The \(p\)-optimal martingale measure and its asymptotic relation with the minimal-entropy martingale measure
Let \(S=(S_k)_{k=0,1,\ldots,N}\) be an \(\mathbb{R}^d\)-valued stochastic process in finite discrete time and assume that \(S\) admits an equivalent martingale measure. The \(p\)-optimal martingale measure \(Q^{(p)}\) is that signed martingale measure for \(S\) whose density with respect to the original measure \(P\) has minimal \(L^p(P)\)-norm, and the minimal-entropy martingale measure \(Q^E\) is that martingale measure for \(S\) which minimizes relative entropy with respect to \(P\). The main result shows that if \(S\) is bounded, the densities \(Z^{(p)}\) converge for \(p\to 1\) to \(Z^E\) in \(L^1(P)\). The proof uses a mixture of probabilistic and functional-analytic arguments and exploits in particular the explicit structure of \(Z^{(p)}\) and \(Z^E\). A counterexample shows that the assertion fails in general if \(S\) is unbounded. An auxiliary result shows that the space of all final values of stochastic integrals of \(S\) is closed in \(L^0(P)\); this requires no boundedness assumptions.
scientific article; zbMATH DE number 1294458