It is not clear to me when you way \omega \in (\Omega, \F), what is that tuple? Wouldn't the meassure (and the sigma algebra) be implied by the random variable?
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! 😅
@@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)
Hey! I hope you enjoyed this video. The really interesting part is coming up next with Brownian motion and Ito calculus. I have many ideas for animations and I’m super excited to share them with you. I will try to find time to work on it🥵 Btw, Thanks for the 100 subscribers! Like, subscribe, and stay tuned!😃
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.
Was on the lookout for one of these series for a while now, this is awesome. I'm still a newbie at quant math so the examples definitely helped me grasp some of these concepts.
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.😄
