Chapter 8 - Stochastic Calculus & Ito Processes

Question 1

What is an Ito Process?

Answer 1

An Ito Process is a stochastic process described in terms of a deterministic part and a random part as a stochastic integral equation.

e.g.
Xt = X0 + int[0,t] mudt + int[0,t]sigmadwt

This can also be written as
dXt = mudt + sigmadWt

Question 2

What is Ito’s Lemma used for?

Answer 2

To differentiate a function f of a stochastic process X

Question 3

In the equation dX = u dt + o dW, what does ‘dX’ represent?

Answer 3

The differential of the stochastic process X

Question 4

What is the form of Itos Lemma df(X,t) when there is explicit time-dependence?

Answer 4

df(X,t) = ∂f/∂t dt + ∂f/∂x dX + (1/2) ∂²f/∂x² dWt

Question 5

Fill in the blank: The second-order terms in Ito’s Lemma can be simplified using the _______.

Answer 5

multiplication table

Question 6

True or False: Ito’s Lemma can only be applied to processes that do not depend on time.

Answer 6

False

Question 7

In the context of stochastic calculus, what is a Wiener process?

Answer 7

A continuous-time stochastic process that represents Brownian motion

Question 8

What is the stochastic differential equation for a General Brownian Motion Process?

Answer 8

dXt = mudt + sigmadWt , X0=X

Xt = X + mut + sigmaWT

(constant drift & diffusion)

Question 9

What is the stochastic differential equation for a Geometric Brownian Motion Process?

Answer 9

dXt = Xtmudt + XtsigmadWt , X0=X

(Proportional drift & diffusion)

Question 10

What is the stochastic differential equation for a Ornstein-Uhlenbeck Process?

Answer 10

dXt = -yXtdt + sigmadWt

(proportional drift, constant volatility)

Question 11

What is the stochastic differential equation for a Ornstein-Uhlenbeck mean reverting Process?

Answer 11

dXt = y(mu-Xt)dt + sigmadWt

Question 12

What is the stochastic differential equation for a Ornstein-Uhlenbeck square root mean reverting Process?

Answer 12

dXt = y(mu-Xt)dt + sigmasqrt(Xt)dWt