Minimal entropy preserves the Lévy property: how and why

This paper defines an optimization criterion for the set of all martingale measures for an incomplete market model when the discounted price process is bounded and quasi‐left continuous. This criterion is based on the entropy–Hellinger process for a nonnegative Doléans–Dade exponential local martingale. We develop properties of this process and establish its relationship to the relative entropy “distance.” We prove that the martingale measure, minimizing this entropy–Hellinger process, is unique. Furthermore, it exists and is explicitly determined under some mild conditions of integrability and no arbitrage. Different characterizations for this extremal risk‐neutral measure as well as immediate application to the exponential hedging are given. If the discounted price process is continuous, the minimal entropy–Hellinger martingale measure simply is the minimal martingale measure of Föllmer and Schweizer. Finally, the relationship between the minimal entropy–Hellinger martingale measure (MHM) and the minimal entropy martingale measure (MEM) is provided. We also give an example showing that in contrast to the MHM measure, the MEM measure is not robust with respect to stopping.

Let χ be a family of stochastic processes on a given filtered probability space (Ω, F, (Ft)t∈T, P) with T⊆R+. Under the assumption that the set Me of equivalent martingale measures for χ is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure that minimizes the relative entropy, with respect to P, in the class of martingale measures. We then provide the characterization of the density of the minimal entropy martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy.

Suppose that discounted asset prices in a financial market are given by a P ‐semimartingale S . Among all probability measures Q that turn S into a local Q ‐martingale, the minimal entropy martingale measure is characterized by the property that it minimizes the relative entropy with respect to P . Via convex duality, it is intimately linked to the problem of maximizing expected exponential utility from terminal wealth. It also appears as a limit of p ‐optimal martingale measures as p decreases to 1. Like for most optimal martingale measures, finding its explicit form is easy if S is an exponential Lévy process and quite difficult otherwise.

A Visual Classification of Local Martingales

An incomplete financial market model is considered, where the dynamics of the assets price is described by an R d -valued continuous semimartingale. We express the density of the minimal entropy martingale measure in terms of the value process of the related optimization problem and show that this value process is determined as the unique solution of a semimartingale backward equation. We consider some extreme cases when this equation admits an explicit solution.

This paper deals with the characterization problem of the minimal entropy martingale measure (MEMM) for a Markov-modulated exponential Levy model. This model is characterized by the presence of a background process modulating the risky asset price movements between different regimes or market environments. This allows to stress the strong dependence of financial assets price with structural changes in the market conditions. Our main results are obtained from the key idea of working conditionally on the modulator-factor process. This reduces the problem to studying the simpler case of processes with independent increments. Our work generalizes some previous works in the literature dealing with either the exponential Levy case or the exponential-additive case.

Asset Price Bubbles in Incomplete Markets

The minimal relative entropy criterion for the selection of an equivalent martingale measure in an incomplete market seems to still hold some mystique in its financial interpretation. In this paper we work toward this interpretation by suggesting and exploring the idea of relating a martingale measure selection criterion to a CAPM-like pricing scheme. We examine this idea in the case of the minimal relative entropy criterion and we present some preliminary results. We work within a one-period financial market and show that the minimal relative entropy pricing criterion is equivalent to some CAPM-like pricing scheme where the classical beta coefficient formula has been replaced by some “entropic beta” and the market portfolio by some “appropriate” reference portfolio. Furthermore, we show that if the assets involved have returns that are jointly normal, then this “entropic beta” formula coincides with the classical beta coefficient. Additionally and for comparison reasons, we briefly illustrate that if our criterion for the choice of the martingale measure was the minimization of the variance of the Radon–Nikodym derivative, then the resulting martingale pricing and the pricing implied by the classical CAPM scheme would be the same.

A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.

Within the setup of continuous-time semimartingale financial markets, we show that a multiprior Gilboa–Schmeidler minimax expected utility maximizer forms a portfolio consisting only of the riskless asset if and only if among the investor’s priors there exists a probability measure under which all admissible wealth processes are supermartingales. Furthermore, we show that under a certain attainability condition (which is always valid in finite or complete markets) this is also equivalent to the existence of an equivalent (local) martingale measure among the investor’s priors. As an example, we generalize a no betting result due to Dow and Werlang.

The similarity between maximum entropy (MaxEnt) and minimum relative entropy (MRE) allows recent advances in probabilistic inversion to obviate some of the shortcomings in the former method. The purpose of this paper is to review and extend the theory and practice of minimum relative entropy. In this regard, we illustrate important philosophies on inversion and the similarly and differences between maximum entropy, minimum relative entropy, classical smallest model (SVD) and Bayesian solutions for inverse problems. MaxEnt is applicable when we are determining a function that can be regarded as a probability distribution. The approach can be extended to the case of the general linear problem and is interpreted as the model which fits all the constraints and is the one model which has the greatest multiplicity or “spreadout” that can be realized in the greatest number of ways. The MRE solution to the inverse problem differs from the maximum entropy viewpoint as noted above. The relative entropy formulation provides the advantage of allowing for non-positive models, a prior bias in the estimated pdf and `hard' bounds if desired. We outline how MRE can be used as a measure of resolution in linear inversion and show that MRE provides us with a method to explore the limits of model space. The Bayesian methodology readily lends itself to the problem of updating prior probabilities based on uncertain field measurements, and whose truth follows from the theorems of total and compound probabilities. In the Bayesian approach information is complete and Bayes' theorem gives a unique posterior pdf. In comparing the results of the classical, MaxEnt, MRE and Bayesian approaches we notice that the approaches produce different results. In␣comparing MaxEnt with MRE for Jayne's die problem we see excellent comparisons between the results. We compare MaxEnt, smallest model and MRE approaches for the density distribution of an equivalent spherically-symmetric earth and for the contaminant plume-source problem. Theoretical comparisons between MRE and Bayesian solutions for the case of the linear model and Gaussian priors may show different results. The Bayesian expected-value solution approaches that of MRE and that of the smallest model as the prior distribution becomes uniform, but the Bayesian maximum aposteriori (MAP) solution may not exist for an underdetermined case with a uniform prior.

In this paper the principle of minimum relative entropy (PMRE) is proposed as a fundamental principle and idea that can be used in the field of AGI. It is shown to have a very strong mathematical foundation, that it is even more fundamental then Bayes rule or MaxEnt alone and that it can be related to neuroscience. Hierarchical structures, hierarchies in timescales and learning and generating sequences of sequences are some of the aspects that Friston (Fri09) described by using his free-energy principle. These are aspects of cognitive architectures that are in agreement with the foundations of hierarchical memory prediction frameworks (GH09). The PMRE is very similar and often equivalent to Friston's free-energy principle (Fri09), however for actions and the defi nitions of surprise there is a diff erence. It is proposed to use relative entropy as the standard definition of surprise. Experiments have shown that this is currently the best indicator of human surprise (IB09). The learning rate or interestingness can be defi ned as the rate of decrease of relative entropy, so curiosity can then be implemented as looking for situations with the highest learning rate.

We give a sufficient condition to identify the q-optimal signed and the q-optimal absolutely continuous martingale measures in exponential Levy models. As a consequence, we find that in the one-dimensional case, the q-optimal equivalent martingale measures may exist only if the tails for upward jumps are extraordinarily light. Moreover, we derive the convergence of q-optimal signed, resp. absolutely continuous, martingale measures to the minimal entropy martingale measure as q approaches one. Finally, some implications for portfolio optimization are discussed.

Downside Risk Neutral Probabilities

Downside Risk Neutral Probabilities