Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Financial Mathematics 378 > Integration, integrals, MCT & DCT > Flashcards

Integration, integrals, MCT & DCT Flashcards

1
Q

Definition of an integral:

A

Given f: (Omega, F,mu) -> Rbar where f is measurable.
Integral of f w.r.t. measure mu:
Integral(f)dmu

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Integral w.r.t Dirac measure:

A

mu = Sx0 is a dirac measure.

Sx0(A) = {1, if x0 element of A & 0 otherwise

Integral(f)dSx0 = f(x0)

~if xi is in set A: if yes, Sxi(A)=1, if no Sxi(A) = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

General rule of integrating w.r.t. dirac measure:

A

If mu = a1Sx1 + a2Sx2+ …
where ai element of [0,infinity) & xi element of omega:

mu(A) = sum(for i)ai*Sxi(A)

Integral(f)dmu = sum(for i)f(xi)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Integral w.r.t. lebesgue-measure:

A

f: (R,B(R),lambda)-> Rbar

Integral(f)dlamba = Integral(from -infinity to infinity) (f(x))dx

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

Integral w.r.t. Lebesgue-Stieltjies measure:

A

If function F: Real->Real is
increasing (a ≤ b => F(a) ≤ F(b) ) &
right continuous (lim(x->a^+)F(x) = F(a)

musubscripF( (a,b)) = musubscripF([a,b]) = musubscripF((a,b]) = F(b)- F(a)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

If X: (Omega,F,Real) -> Real &
F(X) = P(X ≤ x), what is X’s distribution?

A

The Lebesgue-Stieltjies measure, musubscriptF(.).

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Integral w.r.t. pushed-measure:

A

If G: (Omega,F,Real) -> (S, L, muG^-1) & G is measurable.

(thus B element of L,G^-1(B) element of F because G is measurable)

muG^-1(B) = mu(G^-1(B))
(Push-measure = mu-measure of pullback)

muG^-1 is a measure on (S,L)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

If X: (Omega,F,Real) -> Real &
F(X) = P(X ≤ x), what is X?

A

A random variable

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

True or False:
i) If a function G is measurable
<=> G is continuous?

ii) If a function G is continuous => G is measurable?

iii) If a function G is measurable => G is continuous?

A

i) False

ii) True

iii)False

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

General rule of integrating a simple function:

A

Given f: (Omega,F,mu) ->(Rbar, B(Rbar) and assuming f is measurable.

If f = sum(for i)ai*Isubscript(Ai)
where ai element of Real &
Ai element of F, then

Integral(f)dmu = sum(for i)ai*mu(Ai)

Integral(Isubscript(A))dmu = mu(A)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Definition of expectation:

A

Suppose X: (Omega,F,P) -> (Rbar,B(Rbar)) is measurable (thus X is measurable)
then
E[X] = Integral(X)dP

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

Do integrals have to exist?

A

No

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

Properties of integrating simple functions:

A

If f & g are simple functions , then

1) Integral(f)dmu doesn’t depend on the simple interpretation of f.

2) Integral(Isubscript(A))dmu = mu(A) for A element of F

3) Linearity:
If f,g ≥ 0 & a,b ≥ 0, then
Integral(af + bg)dmu = aIntegral(f)dmu +bIntegral(g)dmu

4)Monotonicity:

If f,g ≥ 0 & f ≤ g, then
Integral(f)dmu ≤ Integral(g)dmu

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

What do we call measurable function whose integrals exist?

A

Integrable functions

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

When is a function f: Omega -> Real
simple?

A

If it only takes a finite number of different values.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

If f: (Omega, F,mu) -> Clo Real, what is:
i) sF ?
ii) sF^+ ?
iii) mF ?
iv) mF^+ ?

A

i) sF : set of simple functions

ii) sF^+ : set of simple function ≥ 0

iii) mF : set of measurable functions

iv) mF^+ :set of measurable function ≥ 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

If f: (Omega, F,mu) -> Rbar, what is:
i) sF ?
ii) sF^+ ?
iii) mF ?
iv) mF^+ ?

