2) Martingales

Question 1

What is a filtration in probability theory

Answer 1

Question 2

What is a martingale with respect to a filtration

Answer 2

Question 3

What are submartingales and supermartingale

Answer 3

Question 4

What is the natural filtration

Answer 4

Fn = σ(Z1, Z2, . . . , Zn), the σ-algebra generated
by Z1, Z2, . . . , Zn

Question 5

How does expectation behave for martingales, submartingales, and supermartingales

Answer 5

Question 6

What is Doob decomposition theorem (NOT NEEDED FOR MIDTERM)

Answer 6

Question 7

What is a stopping time with respect to a filtration

Answer 7

Question 8

What are the key properties of stopping times

Answer 8

Question 9

What is a stopped process and how is it defined for a stopping time T

Answer 9

Question 10

If {Zn ,n≥0} is a martingale, what can be said about the stopped process Yn =ZT∧n

Answer 10

Question 11

What is the σ-algebra associated with a stopping time τ

Answer 11

Question 12

What are the key properties of the σ-algebra associated with stopping times

Answer 12

Question 13

What is Doob’s optional stopping theorem

Answer 13