Week 6

Question 1

A stochastic process is a martingale if

Answer 1

Question 2

Show that Brownian motion is a martingale

Answer 2

Question 3

Define a general ODE and rewrite it discretely

Answer 3

Question 4

Relate SDE to ODE

Answer 4

SDEs include some randomness at each time step

Question 5

Show how we use Brownian motion in a SDE

Answer 5

Question 6

Partial deriv notation, meaning and example

Answer 6

Question 7

For SDEs (in this course) we will only ever take deriv of

Answer 7

Lower case Latin letter, which represent standard fxs

capital Latin letters represent stochastic processes

Greek letter are coefficients of SDE (and are FXs of stochastic processes)

Question 8

Ito’s lemma

Answer 8

Question 9

Easy ways to remember ito’s lemma

Answer 9

Question 10

How to use ITO’s

Answer 10

Let dX_t = dW_t

Then take partial of f(t,x(t)) wrt x, t, xx

Question 11

Relate wether a process is a martingale to its PDE through Ito’s

Answer 11

If drift term = 0 (the coefficient on dt) then the process Yt is a martingale

This makes sense as this shows time independence