The anticipatory views of Bruno de Finetti on portfolio theory,
set out by the author in a 1940 article, have recently been discovered by Mark
Rubinstein and reviewed by Harry Markowitz .
An historical note by Mark
Rubinstein on Bruno de Finetti himself and on de Finetti (1940) has been
recently published in the Journal of Investment Management: "it has just
recently come to light (or should I say the attention of economists in the
English-speaking world) that among de Finetti's papers is a treasurer-trove of
results in economics and finance written well before the work of the scholars
that are traditionally credited with these ideas. For example, in 1952,
anticipating Kenneth Arrow and John Pratt by over a decade, he formulated the
notion of absolute risk aversion, used it in connection with risk premia for
small bets, and discussed the special case of constant absolute risk aversion.
He also has early work on martingales (1939), optimal dividend policy (1957) and
Samuelson's consumption loan model of interest rates (1956). But perhaps most
astounding is de Finetti's 1940 paper anticipating much of mean-variance
portfolio theory later developed by Harry Markowitz in a series of three works
(1952), (1956) and (1959) and A.D. Roy (1952)."
Harry Markowitz has kindly
provided a review of de Finetti's paper to accompany the publication of the
English translation of its first chapter on the Journal of Investment
Management: "In 1940, in the context of choosing optimum reinsurance levels,
Bruno de Finetti essentially proposed mean-variance analysis with correlated
risks. It was not until 1952 that Markowitz and Roy introduced mean-variance
analysis with correlated risks into the financial literature. De Finetti solved
the problem of computing mean-variance efficient frontiers for a particular
constraint set (one that describes the reinsurance problem) assuming
uncorrelated risks. While he understood and explained the importance of the case
with correlated risks, he did not provide an algorithm for this case. In fact,
one of his conjectures concerning his solution was incorrect. The present
article summarizes de Finetti's contribution, presents an algorithm for solving
"the de Finetti problem" when risks are correlated and illustrates these matters
with an easily visualized two policy reinsurance problem."
Exchangeability in the Twenty First Century
Persi Diaconis (Stanford University, USA)
I will review recent appearances of exchangeability and its variations and point to some work that remains to be done. Topics include: Exchangeability at the foundations of exploratory data analysis (what is a batch?). Exchangeability and quantum mechanics, exchangeability and the structure of large networks (work of Lovasz) and the search for a Bayesian central limit theorem.
Some highlights from the theory of multivariate
symmetries
Olav H. Kallenberg (Auburn University , USA)
We explain how invariance in distribution under separate or
joint contractions, permutations, or rotations can be defined in a natural way
for d-dimensional arrays of random variables. In each case, the distribution is
characterized by a general representation formula, often easy to state but
extremely tricky to prove. Comparing the representations in the first two cases,
one sees that an array on a tetrahedral index set is contractable iff it admits
an extension to a jointly exchangeable array on the full product set. (No direct
proof of this amazing fact is known.)
Multivariate rotatability is defined
most naturally for continuous linear random functionals on tensor products of
Hilbert spaces. Here the basic examples are the multiple Wiener-It\^o integrals,
and the general representation is essentially a linear combination of such
integrals. The rotatable theory can be used to derive similar representations
for separately or jointly exchangeable or contractable random sheets. The
technical details are all available in a recent monograph, and our present aim
is only to try to convey, by rather elementary means, some of the basic
underlying ideas.
Exchangeable Rasch Models
Steffen L.
Lauritzen (Oxford University, UK)
Rasch (1960) introduced models for finite random binary matrices corresponding to cognitive test results which are widely applied, also to very different areas. The Rasch models have the property that the probability of any specific matrix depends only on its row- and column sums. The lecture describes the structure of such matrices when they also obey exchangeability and other types of symmetry properties. The relation to models for random graphs and social networks, shall be touched upon.
Bruno de Finetti and Economic Theory
Giorgio Lunghini (Università di Pavia, Italy)
Wooziness that knows it is woozy
may tell truths
Logic is
deaf to.
W.H. Auden
Bruno de Finetti gave many important contributions to economic
theory and on economic theory. De Finetti's main contribution is his own concept
of probability, which is both an answer and a critique to the question put by
J.M. Keynes: "[Under uncertainty] there is no scientific basis on which to form
any calculable probability whatever. We simply do not know. Nevertheless, the
necessity for action and for decision compels us as practical men to do our best
to overlook this awkward fact and to behave exactly as we should if we had
behind us a good Benthamite calculation of a series of prospective advantages
and disadvantages, each multiplied by its appropriate probability waiting to be
summed". Unfortunately, a policy - oriented economic theory seems to be
incapable of adopting a Ramsey - de Finetti concept of probability.
