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Martingale decomposition of an L2 space with nonlinear stochastic integrals

Part of: Stochastic processes

Published online by Cambridge University Press:  11 December 2019

Clarence Simard
Clarence Simard*
Affiliation:
Université du Québec à Montréal
*
* Postal address: Département de Mathématiques, Université du Québec à Montréal, C.P. 8888, succ. Centre-ville, Montréal (Québec), H3C 3P8, Canada.
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Abstract

This paper generalizes the Kunita–Watanabe decomposition of an $L^2$ space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$ . This result is also the solution of an optimization problem in $L^2$ . First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.

Keywords

Stochastic calculusoptimizationmartingalesorthogonal decompositionClark–Ocone formula

MSC classification

Primary: 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 1231 - 1243
Copyright
© Applied Probability Trust 2019 

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