Weighted norm inequalities and hedging in incomplete markets

Let be an -valued special semimartingale on a probability space with canonical decomposition . Denote by the space of all random variables , where is a predictable -integrable process such that the stochastic integral is in the space of semimartingales. We investigate under which conditions on the semimartingale the space is closed in , a question which arises naturally in the applications to financial mathematics. Our main results give necessary and/or sufficient conditions for the closedness of in . Most of these conditions deal with BMO-martingales and reverse H?lder inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities, we also extend previous results on the F?llmer-Schweizer decomposition.     

Minimal Hellinger martingale measures of order <Emphasis Type="Italic">q</Emphasis>

This paper proposes an extension of the minimal Hellinger martingale measure (MHM hereafter) concept to any order q≠1 and to the general semimartingale framework. This extension allows us to provide a unified formulation for many optimal martingale measures, including the minimal martingale measure of Föllmer and Schweizer (here q=2). Under some mild conditions of integrability and the absence of arbitrage, we show the existence of the MHM measure of order q and describe it explicitly in terms of pointwise equations in ? ^d . Applications to the maximization of expected power utility at stopping times are given. We prove that, for an agent to be indifferent with respect to the liquidation time of her assets (which is the market’s exit time, supposed to be a stopping time, not any general random time), she is forced to consider a habit formation utility function instead of the original utility, or equivalently she is forced to consider a time-separable preference with a stochastic discount factor.     

The Relations between Minimal Martingale Measure and Minimal Entropy Martingale Measure

We consider incomplete markets, where each risky asset fluctuation is a continuous semimartingale, and study a subset of Equivalent Local Martingale Measures in which Minimal Martingale Measure minimizes relative entropy.We also discuss, as special cases, some models with the risky assetfluctuation represented as a solution of some stochastic differential equations.Finally, we mention that the predictable representation property is essentialin order that Minimal Martingale Measure coincides with Minimal Entropy Martingale Measure.     

An expansion in the model space in the context of utility maximization

In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian), an explicit second-order expansion formula for the power investor’s value function—seen as a function of the underlying market price of risk process—is provided. This allows us to provide first-order approximations of the optimal primal and dual controls. Two specific calibrated numerical examples illustrating the accuracy of the method are also given.     

The Minimal Entropy Martingale Measures for Exponential Additive Processes

In this paper, we will consider exponential additive processes as a financial market model. Under a mild condition, we will determine the minimal entropy martingale measures (MEMMs) for the exponential additive processes. To this end, we will prepare several results on the exponential moment of additive processes and integrals based on them. As an application of our result, we will deduce optimal strategy for exponential utility maximization problem. We will also investigate our result through several examples, such as time-dependent versions of double Poisson model, Merton model and Kou model.