A semimartingale BSDE related to the minimal entropy martingale measure
In complete markets every contingent claim is replicable and the price of the claim is determined using the uniquely defined martingale measure. If the market is incomplete the set of martingale measures is not in general a singleton and there is an interval of arbitrage-free prices associated to different pricing martingale measures. The choice of the pricing measure is not longer preference-tree and depends on the utility functions of the investors or on the criterion relative to which the hedging error is measured. The most popular choices of pricing measures are the minimal martingale measure introduced by \textit{H. Föllmer} and \textit{M. Schweizer} [Applied stochastic analysis, Pap. Workshop, London/UK 1989, Stochastic Monogr. 5, 389-414 (1990; Zbl 0738.90007)] and the variance-optimal martingale measure (see \textit{F. Delbaen} and \textit{W. Schachermayer} [Bernoulli 2, No. 1, 81-105 (1996; Zbl 0849.60042)], \textit{M. Schweizer} [Ann. Probab. 24, No. 1, 206-236 (1996; Zbl 0854.60045)]). The variance-optimal martingale measure is determined by minimizing the \(L_2\)-norm of densities of the martingale measure with respect to the basic measure \(P\) among all signed martingale measures and represents the best choice when the quality of the hedging strategies is measured by the quadratic criterion. An another possibility is the minimal entropy martingale measure which minimizes the relative entropy of a martingale measure with respect to the measure \(P\) and is known to be closely related to the exponential hedging problem. For a locally bounded process \(X\) the minimal entropy martingale measure exists, is unique and if there is a martingale measure with the finite relative entropy, then the minimal entropy martingale measure is equivalent to \(P\). Using the dynamic programming technique the authors of this paper propose a construction of the minimal entropy martingale measure in the case where the dynamics of the discounted assets price is described by an \(\mathbb R^d\)-valued continuous semimartingale. They propose a description of the minimal entropy martingale measure in terms of the value function of a suitable problem of an optimal equivalent change of measure and show that this value process uniquely solves the corresponding semimartingale backward stochastic differential equation which is considered as a stochastic version of the Bellman equation in an optimal control problem. It is shown that in two specific extreme cases this semimartingale backward stochastic differential equation admits an explicit solution which gives an explicit construction of the minimal entropy martingale measure. In particular, a necessary and sufficient condition is presented under which the minimal entropy martingale measure coincides with the minimal martingale measure.
scientific article; zbMATH DE number 2169535