Conic martingales from stochastic integrals. (18th April 2017)

• Record Type: Journal Article
• Authors: Jeanblanc, Monique; Vrins, Frédéric
• Is Part Of: Mathematical finance. Volume 28:Number 2(2018)
• Page Start: 516
• Page End: 535
• Publication Date: 2017-04-18
• Subjects: bounded martingale -- diffusion process -- stochastic differential equation -- stochastic survival probability; Business mathematics -- Periodicals; 332
• DOI: 10.1111/mafi.12147
• Languages: English
• ISSNs: 0960-1627

Abstract

Abstract: 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.