Deck 1 Flashcards
What is a central tool in measuring changes?
A central tool will measure changes, especially of measures with the same null sets.
What is the definition of absolute continuity in measures?
V is absolutely continuous with respect to M, if v(N) = 0, VN E F with u(N) = 0.
What does it mean for measures to be equivalent?
M, v are called equivalent, if u < V and v < M.
What describes the evolution of available information about various factors like prices and politics?
A filtration
A filtration is a sequence of σ-algebras that represents the information available over time.
Define a filtration in the context of probability space.
A sequence (Fn) of σ-algebras in S such that In ⊆ Ik for all n, k ∈ N
This sequence encodes the evolution of information in a filtered probability space.
What is a filtered probability space?
(S, F, P, F) where F = (In)_{n ∈ N}
This space incorporates a filtration that represents the evolution of information.
What does In represent in the context of filtration?
The information known/available at time n
Events A ∈ In can be determined as having happened or not at time n.
What is a stochastic process?
A family X = (Xn)_{n ∈ N} of random variables at each point of time
This definition applies in discrete time.
What condition must a stochastic process satisfy to be called adapted?
Xn is Fn-measurable for all n ∈ N
An adapted process is further classified as a semimartingale.
How can a stochastic process be alternatively viewed?
As a mapping X : S × N0 → R (or R’)
It can also be considered as a random variable X with values in RN (or (Rd) N0).
What is the notation for n- and 0- in the context of stochastic processes?
n- := n - 1 and 0- := 0
This notation is used to denote previous time indices.
What is /_\ Xn in a stochastic process?
/_\ Xn = Xn - Xn-1 is the change of a stochastic process.
What is a predictable process?
A process X is called predictable (previsible) if Xn is Fn-measurable ∀n ∈ No.
What is a filtration generated by a process X?
A filtration F is said to be generated by a process X if Fn = σ(X0, …, Xn) ∀n ∈ No.
What are stopping times?
Random times for which we know whether they already occurred or not are called stopping (optional) times.
What is a stopping time?
A random variable T: S → No = No ∪ {0} is called a stopping time if {T = n} ∈ Fn ∀n ∈ No.
What does Lemma 3.1.8 state about stopping times?
A random variable T: S → No is a stopping time → {T = n} ∈ Fn ∀n ∈ No.
Give an example of a stopping time.
A typical stopping time is the first time the DAX is above 10000 points for the first time.
5 days before this event is not a stopping time.
What does Lemma 3.1.9 state about semimartingales?
Let X be an IRd-valued (discrete-time) semimartingale and B ∈ d (Borel-σ-algebra over Rd). Then T = inf {n ∈ No : Xn ∈ B} is a stopping time.
What does it mean to ‘freeze’ processes at stopping times?
Often we ‘freeze’ processes at stopping times.
What is the process X stopped at time T?
Let X be a stochastic process and T a stopping time. The process XT defined by Xn = Xn ∀n ∈ No is called the process X stopped at time T.
