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Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences—which is closely related to the latter’s foundational contribution to model theory. In this note we show in a model-theoretic framework how Arrow’s use of von Neumann and Morgenstern’s concept of winning coalitions allows to exploit the algebraic structures involved in preference aggregation; this approach entails an alternative indirect ultrafilter proof for Arrow’s dictatorship result. This link also connects Arrow’s seminal result to key developments and concepts in the history of model theory, notably ultraproducts and preservation results.

Cerreia-Vioglio et al. (Econ Theory 48(2–3):341–375, 2011) have proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences—and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory (in the sense of mathematical logic), is employed.

Investigation of the molecular dynamics in lung cancer is crucial for the development of new treatment strategies. Fibroblast growth factor (FGF) 14 belongs to the FGF family, which might play a crucial role in cancer progression. We analyzed lung adenocarcinoma (LUAC) patients samples and found that FGF14 was downregulated, correlating with reduced survival and oncogenic mutation status. FGF14 overexpression in lung cancer cell lines resulted in decreased proliferation, colony formation, and migration, as well as increased expression of epithelial markers and a decreased expression of mesenchymal markers, indicating a mesenchymal to epithelial transition in vitro. We verified these findings using small interfering RNA against FGF14 and further confirmed the suppressive effect of FGF14 in a NOD.Cg-Prkdcscid Il2rgtm1Wjl/SzJ immunodeficient xenograft tumor model. Moreover, FGF14 overexpressing tumor cell RNA sequencing data suggests that genes affected by FGF14 were related to the extracellular matrix, playing a role in proliferation and migration. Notably, newly identified FGF14 target genes, adenosine deaminase RNA specific B1 (ADARB1), collagen and calcium-binding epidermal growth factor domain-containing protein 1 (CCBE1), &alpha, 1 chain of collagen XI (COL11A1), and mucin 16 (MUC16) expression was negatively correlated with overall survival when FGF14 was downregulated in LUAC. These findings led us to suggest that FGF14 regulates proliferation and migration in LUAC.

The hyperfinite $G$-expectation is a nonstandard discrete analogue of $G$-expectation (in the sense of Robinsonian nonstandard analysis). A lifting of a continuous-time $G$-expectation operator is defined as a hyperfinite $G$-expectation which is infinitely close, in the sense of nonstandard topology, to the continuous-time $G$-expectation. We develop the basic theory for hyperfinite $G$-expectations and prove an existence theorem for liftings of (continuous-time) $G$-expectation. For the proof of the lifting theorem, we use a new discretization theorem for the $G$-expectation (also established in this paper, based on the work of Dolinsky et al. [Weak approximation of $G$-expectations, Stoch. Proc. Appl. 122(2), (2012), pp.664--675])., 14 pages

We modify the definable ultrapower construction of Kanovei and Shelah (2004) to develop a ZF-definable extension of the continuum with transfer provable using countable choice only, with an additional mild hypothesis on well-ordering implying properness. Under the same assumptions, we also prove the existence of a definable, proper elementary extension of the standard superstructure over the reals. Keywords: definability; hyperreal; superstructure; elementary embedding., Comment: 8 pages, to appear in Journal of Symbolic Logic

Amartya Sen (1970) has shown that three natural desiderata for social choice rules are inconsistent: universal domain, respect for unanimity, and respect for some minimal rights—which can be interpreted as either “expert rights” (an expert’s right to have her competence respected) or liberal rights. Dietrich and List (2008) have generalised this result to the setting of binary judgement aggregation. This paper proves that the paradox of a Paretian liberal holds even in the framework of probabilistic opinion pooling and discusses options to circumvent this impossibility result: (i) restricting the aggregator domain to profiles with no potential for conflicting rights; (ii) avoiding agendas where all issues are pairwise entangled (interdependent).

Applying a framework of perfect competition under uncertainty, we contribute to the discussion of whether or not ad valorem taxes and specific taxes are equivalent. While this equivalence holds without price uncertainty, we show that ad valorem taxes and specific taxes are "almost never" equivalent in the presence of uncertainty if we demand equivalence to hold pathwise. Since we obtain this result under perfect competition, our analysis also provides a further rationale for why the equivalence must fail under imperfect competition.

