In this section we set the notion of probability measures on a mathematically solid foundation. This will enable us to formulate many problems in a more general setting and to see that lotteries with finitely many outcomes and lotteries with continuous outcome distributions are only two special instances of general lotteries.
Many results can then derived much simpler. There is, however, some more severe mathematics involved, although we will skip a lot of more subtle details. Therefore we need to apologize to the specialists for pretending that things are easier than they actually are, and to the non-specialist for presenting things that are quite theoretical.
Nevertheless we hope that both can forgive us and appreciate the goal of bridging the gulf between measure theory and applications in finance.
We first need to define what we want to “measure” with a probability measure.
Given a set of all possible states, astate space˝, we want to assign probabilities to subsets of this state space, i.e., toevents. As example you may think of the state space as the set of all possible returns of an assetR, and as events like “the asset yields a return larger thanx” which would correspond to the subset.x;1/ofR.
Unfortunately, for measure theoretical reasons it is generally not possible to assign meaningful probabilities toallsubsets of a state space,1therefore we have to restrict ourself to subsets that are in a so-called “-algebra” (sometimes also called
“tribe”). This point is not essential for most of our applications, but it helps to know the properties these subsets satisfy:
Definition A.4 (-algebra) A collectionFof subsets of˝ is called-algebra if the following three conditions are satisfied:
1If you want to know what can happen when you allow for arbitrary subsets, take a look on the creepy Banach-Tarski Paradox, see e.g. [Fre88]: you could decompose a massive ball into five pieces, put them together again and – the new (still massive) ball is twice as large as before!
(i) If a setB is inF, then its complement˝ nB is also in F. (This condition ensures that if an eventBhas a certain probability, then its complement, i.e., the event thatBdoesnothappen, has also a well-defined probability.)
(iii) IfB1;B2; : : :are inF, then also their unionB1[B2[: : :is inF. (This condition ensures that separate events that have a probability, can be summarized to one event with a certain probability.)
(iii) The empty set;and the whole state space˝ are both inF. (This condition simply ensures that the event “nothing happens” and the event “something happens” both have a well-defined probability.)
The mathematically less inclined reader is again ensured that all events he usually encounters are in a -algebra and hence have a mathematically well- defined probability. It is enough to remember that there is a potential mathematically problem hidden that fortunately has been solved in a clever way by the definition of -algebras.
After this technical prelude we can now define probability measures:
Definition A.5 (Probability measure) A probability measurepon a state space˝ is a map that assigns to subsets of ˝ that are in some-algebraF a number in Œ0;1and satisfies
(i) p.;/D0(the empty set has measure zero), (ii) for pairwise disjoint setsBi˝, we have
p [1 iD1
Bi
! D
X1 iD1
.Bi/;
(iii) p.˝/D1(the whole state space has measure one).
In general, not only the empty set will have the probability zero (even if we deal only with a finite numbers of events). In the general case that does not mean that the event can never happen: as example think of the probability to get the number when randomly picking a number between0and10. This probability will be zero if the probability distribution is uniform, however, it is perfectly possible to get.
If we want to restrict our analysis of a certain situation only to the “relevant”
events, i.e., to the events with non-zero probability, it is handy to use the following notation:
Definition A.6 (“Almost” and null sets) Let.˝;F;p/be a probability space. We say that an eventB˝happensalmost surely(abbreviateda.s.) ifp.B/D1, and we say that a property that holds on a setB˝ withp.B/D 1holds foralmost everyelement in˝(abbreviateda.e.). We call all setsBwithp.B/D0nullsets.
338 A Mathematics We are mostly (but not exclusively) interested in probability measures onR. (The outcome distribution of an asset, e.g., is nothing else than a probability measure on R.) It is possible to characterize probability measures onRin a handy way. This can be done by applying Lebesgue’s Decomposition Theorem and the Radon-Nikodym Theorem to get the following result:
Theorem A.7 Let p be a probability measure on R. For A R let jAj denote the usual (Lebesgue) measure of the set A. Then there exists an integrable function fWR!Œ0;1and a singular measure pssuch that pDfdxCps. Heredx denotes the usual (Lebesgue) measure onR, and “singular” means that there exist disjoint sets A;BRwith A[BDRand ps.A/D0andjBj D0.
We can now decompose the singular part even more. For this we need first the following definition:
Definition A.8 (Dirac measure) TheDirac measureıxassigns to every setAR the value one ifx2Aand zero ifx62A.ıxis a probability measure onR.
With this definition we can decompose the singular measure ps into a linear combination of Dirac measures plus a remainder:
psD X1 iD1
iıxiCpc;
wherei > 0, xi 2 R, and the remainder pc is called the “Cantor-part” of the measure. In most of our applications we assume that all probability measures p satisfypcD0.
We finally need to find a way to integrate with respect to a general probability measure. This is quite natural when we want to integrate over a step function, so let us do this first:
Definition A.9 Letf be a step function, i.e., there is an increasing sequence of xi 2 Rsuch thatf is constant on each intervalŒxi;xiC1/with valuefi. Letpbe a probability measure onR. Then we define
Z
fpWDX
i
p
Œxi;xiC1/ fi:
For general integrable functionsf we defineR
fpas limit ofR
fnpwhere.fn/is a sequence of step functions which approximatesf.
How can we make use of this new formalism? We illustrate this with the following example: if we write down the EUT for a lotteryA with finitely many outcomesx1; : : : ;xnand probabilitiespi; : : : ;pnwe get
EUT.A/D Xn
iD1
u.xi/pi:
If a lotteryBhas instead a continuous outcome distribution with densityf, we have to replace this definition with an integral formulation
EUT.B/D Z
u.x/f.x/dx:
But what if we have a mixture of both lotteries? With our new formalism we can express all these cases in one simple formula:
EUT D Z
udp;
wheredp denotes a probability measure. (The little “d” reminds us that we are dealing with an integration.) If, for instance, dp D fdxC Pn
iD1iıxi, a short computation gives
EUT D Z
udp D Xn
iD1
u.xi/iC Z
u.x/f.x/dx:
This new way of dealing with probability measures is not only convenient, but also allows us to discuss situations involving discreteandcontinuous lotteries, for instance when we want to approximate a continuous lottery by finite lotteries. (We do this in Sect.2.4.5and in Sect.2.4.6, when we discuss continuity properties of decision theories.)
We conclude this section with a useful result, the Jensen Inequality:
Theorem A.10 (Jensen Inequality) Let uWR! Rbe a concave function and let dp be a probability measure on R. Let be the expected value of dp, i.e., WD EŒpWDR
xdp. Then we have
u./
Z udp:
Proof Since u is concave, the tangent onu in lies nowhere below u. In other words, there is a lineg.x/ D u./Ca.x/(for some constanta), such that
340 A Mathematics g.x/u.x/for allx2R. Now we can estimate
Z udp
Z gdpD
Z
˛.x/Cu./dp D˛
Z
xdp˛ Z
dpCu./
Z dp D˛˛Cu./Du./:
This concludes the proof. ut
Jensen’s Inequality can be generalized in several ways:
• The inequality holds on a subinterval ofRifuis concave only on this subinterval.
• It can be generalized fromRtoRn.
• We can obtain a strict inequality sign ifuis not only concave, butstrictlyconcave (as long as dp¤ı).
• We obtain the inverse inequalities ifuis not concave, but convex.2 It is a nice little exercise to prove these statements.