Dynamic bid–ask pricing under Dempster-Shafer uncertainty

We deal with the problem of pricing in a multi-period binomial market model, allowing for frictions in the form of bid–ask spreads. We introduce and characterize time-homogeneous Markov multiplicative binomial processes under Dempster-Shafer uncertainty together with the induced conditional Choquet expectation operator. Given a market formed by a frictionless risk-free bond and a non-dividend paying stock with frictions, we prove the existence of an equivalent one-step Choquet martingale belief function. We then propose a dynamic Choquet pricing rule with bid–ask spreads showing that the discounted lower price process of a European derivative contract on the stock is a Choquet super-martingale. We finally provide a normative justification in terms of a dynamic generalized no-arbitrage condition relying on the notion of partially resolving uncertainty due to Jaffray

A Dutch book coherence condition for conditional completely alternating Choquet expectations

Stemming from de Finetti’s coherence for finitely additive (conditional) probabilities, the paradigm of coherence has been extended to other uncertainty calculi. We study the notion of coherence for conditional completely alternating Choquet expectations, providing an avoiding Dutch book like condition

Modeling agent's conditional preferences under objective ambiguity in Dempster-Shafer theory

We manage decisions under “objective” ambiguity by considering generalized Anscombe-Aumann acts, mapping states of the world to generalized lotteries on a set of consequences. A generalized lottery is modeled through a belief function on consequences, interpreted as a partially specified randomizing device. Preference relations on these acts are given by a decision maker focusing on different scenarios (conditioning events). We provide a system of axioms which are necessary and sufficient for the representability of these “conditional preferences” through a conditional functional parametrized by a unique full conditional probability P on the algebra of events and a cardinal utility function u on consequences. The model is able to manage also “unexpected” (i.e., “null”) conditioning events and distinguishes between a systematically pessimistic or optimistic behavior, either referring to “objective” belief functions or their dual plausibility functions. Finally, an elicitation procedure is provided, reducing to a Quadratically Constrained Linear Program (QCLP)

Probability envelopes and their Dempster-Shafer approximations in statistical matching

Many economic applications require to integrate information coming from different data sources. In this work we consider a specific integration problem, called statistical matching, referring to integration of data sets where some variables are separately observed and some others are observed in all the data sets. This problem leads to the issue of non-uniqueness for the compatible (conditional) distributions and so it suggests to deal with sets of probabilities. For that we consider different strategies to get a (conditional) belief function that approximates the lower envelope of the class of compatible (conditional) probabilities. We first analyze the case without logical constraints among the variables and then generalize the obtained results by allowing for logical constraints. We finally show an application to real data

The extent of partially resolving uncertainty in assessing coherent conditional plausibilities

Handling uncertainty and reasoning under partial knowledge are challenging tasks that require to deal with coherent assessments and their extensions. Plausibility theory is shown to rest upon the principle of partially resolving uncertainty due to Jaffray, together with a systematically optimistic behavior. This means that we allow situations in which the agent may only acquire the information that a non-impossible event occurs, without knowing which is the true state of the world. This leads to assume that a target event is plausibly true if it is compatible with the acquired piece of information. The aim of the paper is to provide coherence conditions for a conditional plausibility assessment (namely, Pl-coherence), by referring to a suitable axiomatic definition based on the Dempster's rule of conditioning. We provide different equivalent notions of Pl-coherence in terms of consistency, betting scheme, and penalization that, as a by-product, highlight different interpretations. We then specialize the Pl-coherence conditions to the subclasses of (finitely additive) conditional probabilities and (finitely maxitive) conditional possibilities

Bayesian inference: the role of coherence to deal with a prior belief function

Starting from a likelihood function and a prior information represented by a belief function, a closed form expression is provided for the lower envelope of the set of all the possible “posterior probabilities” in finite spaces. The same problem, removing the hypothesis of finiteness for the domain of the prior, is then studied in the finitely additive probability framework by considering either the whole set of coherent extensions or the subset of disintegrable extensions.Starting from a likelihood function and a prior information represented by a belief function, a closed form expression is provided for the lower envelope of the set of all the possible “posterior probabilities” in finite spaces. The same problem, removing the hypothesis of finiteness for the domain of the prior, is then studied in the finitely additive probability framework by considering either the whole set of coherent extensions or the subset of disintegrable extensions

Conditional submodular Choquet expected values and conditional coherent risk measures

We provide an axiomatic definition of conditional submodular capacity that allows conditioning on “null” events and is the basis for the notions of consistency and of consistent extension of a partial assessment. The same definition gives rise to an axiomatic definition of conditional submodular Choquet expected value, which is a conditional functional defined on conditional gambles, that can be expressed as the Choquet integral with respect to its restriction on conditional indicators. Finally, the notion of conditional submodular Choquet expected value is used to provide a definition of conditional submodular coherent risk measure that, locally on every conditioning event, has an upper expected loss interpretation

Characterization of conditional submodular capacities: Coherence and extension

We provide a representation in terms of a linearly ordered class of (unconditional) submodular capacities of an axiomatically defined conditional submodular capacity. This allows to provide a notion of coherence for a partial assessment and a related notion of coherent extension

Dutch book rationality conditions for conditional preferences under ambiguity

We study preference relations on conditional gambles of a decision maker acting under ambiguity. Dutch book rationality conditions are provided under a linear utility scale, encoding either an optimistic or a pessimistic attitude towards uncertainty. These conditions characterize possibly incomplete preferences representable by totally alternating or monotone conditional functionals. In general, the uniqueness of the representation is not guaranteed, but it can be obtained by adding the hypothesis of existence of a conditional fair price for every conditional gamble. The given rationality conditions have a betting scheme interpretation relying on “penalty fees” for betting on strict preference comparisons