In this talk, we introduce the concept of conic martingales. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. Specific attention is paid to martingales in [0,1]. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with thetrivial constant, Brownian motion and Geometric Brownian motion) having a separable coefficient sigma(t,y)=g(t)h(y) and that can be obtained via a time-homogeneous mapping of Gaussian processes. The approach is exemplified to the modeling of stochastic conditional survival probabilities in the univariate (both conditional and unconditional to survival) and bivariate cases. Our method allows for perfect calibration to any continuous survival probability curve and the implied Azéma supermartingale is proven to belong to [0,1] almost surely.