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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