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October 2000 Explicit form and robustness of martingale representations
Jean Jacod, Sylvie Méléard, Philip Protter
Ann. Probab. 28(4): 1747-1780 (October 2000). DOI: 10.1214/aop/1019160506


Stochastic integral representation of martingales has been undergoing a renaissance due to questions motivated by stochastic finance theory. In the Brownian case one usually has formulas (of differing degrees of exactness) for the predictable integrands. We extend some of these to Markov cases where one does not necessarily have stochastic integral representation of all martingales. Moreover we study various convergence questions that arise naturally from (for example)approximations of “price processes” via Euler schemes for solutions of stochastic differential equations. We obtain general results of the following type: let $U, U^n$ be random variables with decompositions

$$ U = \alpha + \int_{0}^{\infty} \xi_s dX_s + N_\infty, $$

$$ U^n = \alpha_n + \int_{0}^{\infty} \xi^n_s dX^n_s + N^n_\infty, $$

where $X, N, X^n, N^n$ are martingales. If $X^n \to X$ and $U^n \to U$, when and how does $\xi^n\rightarrow\xi?$


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Jean Jacod. Sylvie Méléard. Philip Protter. "Explicit form and robustness of martingale representations." Ann. Probab. 28 (4) 1747 - 1780, October 2000.


Published: October 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1044.60042
MathSciNet: MR1813842
Digital Object Identifier: 10.1214/aop/1019160506

Primary: 60F17 , 60H05

Keywords: Clark –Haussmann formula , Martingale representation , stability , weak convergence

Rights: Copyright © 2000 Institute of Mathematical Statistics

Vol.28 • No. 4 • October 2000
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