Coherence is a key concept in many accounts of epistemic justification within ‘traditional’ analytic epistemology. Within formal epistemology, too, there is a substantial body of research on coherence measures. However, there has been surprisingly little interaction between the two bodies of literature. The reason is that the existing formal literature on coherence measure operates with a notion of belief system that is very different from—what we argue is—a natural Bayesian formalisation of the concept of belief system from traditional epistemology. Therefore, formal epistemology has so far only been concerned with one particular—arguably not even very natural—way of formalising coherence of belief systems; it has by no means refuted the viability of coherentism. In contrast to the existing literature, we formalise belief systems as families of assignments of (conditional) degrees of belief (which may be compatible with several subjective probability measures). Within this framework, we propose a Bayesian formalisation of the thrust of BonJour’s coherence concept in The structure of empirical knowledge (Harvard University Press, Cambridge, 1985), using a combination of Bayesian confirmation theory and basic graph theory. In excursions, we introduce graded notions for both logical and probabilistic consistency of belief systems—the latter being based on certain geometrical structures induced by probabilistic belief systems. For illustration, we reconsider BonJour’s “ravens” challenge (op. cit., p. 95f.). Finally, potential objections to our proposed formal coherence notion are explored.

This paper formally explores the common ground between mild versions of epistemological coherentism and infinitism; it proposes-and argues for-a hybrid, coherentist-infinitist account of epistemic justification. First, the epistemological regress argument and its relation to the classical taxonomy regarding epistemic justification-of foundationalism, infinitism and coherentism-is reviewed. We then recall recent results proving that an influential argument against infinite regresses of justification, which alleges their incoherence on account of probabilistic inconsistency, cannot be maintained. Furthermore, we prove that the Principle of Inferential Justification has rather unwelcome consequences-formally resembling the Sorites paradox-as soon as it is iterated and combined with a natural Bayesian perspective on probabilistic inferences. We conclude that strong versions of foundationalism and infinitism should be abandoned. Positively, we provide a rough sketch for a graded formal coherence notion, according to which infinite regresses of epistemic justification will often have more than a minimal degree of coherence.

It is well known that the literature on judgement aggregation inherits the impossibility results from the aggregation of preferences that it generalises. This is due to the fact that the typical judgement aggregation problem induces an ultrafilter on the set of individuals. We propose a model-theoretic framework for the analysis of judgement aggregation and show that the conditions typically imposed on aggregators induce an ultrafilter on the set of individuals, thus establishing a generalised version of the Kirman-Sondermann correspondence. In the finite case, dictatorship then immediately follows from the principality of an ultrafilter on a finite set. This is not the case for an infinite set of individuals, where there exist free ultrafilters, as Fishburn already stressed in 1970. Following Lauwers and Van Liedekerke's (1995) seminal paper, we investigate another source of impossibility results for free ultrafilters: the domain of an ultraproduct over a free ultrafilter extends the individual factor domains, such that the preservation of the truth value of some sentences by the aggregate model - if this is as usual to be restricted to the original domain - may again require the exclusion of free ultrafilters, leading to dictatorship once again. (C) 2011 Elsevier B.V. All rights reserved.

For a continuous-time financial market with a single agent, we establish equilibrium pricing formulae under the assumption that the dividends follow an exponential Levy process. The agent is allowed to consume a lump at the terminal date; before that, only flow consumption is allowed. The agent’s utility function is assumed to be additive, defined via strictly increasing, strictly concave smooth felicity functions which are bounded below (thus, many CRRA and CARA utility functions are included). For technical reasons we require for our equilibrium existence result that only pathwise continuous trading strategies are permitted in the demand set. The resulting equilibrium asset price processes depend on the agent’s risk aversion (through the felicity functions). Even in our simple, straightforward economy, the equilibrium asset price processes will essentially only be (stochastic) exponential Levy processes when they are already geometric Brownian motions. Our equilibrium asset pricing formulae can also be modified to obtain explicit equilibrium derivative pricing formulae.

