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

Dmitry Kramkov and Silviu Predoiu
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Dmitry Kramkov: Carnegie Mellon and Oxford
Silviu Predoiu: Citigroup

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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.

Date: 2011-10, Revised 2012-10
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Published in Stochastic Processes and their Applications, 124 (1), pages 81-100, 2014

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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:1110.3248

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