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Conic martingales from stochastic integrals
Author(s) -
Jeanblanc Monique,
Vrins Frédéric
Publication year - 2018
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12147
Subject(s) - martingale (probability theory) , mathematics , conic section , brownian motion , stochastic differential equation , geometric brownian motion , martingale representation theorem , stochastic calculus , separable space , mathematical analysis , pure mathematics , diffusion process , stochastic partial differential equation , differential equation , geometry , computer science , knowledge management , statistics , innovation diffusion
In this paper, we introduce the concept of conic martingales . This class refers to stochastic processes that have 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 σ ( t , y ) = g ( t ) h ( y ) 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.