Download A.1 Finite Probability Spaces

A.1 Finite Probability Spaces
A.1
47
Finite Probability Spaces
Assumption A.1.1. Let (⌦, F, P) be a finite probability space,
⌦ = {!1 , !2 , . . . , !n },
A.1.1
n 2 N.
Mean and Variance
Definition A.10. Let X be a real-valued random variable on (⌦, F, P), the
mean (or expectation) of X is defined by
P
E [X] =
m
X
xi P(X = xi ),
where X(⌦) = {x1 , . . . , xm }, m  n.
i=1
Remember that we use the notation
P(X = xi ) := P X
1
({xi }) = P({! : X(!) = xi }).
When there is no ambiguity about the probability measure considered, we
denote simply E[X] ⌘ EP [X] the expectation of X.
Proposition A.11. Let X be a real-valued random variable on (⌦, F, P)
and f : R ! R a measurable function, then f
(⌦, F, P) and
X is a random variable on
EP [f (X)] = EP [f ].
X
Proof. Since X is a random variable and f : (R, B) ! (R, B) is measurable,
for every Borel set H 2 B, we have
f
1
(H) 2 B and (f
X) 1 (H) = X
1
f
1
(H) 2 F.
Then, assuming X(⌦) = {x1 , . . . , xm }, m  n, we have:
P
E [f (X)] =
=
m
X
i=1
m
X
f (xi )P(X = xi )
f (xi )P X (f = f (xi ))
i=1
X
= EP [f ].
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APPENDIX
Example A.12. The distribution of the random variable ⇠ characterizing
the Binomial model is
P⇠ = p
u
+ (1
0 < d < u, p 2 (0, 1).
p) d ,
The mean of ⇠ can be equivalently computed by applying Proposition A.11
with f = Id, that is
EP [⇠] = up + d(1
p).
Definition A.13. Let X be a real-valued random variable on (⌦, F, P), the
variance of X is defined by
h
Var(X) = EP X
EP [X]
2
i
⇥ ⇤
= EP X 2
EP [X]2 .
Let Y be another real-valued random variable on (⌦, F, P), the covariance
of X, Y is defined by
Cov(X, Y ) = EP
⇥
X
EP [X]
Y
EP [Y ]
⇤
.
Example A.14 (Example A.12 continued). The variance of ⇠ characterizing
the Binomial model is
⇥
Var(⇠) = EP (X
up + d(1
⇤
p))2 = p(1
p)(u
d)2 .
Definition A.15. Two events A, B 2 F are independent if
P(A \ B) = P(A)P(B).
Definition A.16. Given a non-negligible event B, i.e. B 2 F such that
P(B) > 0, the conditional probability P(·|B) is the probability measure on
(⌦, F, P) defined as
P(A|B) =
P(A \ B)
.
P(B)
The conditional probability P(A|B) represents the probability that the
event A occurs given that the event B occurred.
Remark A.17. If B is a non-negligible event, A, B 2 F are independent if
and only if P(A|B) = P(A).
A.1 Finite Probability Spaces
49
Remark A.18. If A, B 2 F are independent events, then also Ac , B c , Ac , B
and A, B c are independent events.
Indeed:
P(Ac \ B c ) = P ((A [ B)c )
=1
=1
= (1
P(A [ B)
(P(A) + P(B)
P(A))(1
P(A \ B))
P(B)).
The other cases are proven analogously.
Definition A.19. Two sub- -algebras G, H of F, i.e. G, H ✓ F
-algebras
on ⌦, are independent if
P(A \ B) = P(A)P(B),
8A 2 G, B 2 H.
Two random variables X, Y on (⌦, F, P) are independent if the correspondent
-algebras (X), (Y ) are independent.
Remark A.20. If X, Y are independent random variables on (⌦, F, P), then:
(i) E[XY ] = E[X]E[Y ];
(ii) Cov(X, Y ) = 0, in particular Var(X + Y ) = Var(X) + Var(Y );
(iii) if Z is a (Y )-measurable random variable, then X, Z are independent;
(iv) if f, g : R ! R are measurable functions, then f (X), g(Y ) are independent random variables;
(v) if X is (Y )-measurable, then X is constant P-almost surely, i.e. let
⌦x = X
1
({x}) 6= ;, then P(A) = 0 for any A 2 F, A ✓ (⌦x )c
Note that the converse of the implication (ii) is not true: there are couples
of random variables with zero covariance but that are not independent. The
definition and properties of independence extend to the case of families of
random variables.
