Stochastic Process, Filtration | Part 1 Stochastic Calculus for Quantitative Finance
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- Опубликовано: 21 авг 2024
- In this video, we will look at stochastic processes. We will cover the fundamental concepts and properties of stochastic processes, exploring topics such as filtration and adapted processes, which are crucial for option pricing and financial modeling.
Playlist of the course: • Stochastic Calculus fo...
Reference:
- Éléments de calcul stochastique pour l’évaluation et la couverture des actifs dérivés by Imen Ben Tahar, José Trashorras, and Gabriel Turinici.
- Stochastic Calculus for Finance II: Continuous-Time Models by Steven Shreve
Another banger video!
Finally someone who explains with examples
this channel is gonna pop
looking forward to the next
I'm currently doing my masters' thesis on Stochastic Processes, talk about perfect timing.
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 ?
👍
wtf this is amazing
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.
I looooooved the video. Excellent work, pal :))
Thank you very much 😃!
just don't stop uploading
Hey this is high quality work. How is it free?
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! 😅
@@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)
Great vid, do you think you can cover basic Stochastic Differential Equations?
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.
Nice vid! Is this a series?
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
What do you use to make your animations?
I use Manim (from 3Blue1Brown). Probably the most common tool used for math videos on RUclips 😉