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Conic martingales from Stochastic integrals

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Jeanblanc, Monique [Université d'Evry] Vrins, Frédéric [UCL]

In this paper we introduce the concept of \textit{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 diffusion coefficient $\sigma(t,y)=g(t)h(y)$ and that can be obtained via a time-homogeneous mapping of \textit{Gaussian diffusions}. The approach is exemplified by modeling stochastic conditional survival probabilities in the univariate and bivariate cases.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès restreint
Publication date 2018
Language Anglais
Journal information "Mathematical Finance" - Vol. 28, no. 2, p. 516-535
Peer reviewed yes
Publisher Wiley
issn 1467-9965
Publication status Publié
Affiliations UCL - SSH/IMMAQ/LFIN-Louvain Finance
UCL - SSH/IMMAQ/CORE-Center for operations research and econometrics
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Bibliographic reference Jeanblanc, Monique ; Vrins, Frédéric. Conic martingales from Stochastic integrals. In: Mathematical Finance, Vol. 28, no. 2, p. 516-535
Permanent URL http://hdl.handle.net/2078.1/176590