Відео

Oh no. You are right. I will try to edit that or reupload 😭 edit: I just cut the part that was wrong. Thanks for finding it!

Yes !

The random variable is a function that maps a random event ω in (Ω, F) into a measurable space (ℝ, B(ℝ)). X : (Ω, F) → (ℝ, B(ℝ)) So the function X(ω) is not random by itself. It is the input that is the source of randomness. You can take the example of rolling a dice where we distinguish the event ω = "face two come out" from the numerical value X(ω) = 2. here are some extracts from Øksendal's book Stochastic Differential Equations (6th edition, pages 9 - 10): (Lemma 2.1.2) A random variable X is an F-measurable function X: Ω → ℝⁿ. Every random variable induces a probability measure μₓ on ℝⁿ, defined by μₓ(B) = P(X⁻¹(B)). (Maybe the book I used for the video didn't mention measure because you can change it like for Girsanov theorem) (Definition 2.1.4) Note that for each t ∈ T fixed we have a random variable ω → Xₜ(ω); ω ∈ Ω. On the other hand, fixing ω ∈ Ω we can consider the function t → Xₜ(ω); t ∈ T which is called a path of Xₜ. It may be useful for the intuition to think of t as “time” and each ω as an individual “particle” or “experiment”. With this picture, Xₜ(ω) would represent the position (or result) at time t of the particle (experiment) ω. Sometimes it is convenient to write X(t, ω) instead of Xₜ(ω). Thus we may also regard the process as a function of two variables (t, ω) → X(t, ω). Hope it helps! 😅

@@stochastip as I understand, it maps from Omega to R, and it is the measure (the probability) that is a map from F to R

​@@martinsanchez-hw4fi Yes. And carefull, F and B(ℝ) have their own probability measure Like P for F and μₓ for B(ℝ). You can link both with : μₓ(B) = P(X⁻¹(B)) with B∈ F and X⁻¹(B) ∈ F (because X is measurable) Also careful a measure maps to [0,∞] and a probability measure maps to [0,1] (not ℝ)

I use Manim (from 3Blue1Brown). Probably the most common tool used for math videos on UA-cam 😉

I went to Dauphine for my undergrad. Do you know them?

@@stochastip somehow yes

Thanks! I will try to finish this series before the end of the year😅. After this, I thought about Lebesgue Integrals but SDE may also be an interesting topic.

Thank you very much 😃!

Same here 😂

@@superman39756 Nice 😁, if you don't mind me asking what university do you go to ?

And how are you applying stochastic processes within your dissertation in a practical sense ?

Yes! I already have some animations done, but I need to find time to finish and record. Part 2 will cover the Martingale and Gaussian Characteristic function. Then Brownian motion, Ito's integral, and more.😄

@@stochastip this is next level content i am excited to see other video