A

i) sF : set of simple functions

ii) sF^+ : set of simple function ≥ 0

iii) mF : set of measurable functions

iv) mF^+ :set of measurable function ≥ 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

Integration for non-negative measurable functions, mF^+:

A

Given f: (Omega,F,mu) -> Rbar is measurable & f ≥ 0 then

Integral(f)dmu = sup{Integral(T)dmu:
T ≤ f & T element of sF^+}

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Integration steps:
Given f: (Omega, F,mu)->(Rbar,B(Rbar)) is measurable

A

1) If f is simple, i.e.,
f = sumofi(aiIsubscriptAi)
then
Integral(f)dmu = sumofi(aimu(Ai))

2) If f an element of mF^+ & f ≥ 0
then
Integral(f)dmu = sup{Integral(T)dmu:
T ≤ f & T element of sF^+}

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

Integration for non-negative simple functions, sF^+:

A

If f simple,f = sumofi(aiIsubscriptAi), & f ≥ 0
then
Integral(f)dmu = sumofi(aimu(Ai))

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

Facts about if f element of mF^+:

A

1) Integral(f)dmu always exists
~can be infinity

2) Integral(f)dmu ≥ 0

3) There exists Tn element of sF^+ s.t.:

(i) Tn is increasing (T1 ≤ T2 ≤ …)

(ii)Tn -> T as n -> infinity
(pointwise: lim(n to infinity)Tn(w) = f(w) , for all w element of Omega

(iii) Tn ≤ f for all n

then
Integral(f)dmu = lim(n to infinity)Integral(Tn)dmu

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

Monotone convergence theorem:

A

Suppose fn: (Omega,F,mu) -> Rbar is measurable for all n element of N & f ≥0.

Assume that: lim n to infinity(fn) = f & f1 ≤ f2 ≤ …
(thus fn upwards arrow f)

then
lim n to infinity (Integral(fn)dmu)) = Integral(f)dmu

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

MCT steps

A

Given fn(x) determine lim n to infinity Integral(fn)dmu

1) Find f(x) = lim n to infinity (fn(x))

2) Check if fn(x) is increasing as a sequence:
fn(x) ≤ fn+1(x) for all n

3) Use MCT results:

lim n to infinity (Integral(fn(x))dmu)) = Integral(f(x))dmu

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

Integrating general functions:

Given f: (Omega,F,mu) -> Rbar is measurable
& recalling:
i) f^+ = max(x,0) ; f^- = -min(x,0)
thus f^+,f^- element of mF^+
ii) f = f^+ + f^-
what is Integral(f)dmu & when if f intergrable?

A

Integral(f)dmu = Integral(f^+)dmu + Integral(f^-)dmu
&
f is integrable if Integral(f)dmu exists & finite

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

When is a function integrable?

A

If Integral(f)dmu exists and is finite.

26
Q

When is a function f element of mF mu-integrable?

A

IFF mu(|f|) = Integral(|f|)dmu < infinity
OR
IFF Integral(f^+)dmu < infinity &
Integral(f^-)dmu < infinity

27
Q

Definition of mu-integrable functions, Ł^1(Omega,F,mu):

A

Ł^1(Omega,F,mu) = {f: Integral(|f|)dmu < infinity}
and is a vector space.

28
Q

If f = g - h & g,h element of mF^+ then Integral(f)dmu=?

A

Integral(f)dmu = Integral(g)dmu - Integral(h)dmu

29
Q

True or false:
Given Integral(f+g)dmu
= Integral((f+g)^+)dmu - Integral((f+g)^-)dmu

A

True

30
Q

True or false:
(f+g)^+ = f^+ + g^+

A

False

31
Q

Dominated Convergence theorem:

A

Let (fn)n be a sequence of integrable functions s.t.

i)lim n to infinity fn(w) exists for all w element of Omega

ii) There exists a function, g element of Ł^1(Omega,F,mu) s.t. |fn| ≤ g for all n element of N
(thus its bounded by something independent of n)

then
lim n to infinity Integral(fn)dmu = Integral(lim n to infinity)(fn)dmu

32
Q

Given f is an integrable function and A is a measurable set, what is the integral of f over set A, IntegraloversetA(f)dmu =?

A

IntegraloversetA(f)dmu = Integral(f*IsubscriptA)dmu

33
Q

Fatou’s lemma:

A

If fn element of mF^+ for all n element of N, then
mu(liminf(fn)) ≤ liminf(mu(fn))

34
Q

Reverse Fatou’s lemma:

A

If fn element of mF^+ fo all n element of N
then
limsup(mu(fn)) ≤ mu(limsup(fn))

35
Q

Fatou’s lemma results ordered by size:

A

mu(liminf(fn)) ≤ liminf(mu(fn)) ≤ limsup(mu(fn)) ≤ mu(limsup(fn))

~terms with limits on the inside is outside

36
Q

DCT Steps:

A

Given fn(x) determine lim n to infinity Integral(fn)dmu

1) Find lim n to infinity fn(x)
~get for different cases: x < 0; x = 0 ; x > 0

2) Find binding function g(x) for cases: x < 0; x = 0 ; x > 0

3) Integrate Integral(g(x))dmu

4) Use DCT results:

lim n to infinity Integral(fn(x))dmu ≤ Integral ( lim n to infinity fn(x))dmu

37
Q

Given measurable function X: (Omega,F,P) -> (Rbar, B(Rbar), PX^-1) & X, what is:

i) PX^-1?
ii) E[X] = ?

