Conic martingales from stochastic integrals

Conic martingales from stochastic integrals 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Finance Wiley

Conic martingales from stochastic integrals

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Publisher
Wiley Subscription Services, Inc., A Wiley Company
Copyright
© 2018 Wiley Periodicals, Inc.
ISSN
0960-1627
eISSN
1467-9965
D.O.I.
10.1111/mafi.12147
Publisher site
See Article on Publisher Site

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.

Journal

Mathematical FinanceWiley

Published: Jan 1, 2018

Keywords: ; ; ;

References

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