Outline

Who is this talk primarily for?

Those of you who:

Things that we will need:

Things that will be helpful but not necessary:

https://stack-demo.maths.ed.ac.uk/demo/question/type/stack/doc/doc.php/Authoring/Authoring_quick_start.md

What are we going to do?

Longish exercise (Itô’s Lemma) in a general case

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:

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

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

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

Longish exercise (Itô’s Lemma) in a specific case

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

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

a) Label the answer boxes 1-14 (on a piece of paper)

b) Code it in MAXIMA (screen-shot from MAXIMA)

c) ‘Code’ it as a STACK question on Moodle (screen-shot from Moodle)

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