In a recent paper, Jeanne Peijnenburg and David Atkinson [Studia Logica, 89(3):333-341 (2008)] have challenged the foundationalist rejection of infinitism by giving an example of an infinite, yet explicitly solvable regress of probabilistic justification. So far, however, there has been no criterion for the consistency of infinite probabilistic regresses, and in particular, foundationalists might still question the consistency of the solvable regress proposed by Peijnenburg and Atkinson.

A jump diffusion decomposition theorem for hyperfinite Levy processes is proven; a counterexample to a previous attempt to phrase such a theorem is provided.

A non-Archimedean utility representation theorem for independent and transitive preference orderings that are partially continuous on some convex subset and satisfy an axiom of incommensurable preference for elements outside that subset is proven. For complete preference orderings, the theorem is deduced directly from the classical von Neumann-Morgenstern theorem; in the absence of completeness, Aumann's [Aumann, R.J., 1962. Utility theory without the completeness axiom. Econometrica 30 (3), 445-462] generalization is utilized. (C) 2009 Elsevier B.V. All rights reserved.

Consider an L 1 -continuous functional l on the vector space of polynomials of Brownian motion at given times, suppose l commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, f 1 ( b → ) , … , f m ( b → ) , mapping the Wiener space to R . In the spirit of Schmudgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which l can be written in the form ∫ ⋅ d μ for some probability measure μ on the Wiener space such that μ-almost surely, all the random variables f 1 ( b → ) , … , f m ( b → ) are nonnegative.

The minimisation problem for a functional P bar right arrow u(Gamma,P,(g) over tilde) is considered, where (g) over tilde is an R-n-valued stochastic process, defined on some filtered probability space Gamma = (Gamma(G(t))(t is an element of[0,1]), P), and P is an admissible probability measure in the sense that it obeys (1) some uniform equivalence condition with respect to the given measure P on Gamma, and (2) a finite number (possibly zero) of arbitrarily given other conditions that require the expectation (with respect to P) of some continuous bounded function phi of ((g) over tilde (t1),..., (g) over tilde (tk)), for t(1),..., t(k) is an element of [0,1], to lie within some closed set. We assume that u can be formulated through finite compositions of conditional expectations and bounded continuous functions. Under the assumption of vertical bar phi vertical bar being uniformly bounded from below and some condition on the dimension of phi ((g) over tilde (t1),..., (g) over tilde (tk)) the existence of a solution on hyperfinite adapted probability spaces, as well as its minimality among admissible measures on any other adapted probability space, is proven. Also, a coarseness result for the Loeb operation is established. The main result of this paper, however, is a "standard result": It does not include any reference to nonstandard analysis and can be perfectly understood without any familiarity with nonstandard analysis.

In a recent paper [Journal of Logic and Computation, forthcoming (2014); doi:10.1093/logcom/ext009], it was claimed that a universal algebraic approach to general aggregation theory based on MV-homomorphisms could even cover linear probabilistic opinion pooling. This is not so, however. The reason is that there are no non-trivial homomorphisms from direct powers of the standard MV-algebra onto the standard MV-algebra itself; an analytic proof thereof is given in this note.

An internal lifting for an arbitrary measurable Levy process is constructed. This lifting reflects our intuitive notion of a process which is the infinitesimal sum of its infinitesimal increments, those in turn being independent from and closely related to each other – for short, the process can be regarded as some kind of random walk (where the step size generically will vary). The proof uses the existence of cadlag modifications of Levy processes and certain features of hyperfinite adapted probability spaces, commonly known as the “model theory of stochastic processes”.

(2006). A combinatorial infinitesimal representation of Levy processes and an application to incomplete markets. Stochastics: Vol. 78, No. 5, pp. 301-325.

Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern theories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions., 35 pages, 2 figures, to appear in Notices of the American Mathematical Society 61 (2014), no. 8

The defence of The No Alternatives Argument in a recent paper by R. Dawid, S. Hartmann and J. Sprenger (forthcoming in the British Journal for the Philosophy of Science; latest version: February 2013) rests on the assumption (among others) that the number of acceptable alternatives to a scientific hypothesis is independent of the complexity of the scientific problem. This note proves a generalisation of the main theorem by Dawid, Hartmann and Sprenger, where this independence assumption is no longer necessary. Some of the other assumptions are also discussed, and the limitations of (this formalisation of) the no-alternatives argument are explored.

This paper builds on a recent proposal for microeconomic foundations for "representative agents". Herzberg [Journal of Mathematical Economics, vol. 46, no. 6, 1115-1124 (2010)] constructed a representative utility function for infinite-dimensional social decision problems and since the decision problems of macroeconomic theory are typically infinite-dimensional, Herzberg's original result is insufficient for many applications. We therefore generalise his result by allowing the social alternatives to belong to a general reflexive Banach space and provide sufficient conditions for our new results to be satisfied in economic applications.

We refine the discretization of G-expectation by Y. Dolinsky, M. Nutz, and M. Soner (Stochastic Processes and their Applications, 122 (2012), 664-675), in order to obtain a discretization of sublinear expectation where the martingale laws are defined on a finite lattice rather than the whole set of reals.

We prove, via the Borel-Cantelli lemma, that for every sequence of Gaussian random variables the combination of convergence in expectation and decreasing variances at fractional-polynomial rate implies strong convergence. This result has an important consequence for macroeconomic stochastic infinite-horizon models: The almost sure transversality condition (i.e., fiscal sustainability with probability one) is satisfied if (a) the discounted levels of net liabilities are Gaussian-distributed with fractional-polynomially decaying variances and (b) their means converge to zero. If (a) holds but (b) fails, the transversality condition will be almost surely violated. Hence, (a) and (b) constitute a test for almost sure fiscal sustainability.

This note employs the recently established consistency theorem for infinite regresses of probabilistic justification (Herzberg in Stud Log 94(3):331-345, 2010) to address some of the better-known objections to epistemological infinitism. In addition, another proof for that consistency theorem is given; the new derivation no longer employs nonstandard analysis, but utilises the Daniell-Kolmogorov theorem.

Cerreia-Vioglio, Ghirardato, Maccheroni, Marinacci and Siniscalchi (Economic Theory, 48:341–375, 2011) have recently proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean (MBA) preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences — and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory (in the sense of mathematical logic), is employed.

This paper studies collective decision making when individual preferences can be represented by convex risk measures. It addresses the question whether there exist non-dictatorial aggregation functions of convex risk measures satisfying the following Arrow-type rationality axioms: a weak form of universality, systematicity (a strong variant of independence), and the Pareto principle. Herein, convex risk measures are identified with variational preferences on account of the Maccheroni et al. (Econometrica 74(6):1447–1498, 2006) axiomatisation of variational preference relations and Follmer’s and Schied’s (Finance Stoch 6(4):429–447, 2002; Stochastic Finance. An Introduction in Discrete Time. de Gruyter Studies in Mathematics, vol. 27, 2nd edn. de Gruyter, Berlin, 2004) representation theorem for concave monetary utility functionals. The cases of both finite and infinite electorates are considered. For finite electorates, we prove a variational analogue of Arrow’s impossibility theorem. For infinite electorates, the possibility of rational aggregation depends on an equicontinuity condition for the variational preference profiles; we prove a variational analogue of Fishburn’s possibility theorem and point to potential analogues of Campbell’s impossibility theorem. The proof methodology is based on a model-theoretic approach to aggregation theory inspired by Lauwers and Van Liedekerke (J Math Econ 24(3):217–237, 1995). Diverse applications of these results are conceivable, in particular (a) the constitutional design of panels of risk managers and (b) microfoundations for (macroeconomic) multiplier preferences.

We prove that in smooth Markovian continuous-time economies with potentially complete asset markets, Radner equilibria with endogenously complete markets exist. (C) 2013 Elsevier B.V. All rights reserved.

