Who is this talk primarily for?

What are we going to do?

What sort of things to look out for?

What are your suggestions/comments/questions?

**Those of you who:**

use Moodle as a virtual learning environment or have a possibility to use it

use STACK (question type for Moodle) or would like to use STACK

would like to implement a longish exercise (with several answer boxes) in STACK; an example of a longish exercise is an application of Itô’s Lemma

**Things that we will need:**

Moodle with STACK plug-in (you can use EMS2022 Moodle to practice)

MAXIMA installation (MAXIMA is the computer algebra system behind STACK)

**Things that will be helpful but not necessary:**

- basic experience with/knowledge of STACK; if you don’t have it, no problem, you can follow ‘Authoring quick start’:

- basic knowledge of MAXIMA

**We will go over a longish exercise (Itô’s Lemma) in a general case****We will go over a longish exercise (Itô’s Lemma) in a specific case, for which:**We will label the answer boxes from 1 to 14 (on a piece of paper)

Code it in MAXIMA

‘Code’ it as a STACK question on Moodle

Randomize its sub-parts/elements (MAXIMA code)

Refine and finish

Itô‘s Lemma: Given an Itô process \(\{X_t\}\), \(dX_t=a(X_t,t) dt+ b(X_t,t) dZ_t\), where \(\{Z_t\}\) is the Wiener Process, \(a(\cdot,\cdot)\) and \(b(\cdot,\cdot)\) are functions of two variables satisfying some regularity conditions, the ’new’ Itô process \(\{G_t\}\), \(G_t=G(X_t,t)\), satisfies the following stochastic differential equation

\[ dG_t=\left( \frac{\partial G}{\partial X_t}a(X_t,t) + \frac{\partial G}{\partial t}+ \frac{1}{2} \frac{\partial^2 G}{\partial X_t^2} [b(X_t,t)]^2 \right) dt+ \frac{\partial G}{\partial X_t} b(X_t,t) dZ_t \]

To apply this lemma we need to know how to:

Identify functions \(a(\cdot,\cdot)\) and \(b(\cdot,\cdot)\) in a given stochastic differential equation.

Find partial derivatives \(\frac{\partial G}{\partial X_t}\), \(\frac{\partial^2 G}{\partial X_t^2}\), \(\frac{\partial G}{\partial t}\)

Plug-in the elements from parts 1. and 2. into the formula from the lemma and then simplify the expression.

Let \(\{S_t\}\) be an Itô process satisfying the following stochastic differential equation

\[ dS_t=\mu S_tdt+\sigma S_tdZt \] where \(\mu\) and \(\sigma\) are constants. (\(\{S_t\}\) is called Geometric Brownian Motion.)

Apply Itô’s Lemma to derive a stochastic differential equation for the process \(\{G_t\}\), where

\[ G_t=G(S_t,t)=S_t^2 \]

- In STACK’s \(\fbox{Question variables}\) we insert MAXIMA code