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

TL;DR

This paper provides conditions under which all local martingales under a probability measure can be represented as stochastic integrals with respect to a specific martingale, addressing endogenous completeness in financial economics.

Contribution

It establishes new conditions for martingale representation in models driven by stochastic differential equations with minimal spatial regularity but analytic time dependence.

Findings

Martingale representation holds under specified conditions.

Counter-example shows necessity of time-analyticity of volatility.

Minimal regularity in spatial variables suffices for the main result.

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.