially observed randomly changing covariates is considered. The general formula for a hazard rate as a function of the value of the observed covariate in the case of one measurement is derived for a wide class of stochastic processes associated with covariates. When covariates are described by the Gauss Markov type stochastic process, the conditional Gaussian property of the distribution of the unobservables is used to get the parametric specification of the hazard rate as a function of observed covariates