Ito's Lemma -- Some intuitive explanations on the solution of stochastic differential equations

Table of contents below, if you just want to watch part of the video.
🌎🌍🌏 subtitles available, German version:    • Ito's Lemma -- Einige intuitive Erläu...  
Prof. Hakenes teaches Finance and Mathematics in Bonn (https://www.econ.uni-bonn.de).

We consider an stochastic differential equation (SDE), very similar to an ordinary differential equation (ODE), with the main difference that the increments are stochastic. We also simulate it numerically, and make a guess for its solution. The guess is incorrect, because it does not take the #volatility (σ) correctly into account. Of course, we then show the correct solution. Here, "solution" means that we get a closed equation for the process that depends only on the initial value, time, and the underlying #Wiener process (W).

No finally, we want to prove that our solution is indeed correct. We therefore need to take a the derivative, but this involves the stochastic increments. We must use #Ito's Lemma, which is essentially an extension of the ordinary chain rule. The proof, actually, is just one line or two.

What you need to watch this video:
* Calculus, and some knowledge of ordinary differential equations,
* Knowledge of Excel to follow the numerical examples.

What you DO get from this video:
* An intuition of what stochastic processes are (like the Wiener process),
* An intuition of what stochastic differential equations are are,
* An intuition of what it means to solve an SDE,
* A relatively simple application of Ito's lemma,
* Some understanding about what Ito's lemma does.

What you DON'T get from this video:
* A proof of Ito's lemma,
* A thorough introduction to stochastic calculus (with measure spaces, filterings, ...).

Comments:
At 04:12, we have chosen a tiny beta. Typically, to numerically solve an ODE, on let's dt converge to zero. We have dt constant at dt = 1, so to comensate for that, we choose a function that does not move a lot (tiny beta), such that the tracking error does not become too large.

Thanks @ Prof. Dr. Schweizer for very helpful comments.
Thanks @ Prof. Dr. Bühler for teaching me this material.
Thanks @ Prof. Dr. Sandmann for teaching me even more of this stuff.
Thanks @ all of you for your positive feedback. I am therefore planning to make more videos, also answering to some requests. Please *let me know*, in the comments, what topics you would be most interested in. Option pricing, like Black Scholes? Other processes, like Vasicek, Ornstein-Uhlenbeck or the Brownian bridge? Or what else? I am willing to put in some effort. The underlying theme would still be: I try to create a bridge between the mathematical theory, which is beautiful, and (economic) intuition, which would typically also include Excel examples.

Here is a link to our Excel example: https://docs.google.com/spreadsheets/...

Table of Contents
00:01 Introduction
00:34 What is Ito's Lemma about, in words?
01:49 Comparison to Ordinary Stochastic Equation (ODE): What is the "solution" of an ODE?
04:03 Excel simulation of the ODE (not yet the SDE)
06:21 Excel simulation of an SDE
08:40 Geometric Brownian motion (in Excel)
11:05 What is a "solution" of an SDE?
11:57 Educated guess, but without the quadratic term
14:12 True solution, with the term σ^2/2
16:02 Formal solution, using Ito's lemma (finally!)
21:18 Recapitulation