2011/10/31 by Dmitry Kramkov, Silviu Predoiu · 1 citation
Mathematics · Economics, Econometrics and Finance · #math.PR #math.AP #q-fin.GN #msc:60G44 #msc:91B51 #msc:91G99 #msc:35K15 #msc:35K90
paper · pdf · doi:10.1016/j.spa.2013.06.017
published as Stochastic Processes and their Applications, 124 (1), pages 81-100, 2014 · A stronger version of the main theorem is obtained. The "financial" part of the previous version is removed
arxiv created 2012/10/26 · arxiv updated 2014/10/21
Let ℚ and ℙ be equivalent probability measures and let ψ be a J-dimensional 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 every local martingale under ℚ is a stochastic integral with respect to the J-dimensional martingale St \set 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.