This is
the first topic briefly discussed in this paper, the others being de Finetti's
(and Ettore Majorana's) refusal of determinism after the developments of physics
in the 20's; de Finetti's critique of the (mis)use of mathematics in economic
analysis; of the asserted neutrality of economic theory; and of econometrics as
play-o-metrics. On this issue, de Finetti's point seems to be the same of
Keynes's point, when Keynes argues (against Tinbergen) that "One has to be
constantly on guard against treating the material as constant and homogeneous in
the same way that the material of the other sciences, in spite of its
complexity, is constant and homogeneous. It is as though the fall of the apple
to the ground depended on the apple's motives, on whether it is worth while
falling to the ground, and whether the ground wanted the apple to fall, and on
mistaken calculations on the part of the apple as to how far it was from the
centre of the earth".
Bruno de Finetti hero of the two worlds:
(applied) mathematician and (quantitative) economist
Flavio
Pressacco (Dipartimento di Finanza Università di Udine, Presidente Comitato
Scientifico AMASES)
Bruno de Finetti was a lofty mathematician and a refined scholar of actuarial sciences. He is universally known as the founder of subjective probability, while only recently has been credited of path breaking contributions to the foundations of modern theory of finance in such topics as mean-variance, expected utility and arbitrage free pricing. Here we show how some of these results, whose importance the author himself did not realize, came, while contingently studying a specific practical actuarial problem, as a someway unexpected by product of a long effort of research in other topics (foundations of economic theory and the gamblers' ruin theory). In other words it could be said that a reinsurance problem and a long lasting research in foundations of economic theory are parents of the author's primacy in mean-variance and expected utility approach. But behind all these results there is the basic idea that mathematics is the science which makes possible (may be applied to) the comprehension of world, including the behaviour of economic agents under uncertainty.
De Finetti's contribution to the theory of random
functions
Eugenio Regazzini (Università di Pavia, Italy)
Bruno de Finetti initiated the study of processes with
independent increments at the end of Twenties, at the same time when he was
laying the foundations of the subjectivistic theory of probability. The
directions of his research into specific fields of probability would have been
widely influenced by his own mathematical formulation of the subjectivistic idea
of probability, which prescribes only finite additivity.
In this talk, the
peculiarity of de Finetti's contributions to processes with independent
increments are examined, both with respect to previous studies by Bachelier,
Einstein, Wiener, Paley, Zygmund, and in connection with fundamental later
investigation of Kolmogorov and Lévy.
Finally, one touches on the
reformulation of certain propositions, relating to the aforesaid processes, when
probabilities are not thought of as completely additive set functions.
Exchangeability and semigroups
Paul
Ressel (Katholische Universitat Eichstatt, Germany)
Exchangeability of a "random object" is a strong symmetry condition, leading in general to a convex set of distributions not too far from a "simplex" - a set easily described by its extreme points, in this case distributions with very special properties as for example iid coin tossing sequences in De Finetti's original result. Although in most cases of interest the symmetry is defined via a non-commutative group acting on the underlying space, it very often can be described by a suitable factorization involving an abelian semigroup. The factorizing function typically turns out to be positive definite, and results from Harmonic Analysis on semigroups become applicable. In this way many known theorems on exchangeability can be given another proof, more analytic/algebraic in a sense, but also new results become available.
Exchangeability, concepts of dependence, and
statistical implications
Yoseph Rinott (Hebrew University,
Jerusalem)
Exchangeability is a key notion in statistical inference and in probability. de Finetti's Theorem connects it to positive dependence, but in statistics it is often connected to negative dependence, for example in sampling (without replacement) and rank distribution. I will review notions of dependence and connections to exchangeability, and some related ideas on role of exchangeability in statistical inference and causality.
Dependence structures of some infinite variance
stochastic processes
Murad S. Taqqu (Boston University)
Fractional Gaussian noise is a Gaussian process whose increments exhibit long-range dependence. There are many extensions of that process in the infinite variance stable case. Log-fractional stable noise (log-FSN) is a particularly interesting one. I t is a stationary mean-zero stable process with infinite variance, parametrized by a number alpha between 1 and 2. The lower the value of alpha, the heavier the tail of the marginal distributions. The fact that alpha is less than 2 renders the variance infinite. Therefore dependence between past and future cannot be measured using the correlation. There are other dependence measures that one can use, for instance the "codifference" or the "covariation". Since log-FSN is a moving average and hence "mixing", these dependence measures converge to zero as the lags between past and future become very large. We show that the codifference decreases to zero like a power function as the lag goes to infinity. The value of the exponent, which depends on alpha, measures the speed of the decay. There is also a multiplicative constant of asymptoticity c which depends also on alpha and plays an important role. This constant c turns out to be positive for symmetric alpha-stable log-FSN, and the rate of decay of the codifference is such that one has long-range dependence. We also show that a second measure of dependence, the "covariation", converges to zero with the same intensity and that its constant of asymptoticity is positive as well. This is joint work with Joshua B. Levy.
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