  In a general counting process setting, we consider the problem of obtaining a
prognostic on the survival time adjusted on covariates in high-dimension.
Towards this end, we construct an estimator of the whole conditional intensity.
We estimate it by the best Cox proportional hazards model given two
dictionaries of functions. The first dictionary is used to construct an
approximation of the logarithm of the baseline hazard function and the second
to approximate the relative risk. We introduce a new data-driven weighted Lasso
procedure to estimate the unknown parameters of the best Cox model
approximating the intensity. We provide non-asymptotic oracle inequalities for
our procedure in terms of an appropriate empirical Kullback divergence. Our
results rely on an empirical Bernstein's inequality for martingales with jumps
and properties of modified self-concordant functions.
