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

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 $S_t \set \mathbb{E}^{\mathbb{Q}}[ψ|\mathcal{F}_t]$. 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.

A stronger version of the main theorem is obtained. The "financial" part of the previous version is removed

References in corpus (3)

• A model for a large investor trading at market indifference prices. II: Continuous-time case
• Existence of an endogenously complete equilibrium driven by a diffusion
• Existence of Financial Equilibria in Continuous Time with Potentially Complete Markets

Cited by in corpus (5)

• Existence of an endogenously complete equilibrium driven by a diffusion
• Market Completion with Derivative Securities
• Existence of Financial Equilibria in Continuous Time with Potentially Complete Markets
• On the Heston Model with Stochastic Volatility: Analytic Solutions and Complete Markets
• Density of the set of probability measures with the martingale representation property