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

Dmitry Kramkov and Silviu Predoiu

Stochastic Processes and their Applications, 2014, vol. 124, issue 1, 81-100

Abstract: Let Q and P be equivalent probability measures and let ψ be a J-dimensional vector of random variables such that dQdP and ψ are defined in terms of a weak solution X to a d-dimensional stochastic differential equation. Motivated by the problem of endogenous completeness in financial economics we present conditions which guarantee that every local martingale under Q is a stochastic integral with respect to the J-dimensional martingale St≜EQ[ψ|Ft]. While the drift b=b(t,x) and the volatility σ=σ(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 σ cannot be removed.

Keywords: Integral representation; Martingales; Diffusion; Parabolic equations; Analytic semigroups; Real analytic functions; Krylov–Ito formula; Dynamic completeness; Equilibrium (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (10)

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DOI: 10.1016/j.spa.2013.06.017

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