Integral representation of martingales and endogenous completeness of financial models D. Kramkov and S. Predoiu Carnegie Mellon University, Department of Mathematical Sciences, 5000 Forbes Avenue, Pittsburgh, PA, 15213-3890, USA May 17, 2011
Abstract Let Q and P be equivalent probability measures and let ψ be a Jdimensional vector of random variables such that dQ dP 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 any local martingale under Q is a stochastic integral with respect to the Jdimensional 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 can not be removed. Our results complement and generalize those in Anderson and Raimondo [1] and Hugonnier, Malamud, and Trubowitz [2].
References [1] Robert M. Anderson and Roberto C. Raimondo. Equilibrium in continuous-time financial markets: endogenously dynamically complete markets. Econometrica, 76(4):841–907, 2008. ISSN 0012-9682. [2] Julien Hugonnier, Semyon Malamud, and Eugene Trubowitz. Endogenous completeness of diffusion driven equilibrium market. Preprint, 2010.
Integral representation of martingales and endogenous ...
May 17, 2011 - While the drift b = b(t, x) and the volatility Ï = Ï(t, x) coefficients for X need to have only minimal regularity properties with respect to x, they are ...
Available online 1 November 2007. Abstract. We axiomatize in the AnscombeâAumann setting a wide class of preferences called rank-dependent additive ...
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