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

## Author

Listed:
• Dmitry Kramkov

(Carnegie Mellon and Oxford)

• Silviu Predoiu

(Citigroup)

## 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.

## Suggested Citation

• Dmitry Kramkov & Silviu Predoiu, 2011. "Integral representation of martingales motivated by the problem of endogenous completeness in financial economics," Papers 1110.3248, arXiv.org, revised Oct 2012.
• Handle: RePEc:arx:papers:1110.3248
as

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

## References listed on IDEAS

as
1. J. Hugonnier & S. Malamud & E. Trubowitz, 2012. "Endogenous Completeness of Diffusion Driven Equilibrium Markets," Econometrica, Econometric Society, vol. 80(3), pages 1249-1270, May.
2. Riedel, Frank & Herzberg, Frederik, 2013. "Existence of financial equilibria in continuous time with potentially complete markets," Journal of Mathematical Economics, Elsevier, vol. 49(5), pages 398-404.
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