Financial Engineering with Stochastic Calculus I
40 / 62
Background in probability theory
September 10, 2019
Outline
Construction of probability spaces
Random variables and measurability

Financial Engineering with Stochastic Calculus I
41 / 62
Background in probability theory
Probability spaces
Definition 3.1
A
probability space
consists of a set
Ω
, a family
F
of subsets of
Ω
(
σ
-algebra
), and a
function
P
(
probability measure
) which assigns to every
A
∈ F
a
probability
P
[
A
]
∈
[0
,
1]
.
The
σ
-algebra
F
specifies those sets of outcomes for which we can assign a meaningful
likelihood. Those sets are called
events
.
In general, we cannot assign likelihood in a meaningful way to all such possible sets of
outcomes, but in this course we will encounter only sets for which we are able to!

Financial Engineering with Stochastic Calculus I
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Background in probability theory
For
F
we demand
a)
Ω
∈ F
b)
if
A
∈ F
, then the complement
A
c
∈ F
c)
if a sequence of events
A
1
,
A
2
, ...
∈ F
, then
∪
∞
j
=1
A
j
∈ F
.

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