A

i) PX^-1 : the distribution of X

ii) E[X] = Integral over Rbar (x)dPX^-1(x)
= Integral(X)dP

38
Q

Given X: (Omega,F,P) -> (Rbar,B(Rbar),PX^-1)
& g: (Rbar,B(Rbar),PX^-1)-> (Rbar,B(Rbar),PX^-1)
thus (Omega,F,P) -> (Rbar,B(Rbar),PX^-1) -> (Rbar,B(Rbar),PX^-1),
change the variables of the integral & find E[g(X)]:

A

Change of variables:

Integral(g)dPX^-1 = Integral(g o X)dP

In general:
E[g(X)] = Integral(g(x))dPX^-1(x)

39
Q

Inequality properties of Expectation:

A

1) Markov’s inequality:

If g ≥ 0 & increasing then
E[g(X)] ≥ g(c)*P({X ≥ c}) for any c element of R

2) Chebyshev’s inequality:

For v > 0, then
P({|X-E[X]|≥ v }) ≤ Var(X)/(v^2)

where Var(X) = E[ (X-E[X])^2 ]

40
Q

If f is integrable, what is mu{f = ±infinity} ?

A

mu{f = ±infinity} = 0
thus f is finite mu-a.e.

41
Q

True or false:
If f is integrable, |Integral(f)dmu| ≤ Integral(|f|)dmu

A

True

42
Q

Definition of changing the measure with density:

A

Given (Omega,F,mu) -> Rbar is measurable and f ≥ 0.
Assume f element of Ł^1(Omega,F,mu),
thus Integral(f)dmu < infinity

Then define v = fmu : F -> [0,infinity] s.t.
v(A) = (fmu)(A) = Integral over A(f)dmu = Integral(f*IsubscriptA)dmu.

v is a measure on (Omega,F)
& v «mu

f is the density of v w.r.t. mu: f = dv/dmu

43
Q

Chain rule:

A

Let v = f*mu & g: (Omega,F) -> Rbar s.t. g element of Ł^1(Omega,F,mu) then

Integral(g)dv = Integral(gf)dmu
since f = dv/dmu => dv = f*dmu

44
Q

Change of variables:

A

Given f: (Omega,F,mu) -> (T,S,mu(f^-1))
& g: (Rbar,B(Rbar),mu(f-1))-> Rbar & f,g are measurable.
Then

Integral(g)dmuf^-1 = Integral(g o f)dmu

45
Q

True or false:

g o f = g(f)?

A

True

46
Q

Given (Omega,F,mu) is a measure space & A subset Omega:

i) When is A a mu-null (a null set)?
ii)When is the space (Omega,F,mu) complete?
iii) What is the completion of F, Fbar?

A

i) If there exists a B element of F s.t. A subset B & mu(B)=0;

ii) The space (Omega,F,mu) is complete if every mu-null set is in F.

iii) Fbar = sigma( F or {A subset Omega: A is mu-null })

pg.8 notes L27-L29

47
Q

If (Omega, F, mu) is a measure space, when does a statement, I0(w), hold mu-almost everywhere (mu-a.e.)?

A

If { w element of Omega: IO(w) is false} is mu-null

48
Q

If (Omega, F, P) is a probability space, when does a statement, I0(w) is true P-almost surely (P-a.s.)?