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APPENDIX
Definition A.21. Let {Xi }i2I be a finite family of real-valued random variables on (⌦, F, P), to any subset of indexes F ✓ I, #F = k 2 N, we associate
a random variable
X F : ⌦ ! Rk ,
XF : ! 7! (Xi (!))i2F .
The family {Xi }i2I is said independent if, for all disjoint subsets of indexes
L, M ⇢ I, L \ M = ;, the random variables XL , XM are independent.
Note that the independence of a family of random variables is stronger
than the pairwise independence. In particular, we have the following equivalent condition.
Lemma A.22. The family of random variables {Xi }i2I is independent if and
only if: for any finite subset of indexes F ⇢ I and any series of Borel sets
(Hi )i2F , Hi 2 B,
P
\
i2F
!
{Xi 2 Hi }
=
Y
i2F
P (Xi 2 Hi ) .
The proof is based on the Dynkin’s Lemma, which we skip at the moment.
A.1.2
Conditional Expectation
Assumption A.1.1 is equivalent to having a generic (possibly not finite)
sample space and only consider simple random variables, that is random
variables X taking only a finite number of values: X(⌦) = {x1 , . . . , xn }.
Such a random variable can be written as
X=
n
X
xi 1A i ,
where Ai = X
i=1
1
(xi ) 2 F.
In the present section, we do not require Assumption A.1.1, and so (⌦, F, P) is
a generic probability space, but we mostly consider simple random variables.
The integral of a simple random variable X on (⌦, F, P) is defined as
Z
Z
n
n
X
X
XdP ⌘
X(!)P(d!) :=
xi P(Ai ), where X =
x i 1A i ,
⌦
⌦
i=1
i=1
A.1 Finite Probability Spaces
51
thus we write
P
E [X] =
Z
P
⌦
XdP and E [X1A ] =
Z
A
XdP 8A 2 F.
Assumption A.1.2. The probability space (⌦, F, P) is complete, that is such
that: for any event E 2 F with null probability, P(E) = 0, all its subsets
A ✓ E are also measurable events, i.e. A 2 F, and consequently P(A) = 0.
All events with null probability are called negligible events. Note that, in
case (⌦, F, P) was not complete, we could always extend it to a probability
space (⌦, F̄, P̄) that includes the negligible sets in the following way: denote
N := {A ⇢ E | E 2 F, P(E) = 0} and let F̄ be the smallest
-algebra
containing both F and N , i.e.
F̄ := (F [ N ) = {B ✓ ⌦ | B = E [ A, E 2 F, A 2 N },
and finally extend P to a probability measure P̄ on F̄ by defining, for all
B 2 F̄, P̄(B) = P(E), where B = E [ A and E 2 F, A 2 N .
In financial applications, the available information plays an important
role and needs to be considered when computing the expectation of a random
variable, e.g. the future price of a security. From this, the introduction of the
concept of ‘conditional expectation’ is natural. In particular, given a random
variable X and a -algebra G, we would like to compute an estimate of X
which takes into account the set of information contained in G. To define
this object, we go through gradual steps.
Let X be a simple random variable on (⌦, F, P).
• Given an event B 2 F, the conditional expectation of X with respect
to the event B is defined as
1
E[X|B] :=
P(B)
Z
XdP,
(A.2)
B
which is equal to the expectation of X with respect to the conditional
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APPENDIX
probability P(·|B), i.e. E[X|B] = EP(·|B) [X]. Indeed:
Z
P(·|B)
E
[X] =
X(!)P(d!|B)
⌦
Z
1
=
X(!)1B (!)P(d!)
P(B) ⌦
Z
1
=
X(!)P(d!).
P(B) B
Note that in this case the conditional expectation is a scalar.
