On the minimal entropy martingale measure.
Let \(X=(X_t)\) \((0\leq t\leq T)\) be a locally bounded semimartingale. Let \(P\) be a reference measure. The authors show that an equivalent martingale measure \(\overline Q\) for \(X\) satisfying the following three conditions minimizes relative entropy \(I(Q,P)\) among all martingale measures \(Q\): (i) \(dP/d\overline Q\in L^\varepsilon(P)\) for some \(\varepsilon>0\); (ii) \(d \overline Q/dP= c\exp ((\int \eta dX)_T)\) for a constant \(c\) and an \(X\)-integrable \(\eta\); (iii) \(\int\eta \,dX\in\text{BMO}(\overline Q)\). If \(X\) is continuous, the preceding result is used to obtain sufficient conditions for \(Q^{(q)}\) to converge to \(Q^E\) in entropy as \(q\to 1\) (here, \(Q^{(q)}\) is the \(q\)-optimal martingale measure, and \(Q^E\) minimizes \(I(Q,P))\).
scientific article; zbMATH DE number 1906077