  In this paper 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. We first review some results
about the martingale property of solution to driftless stochastic differential
equations. We then provide a simple way to construct and handle such processes.
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 the trivial 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 diffusions. The approach is exemplified to the modeling of
stochastic conditional survival probabilities in the univariate (both
conditional and unconditional to survival) and bivariate cases.
