The Phi-martingale

Vrins, Frédéric [UCL] Jeanblanc, Monique [CNRS & Université d’Evry Val d’Essone] et al.

In this paper we focus on continuous martingales evolving in the unit interval $[0,1]$. We first review some results about the martingale property of solution to one-dimensional driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. One of these martingales proves to be analytically tractable, and received the specific name of $Phi$-martingale. 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 $sigma(t,x)=g(t)h(x)$ that can be obtained via a time-homogeneous mapping of Gaussian processes. The approach is applied to the modeling of stochastic survival probabilities.