Conic Martingales from Stochastic Integrals

34 Pages Posted: 23 Mar 2016 Last revised: 17 May 2017

See all articles by Frédéric D. Vrins

Frédéric D. Vrins

Louvain Finance Center (LFIN), UC Louvain; Center for Operations Research and Econometrics (CORE), UC Louvain

Monique Jeanblanc

Université d'Évry - Departement de Mathematiques

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Date Written: February 23, 2016

Abstract

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 (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.

Keywords: bounded martingale, stochastic differential equation, diffusion process, stochastic survival probability

Suggested Citation

Vrins, Frederic Daniel and Jeanblanc, Monique, Conic Martingales from Stochastic Integrals (February 23, 2016). Available at SSRN: https://ssrn.com/abstract=2753529 or http://dx.doi.org/10.2139/ssrn.2753529

Frederic Daniel Vrins (Contact Author)

Louvain Finance Center (LFIN), UC Louvain ( email )

Voie du Roman Pays 34
Louvain-la-Neuve, 1348
Belgium

HOME PAGE: http://www.uclouvain.be/frederic.vrins

Center for Operations Research and Econometrics (CORE), UC Louvain ( email )

Voie du Roman Pays 34
Louvain-la-Neuve,, B-1348
Belgium

Monique Jeanblanc

Université d'Évry - Departement de Mathematiques ( email )

rue du pere Jarlan
F-91025 Evry Cedex
France
33 (0) 1 69 47 02 05/ 02 01 (Phone)
33 (0) 1 69 47 02 18 (Fax)

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