• Given an event B 2 F such that 0 < P(B) < 1, the conditional expectation of X with respect to the
-algebra
({B}) generated by the
singleton {B} is defined as
8
<E[X|B] if ! 2 B,
E[X| (B)](!) :=
:E[X|B c ] if ! 2 B c ,
8! 2 ⌦
(A.3)
Note that E[X| (B)] is a random variable.
• Given another simple random variable Z on (⌦, F, P), taking values
Z(⌦) = {z1 , . . . , zm }, the conditional expectation of X with respect to
the -algebra (Z) generated by Z is defined as
E[X| (Z)](!) :=
m
X
j=1
E[X|{Z = zj }]1{Z=zj } (!),
! 2 ⌦.
(A.4)
We will use the notation E[X|Z] := E[X| (Z)]. Note that the definition
in (A.4) is only possible in the discrete setting (i.e. finite probability
space or simple random variables).
Before introducing the general definition of conditional expectation, let
us look at the properties of the objects defined in (A.3)-(A.4).
Lemma A.23. Let G be the
-algebra generated by either the event B or
the random variable Z. Then: a random variable Y is a version of the
conditional expectation with respect to G, i.e. Y = E[X|G] P-almost surely,
if and only if it satisfies
A.1 Finite Probability Spaces
53
(i) Y is G-measurable;
(ii)
R
A
XdP =
R
A
Y dP for all A 2 G.
Proof. ()) The measurability with respect to G is a direct consequence of
the definitions in (A.3)-(A.4). Regarding the property (ii), we distinguish
the two cases.
If G = (B) = {;, B, B c , ⌦}, then
Z
Z
E[X|G]dP =
(E[X|B]1B (!) + E[X|B c ]1B c (!)) P(d!)
A
A
8
>
0
if
>
>
>
>
R
>
< E[X|B]P(d!)
if
= RB
>
>
E[X|B c ]P(d!)
if
>
Bc
>
>
R
R
>
:
E[X|B]P(d!) + B c E[X|B c ]P(d!) if
B
and
Z
B
E[X|B]P(d!) = E[X|B]P(B) =
(analogously for B c ).
Z
A = ;,
A = B,
A = Bc,
A=⌦
X(!)P(d!)
B
If G = (Z) = ({Z = zj }, j = 1, . . . , m), then, for all j = 1, . . . , m,
Z
Z
E[X|G]dP =
E[X|{Z = zj }]P(d!)
{Z=zj }
{Z=zj }
= E[X|{Z = zj }]P({Z = zj })
Z
=
XdP
{Z=zj }
(() Since both Y and E[X|G] are G-measurable, also Y
measurable random variable. Thus the event {Y
E[X|G] is a G-
E[X|G] > 0} = {(Y
E[X|G]) ((0, +1))} 2 G is contained in G, so that we can apply the prop1
erty (ii) on it:
Z
Y
E[X|G]>0
This imply that P(Y
prove P(Y
(Y
E[X|G])dP =
Z
(X
Y
X)dP = 0.
E[X|G]>0
E[X|G] > 0) = 0. Through an analogous procedure we
E[X|G] < 0) = 0. So eventually we have P(Y 6= E[X|G]) = 0.
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APPENDIX
Another property of the conditional expectation of a simple random variable X defined in (A.4) is that it can be rewritten as a sum of the possible
values of X weighted with the respective conditional probabilities, analogously as the expectation of X is the sum of its values weighted with the
respective probabilities.
Lemma A.24. Let X, Z be two simple random variables on (⌦, F, P), then
E[X|Z] =
n
X
i=1
xi P({X = xi }|Z),
where X(⌦) = {x1 , . . . , xn } and we denote P(F |Z) := E[1F |Z] for any F 2
F.
Proof. Suppose Z(⌦) = {z1 , . . . , zm }. For all j = 1, . . . , m, for any ! 2 {Z =
zj }, we have
E[X|Z](!) = E[X|{Z = zj }]
Z
1
=
XdP
P({Z = zj }) {Z=zj }
n
X
1
=
xi P({X = xi } \ {Z = zj })
P({Z = zj }) i=1
=
n
X
i=1
xi P({X = xi }|{Z = zj }).