For any two processes \(\xi ,\eta \), the stochastic integral of \(\eta \) with respect to ξ is the process \(\int \eta \mathrm{d}\xi \) defined by \(\int_{0}^{s}\eta \mathrm{d}\xi =\int_{0}^{s}\eta (t)\mathrm{d}\xi (t) =\sum\limits_{t

The expressive power of minIST comes from the fact that it allows for the notions of finite sets with unlimited cardinality, and finite subsets of the reals whose distance is at most an infinitesimal from every point in some non-empty open interval.

Let η be a limited, \(\mathcal{F}\) -adapted stochastic process, let ξ be the process defined by ξ(0) = 1 and \(\mathrm{d}\xi (t) = \xi (t)\eta (t)\ \mathrm{d}W(t)\) for all \(t \in \mathbf{T} \setminus \{ 1\}\).

In this excursion, which is inspired by Albeverio et al. [3, Sect. 6.6] and the classical article by Nelson [58], we give another demonstration of the usefulness of radically elementary mathematics in mathematical physics, by providing a rigorous, radically elementary definition of Feynman path integrals in Minimal Internal Set Theory. A summary of these ideas—combined with a brief introduction to radically elementary mathematics for mathematical physicists and some references to previous attempts at formalizing the Feynman path integral by means of nonstandard analysis—can be found in [35].

Let \({t}_{0} \in \mathbf{T} \setminus \{ 1\}\) and \(a,b : \mathbf{R} \times [0,1] \rightarrow \mathbf{R}\).

It is well-known that the diffusion invariance principle conveys an important insight for financial economics: when (logarithmic) Ito processes are used as models of stock prices, the drift coefficient of the logarithmic price process is interpreted as a measure for the expected return, and its diffusion coefficient is interpreted as a measure for the volatility. In this context, the diffusion invariance principle asserts roughly that under an equivalent change of probability measure, only the expected returns will be affected, but not the volatilities. In particular, a price process will have the same volatility under the real-world probability measure as under an equivalent risk-neutral (i.e. arbitrage-free) probability measure; changing the probability measure corresponds to changing the expected return (and vice versa).

What follows in this excursion is the attempt to construct a radically elementary version of continuous-time financial economics. Mathematicians sometimes confuse financial economics with mathematical finance (also known as financial mathematics) or even financial engineering. There is however, a profound difference in interest and methodology between the two: While mathematical finance and financial engineering are concerned with technical mathematical problems arising from the analysis of quantitative models of financial markets (in particular, models used at financial institutions), financial economics is a subdiscipline of economic theory and has a conceptual interest in understanding how financial markets work.

The most important feature of Nelson’s [60] radically elementary analysis is the discretization of the continuum. The crucial step herein is the consistent use of infinitely large (“nonstandard”) numbers and infinitesimals, in a manner which was first proposed by Nelson through the axiom system of Internal Set Theory [59], motivated by the groundbreaking work of Abraham Robinson [66, 67]. One decade on, Nelson [60] introduced an even more elementary, yet still very powerful, formal system, which we shall review presently.

An important part of this book has been concerned with the investigation of hyperfinite Markov chains \( {X\,=\,(X_t)_{t\in T}}\) on an internal probability space Ω. If we confine ourselves to a finite time horizon – say, 1 – and if we assume \({\Delta {t}}\, \)to be a reciprocal hypernatural number, then \( {H} \,=\,{1/{\Delta{t}}}\in\,{^*}\mathbb{N}\,\backslash\mathbb{N}\, \)will be the number of possible transitions occurring between \( {0\,\in\,T}\,\)and \( {1\,\in\,T\,.}\,\)Because X was assumed to be a hyperfinite Markov chain starting at a given point x0 in the state space, the number of possible paths between 0 and 1 will be hyperfinite. Thus, provided we are only interested in studying events that lie in the filtration generated by X, we may assume Ω to be hyperfinite.

