The Phi-martingale

Vrins, Frédéric;Jeanblanc, Monique;et.al.
(2015) International Actuarial Association Colloquium — Location: Oslo, Norway (7.June.2015)

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  • Vrins, Frédéricorcid-logoUCLouvain
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  • Jeanblanc, MoniqueCNRS & Université d’Evry Val d’Essone
    Author
  • et. al.
Abstract
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.
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Vrins, F., Jeanblanc, M., & et al. (2015). The Phi-martingale. International Actuarial Association Colloquium, Oslo, Norway. https://hdl.handle.net/2078.5/194467