A

If { w element of Omega: IO(w) is false} is P-null

49
Q

True or false:

i) If F is complete, Fbar, then
mu({w element of Omega: IO(w) false}) = 0
=> IO(w) true a.e.

ii) If F is complete, Fbar, then
IO(w) true a.e.
=> mu({w element of Omega: IO(w) false}) = 0

iii) If F is complete, Fbar, then
IO(w) true a.e. <=>
mu({w element of Omega: IO(w) false}) = 0

A

i) TRUE
ii) TRUE
iii) TRUE

50
Q

Radon-Nikodym theorem:

A

If v & mu are sigma-finite measure on (Omega,F) &
v«mu (v abs cont w.r.t. mu), then

there exists a function f: omega-> Rbar s.t.
f = dv/dmu <=> dv = f.dmu

f: density/Radon-Nikodym derivative

51
Q

Definition of a measure, v, being absolutely continuous w.r.t. another measure, mu
(v«mu)

A

A measure v on (Omega,F) is absolutely continuous w.r.t. mu
if mu(A) = 0 => v(A) =0;

thus v«mu

52
Q

Using Radon-Nikodym theorem:

A

Find f(x) = lim h to o^+ (mu([x, x+h])/(lambda([x, x+h])

53
Q

Definition of product of measure spaces:

A

Let (Omega1,F1) & (Omega2,F2) be measurable spaces.

Define projections maps:
pie1 = Omega1 x Omega2 -> Omega1
( (w1,w2) -> pie1(w1,w2) = w1 )
&
pie2 = Omega1 x Omega2 -> Omega2
( (w1,w2) -> pie2(w1,w2) = w2 )

Define F1 circle* F2 = sigma(pie1,pie2) to be the smallest sigma-algebra that makes pie1 & pie2 measurable.

~(w1,w2): sample point in a space of combined outcomes (w1 element of Omega1 occurred & w2 element of Omega2 occurred)

~thus given combined outcome (w1,w2),
pie1(w1,w2)=w1 measures which outcome occurred in Omega1 &
pie2(w1,w2)=w2 measures which outcome occurred in Omega2

~projection maps pie1 & pie2 are measurable given that if we know a combined outcome (w1,w2), we should know the component outcomes, w1 & w2.

54
Q

True or false:

B(R) circlex B(R) = B(R^2)

A

True

55
Q

Definition of product of measures:

A

Let (S,F1,mu1) & (T,F2,mu2) be sigma-finite measure spaces.
Define map
mu1 circlex mu2: F1 circlex F2 ->Rbar^+

Then product measure of mu1 & mu2 on
(S circlex T, F1 circlex F2) for B element of F1 circlex F2:

mu1 circlex mu2 = Int(Int (IsubB(s,t)) mu2(dt))mu1(ds)

56
Q

Fubini’s theorem:

A

Assume (Omega1,F1,mu1) & (Omega2,F2,mu2) are sigma-finite measure spaces then
if f element of Ł^1(Omega1 x Omega2, F1 cricx F2, mu1 circx mu2)

(mu1 circx mu2)(f) = Int(f)dmu1circx mu2
= Int( Int(f(x,y)dmu1(x)) dmu2(y)
=Int( Int(f(x,y)dmu2(y)) dmu1(x)

~ Int( Int(f)dv)du = Int( Int(f)du)dv

~f element of Ł^1 <=> abs. conv. Int(|f|)dmu1dmu2 < inf

57
Q

Jensen’s inequality for E[X]:

A

If g: R->R is convex (function as a + 1st derivative: bend upwards), then
E[g(X)] ≥ g(E[X])

58
Q

Properties of expectation:

A

1) If X ≤ Y a.s. then E[X] ≤ E[Y]

2) E[aX+bY] = aE[X] + bE[Y]

3) If X = Y a.s. then E[X]=E[Y]

4) Continuity:
Assume X -> X a.s. , then

(i) Xn≥ 0 & Xn upwardsarrow X a.s. (non-neg & increasing)
then E[Xn] -> E[X]

(ii) There exists Y s.t. E[Y] < inf & |Xn| ≤ Y a.s. then
E[Xn] -> E[X]

(iii) & Xn is bounded, exists a constant c s.t. |Xn| ≤ c then
E[Xn] -> E[X] & E[Xn^k] -> E[X^k] for k element of N

5) E[c] = c if c a constant

59
Q

Definition of correlation:

A

Let X,Y element of Ł^1(omega,F,P) then correlation
px,y = cov(X,Y)/(sd(X)*sd(Y))
&
-1 ≤ px,y ≤ 1

~X,Y are correlated if cov(X,Y) = 0 or px,y = 0.

60
Q

Definition of covariance:

A

Let X,Y element of Ł^1(omega,F,P) then covariance,
cov(X,Y) = E[ (X-E[X]) (Y-E[Y]) ) = E[XY]-E[X]E[Y]

61
Q

True or false:

Uncorrelated ≠ Independent

A

True

62
Q

True or false:
Cov & correlation measures only LINEAR dependence (Y = aX+b)

A

True

Financial Mathematics 378 (2 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions