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.

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.

*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

[sipholukhozi]

In 2023 - this is still the most powerful explanation I have ever came across regarding Ito and SDEs. Thanks a lot!

[kostas]

one the simplest and most excellent expositions I ve seen. Bravo!

[manueljoaquincerezodelaroc]

This is the first time I got SDE's and how to use Ito's Lemma. Thank you!

[tvlobo]

Good job sir, i always try to watch intuitive videos of math and the solve the equations understanding why you use that

[xddxd]

I wish all your videos were on english, because your explanations are just excellent. I was familiar with Ito but u just gave me a new intuition, Thank you so much

[oigzqtg]

I would love to see more financial mathematics videos covered in english!!! This was really helpful. Thank you :)

[idealized_]

I’m taking a financial mathematics course this semester. Thanks for this

[wenzhang]

Great explanation with Excel. Good job, thank you!

[codyfintech]

Very good, although I do think its useful to include a note on the quadratic variance of the Wiener process being equal to dt, for the application of Ito's lemma.

[syng]

Thanks for the clear explanation! greeting from malaysia👍👍👍

[who]

It took you 25 mins to explain what my teacher tried to explain in 6 months

[MarineLefarge]

The video is very interesting, thank you! However, I didn't quite understand how Ito's lemma allows to take into account continuous variations of the interest rate

[cdenn]

Would love you to go through ito integration in similar detail

[Tyokok]

Thanks for the great video! One question if I may, at 8:29 if your delta t is not 1, your dWt still using standard normal? I just want to clarify the relation between dWt, standard normal, and dt. Is dWt always ~N(0, 1) under any dt? Many thanks in advance if anyone can advise.

[yann]

Very instructive video thanks sir. May I know where the intuition comes for adding the variance term in order to correct the solution of the PDE for Wienner process ?

[paulbirs]

Just stumbled upon this explanation. Really nice, thanks a lot! I have a question though. At 15:00 you show the formula for the correct solution and there I can see that there is a term -A2 which is "t". But in this case we basically end up with the function similar to exp(-t), because "t" is increasing and Wt is not. And the solution can't look like what we need. So did you correct the formula afterwards? How does "t" actually contribute to exp(sigma*Wt - t*sigma^2/2) formula?

[Manik_]

Sir, please more videos in english on Financial Mathematics

[bhavinmoriya]

Dear Hendrik, Thanks very much for an awesome video. Could you please share the Excel sheet which you produced in the video?

[wishuahappyday]

Professor, I have tested the equations on my computer, and I found that the denotation "t" used is actually the step size, instead of the actual t {0, 1, 2, 3, 4, 5, 6, 7, 8....} . may I ask why is that ?

[mileknz]

Hi, very basic question: how you get dWt from Wt? And Wt is just ~N(0, 1)?