Received: 1 October 2015 Accepted: 1 August 2016
Conic martingales from stochastic integrals
Université d’Évry Val d’Essonne, Évry,
Université catholique de Louvain,
Frédéric Vrins, Université catholiquede
& CORE,Voie du RomanPays 34,
1348 Louvain-la-Neuve, Belgium.
The researchof Monique Jeanblancwassup-
ported by ChaireMarketsin Transition,(French
Banking Federation)InstitutLouis Bachelier
and LabexANR 11-LABX-0019.
In this paper, we introduce the concept of conic martin-
gales. This class refers to stochastic processes that have
the martingale property but that evolve within given (pos-
sibly 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 atten-
tion is paid to martingales in [0, 1]. One of these martin-
gales proves to be analytically tractable. It is shown that
up to shifting and rescaling constants, it is the only mar-
tingale (with the trivial constant, Brownian motion, and
geometric Brownian motion) having a separable diﬀusion
coeﬃcient (, )=()() and that can be obtained via
a time-homogeneous mapping of Gaussian diﬀusions.The
approach is exempliﬁed by modeling stochastic conditional
survival probabilities in the univariate and bivariate cases.
bounded martingale, diﬀusion process, stochastic diﬀerential equation,
stochastic survival probability
1 INTRODUCTION AND MOTIVATION
Mathematical ﬁnance relies extensively on martingales, mainly due to the fundamental theorem of
asset pricing. Martingales are also central in measure change techniques. Depending on the situation,
the martingale processes may be subjected to some constraints. For instance, discounted stock prices
and Radon–Nikodym derivative processes are positive. Therefore, continuous exponential martingales,
which meet the nonnegativity constraint, are very popular tools.
But ﬁnancial processes can also be subjected to other constraints. Under the risk-neutral mea-
sure ℚ associated to the bank account numéraire, for instance, discounted zero-coupon bond prices
are martingales evolving in [0, 1] when interest rates are nonnegative. Similarly, conditional survival
516 © 2017 Wiley Periodicals, Inc. wileyonlinelibrary.com/journal/maﬁ Mathematical Finance. 2018;28:516–535.