# Integral representation of martingales motivated by the problem of endogenous completeness in financial economics

## Author Info

• Dmitry Kramkov

(Carnegie Mellon and Oxford)

• Silviu Predoiu

(Citigroup)

Registered author(s):

## Abstract

Let $\mathbb{Q}$ and $\mathbb{P}$ be equivalent probability measures and let $\psi$ be a $J$-dimensional vector of random variables such that $\frac{d\mathbb{Q}}{d\mathbb{P}}$ and $\psi$ are defined in terms of a weak solution $X$ to a $d$-dimensional stochastic differential equation. Motivated by the problem of \emph{endogenous completeness} in financial economics we present conditions which guarantee that every local martingale under $\mathbb{Q}$ is a stochastic integral with respect to the $J$-dimensional martingale $S_t \set \mathbb{E}^{\mathbb{Q}}[\psi|\mathcal{F}_t]$. While the drift $b=b(t,x)$ and the volatility $\sigma = \sigma(t,x)$ coefficients for $X$ need to have only minimal regularity properties with respect to $x$, they are assumed to be analytic functions with respect to $t$. We provide a counter-example showing that this $t$-analyticity assumption for $\sigma$ cannot be removed.

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File URL: http://arxiv.org/pdf/1110.3248

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1110.3248.

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 Length: Date of creation: Oct 2011 Date of revision: Oct 2012 Publication status: Published in Stochastic Processes and their Applications, 124 (1), pages 81-100, 2014 Handle: RePEc:arx:papers:1110.3248 Contact details of provider: Web page: http://arxiv.org/

## References

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1. Frederik Herzberg & Frank Riedel, 2012. "Existence of Financial Equilibria in Continuous Time with Potentially Complete Markets," Papers 1207.2010, arXiv.org.
2. J. Hugonnier & S. Malamud & E. Trubowitz, 2012. "Endogenous Completeness of Diffusion Driven Equilibrium Markets," Econometrica, Econometric Society, vol. 80(3), pages 1249-1270, 05.
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