Journal of the Mathematical Society of Japan

From an Itô type calculus for Gaussian processes to integrals of log-normal processes increasing in the convex order

Francis HIRSCH, Bernard ROYNETTE, and Marc YOR

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Abstract

We present an Itô type formula for a Gaussian process, in which only the one-marginals of the Gaussian process are involved. Thus, this formula is well adapted to the study of processes increasing in the convex order, in a Gaussian framework. In particular, we give conditions ensuring that processes defined as integrals, with respect to one parameter, of exponentials of two-parameter Gaussian processes, are increasing in the convex order with respect to the other parameter. Finally, we construct Gaussian sheets allowing to exhibit martingales with the same one-marginals as the previously defined processes.

Article information

Source
J. Math. Soc. Japan, Volume 63, Number 3 (2011), 887-917.

Dates
First available in Project Euclid: 1 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1312203805

Digital Object Identifier
doi:10.2969/jmsj/06330887

Mathematical Reviews number (MathSciNet)
MR2836749

Zentralblatt MATH identifier
1233.60008

Subjects
Primary: 60E15: Inequalities; stochastic orderings 60G15: Gaussian processes
Secondary: 60G44: Martingales with continuous parameter 60G48: Generalizations of martingales 60G60: Random fields

Keywords
convex order 1-martingale Gaussian process Gaussian sheet log-normal process Itô type formula Itô's calculus

Citation

HIRSCH, Francis; ROYNETTE, Bernard; YOR, Marc. From an Itô type calculus for Gaussian processes to integrals of log-normal processes increasing in the convex order. J. Math. Soc. Japan 63 (2011), no. 3, 887--917. doi:10.2969/jmsj/06330887. https://projecteuclid.org/euclid.jmsj/1312203805


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