Probabilistic potential theory has been a very important component in the study of hyperfinite Dirichlet space theory. It provides a probabilistic inter- pretation of potential theory; and, more generally, it establishes a beautiful bridge between functional analysis and the theory of Markov processes. There are many applications of this theory, especially in the area of infi- nite dimensional stochastic analysis and mathematical physics. Our purpose in this chapter is to develop the probabilistic potential theory associated with hyperfinite Dirichlet forms and the related Markov chains. The motivation is twofold. On the one hand, we want to establish a relationship between the standard Dirichlet space theory and the hyperfinite counterpart. On the other hand, we want to provide new methods for the theory of hyperfinite Dirichlet forms itself. Infinite dimensional stochastic analysis has been devel- oped extensively in the last decades. We hope to convince the reader that nonstandard analysis can provide a new tool to deal with problems in this exciting area, see particularly Chap. 4, for example.

In this chapter, we consider the construction of tight strong Markov processes associated with standard quasi-regular Dirichlet forms by using the language of nonstandard analysis. In Fan [154], the construction of strong Markov pro- cesses associated with standard symmetric Dirichlet forms was considered. In this chapter, we consider nonsymmetric Dirichlet forms. We consider a standard Dirichlet form (F(·, ·),D(F)) whose state space is a regular Hausdorff space Y. We impose conditions directly on the Dirichlet form (F(·, ·), D(F)) to obtain associated tight strong Markov processes. The motivation comes from the desire to find the relationship between the family of Dirichlet forms and the family of strong Markov processes.

The purpose of this chapter is to study the standard projection of hyperfinite Markov chains, and the standard projection of hyperfinite Markov chains associated with hyperfinite quadratic forms. At first, we shall introduce in Sect. 3.1 a concept of irregularity; and then we shall prove that if a hyperfinite Markov chain X(·, t) has a set of irregularities, its standard part x(·, t) is a strong Markov process. In Sect. 3.2, we shall find conditions on hyperfinite quadratic forms which guarantee that the modified standard parts of associated hyperfinite Markov chains are strong Markov processes.

So far, we have concentrated in this book on studying arbitrary Markov pro- cesses which admit a modification whose paths are right-continuous with left limits (i.e. cadlag Markov or Feller processes, cf. e.g. [294, Chap. 3, Theorem 2.7]), and their relations to energy forms and Hamiltonians from the perspective of nonstandard analysis.

This article investigates the representative-agent hypothesis for a population which faces a collective choice from a given finite-dimensional space of alternatives. Each individual’s preference ordering is assumed to admit a utility representation through an element of an exogenously given set of admissible utility functions. In addition, we assume that (i) the class of admissible utility functions allows for a smooth parametrization and only consists of strictly concave functions, (ii) the population is infinite, and (iii) the social welfare function satisfies Arrovian rationality axioms. We prove that there exists an admissible utility function r , called representative utility function, such that any alternative which maximizes r also maximizes the social welfare function. Given the structural similarities among the admissible utility functions (due to parametrization), we argue that the representative utility function can be interpreted as belonging to an – actual or invisible – individual. The existence proof for the representative utility function utilizes a special nonstandard model of the reals, viz. the ultrapower of the reals with respect to the ultrafilter of decisive coalitions; this construction explicitly determines the parameter vector of the representative utility function.

The theory of Boolean algebras can be fruitfully applied to judgment aggregation: assuming universality, systematicity and a sufficiently rich agenda, there is a correspondence between (i) non-trivial deductively closed judgment aggregators and (ii) Boolean algebra homomorphisms defined on the power-set algebra of the electorate. Furthermore, there is a correspondence between (i) consistent complete judgment aggregators and (ii) 2-valued Boolean algebra homomorphisms defined on the power-set algebra of the electorate. Since the shell of such a homomorphism equals the set of winning coalitions and since (ultra)filters are shells of(2-valued) Boolean algebra homomorphisms, we suggest an explanation for the effectiveness of the (ultra)filter method in social choice theory. From the (ultra)filter property of the set of winning coalitions, one obtains two general impossibility theorems for judgment aggregation on finite electorates, even without assuming the Pareto principle. (C) 2009 Elsevier B.V. All rights reserved.