Probability Space Fundamentals
A stochastic process is built on a probability space with three components:
- Sample Space (Ω): All possible outcomes. Example: dice roll = {1, 2, 3, 4, 5, 6}
- Sigma Algebra (F): Collection of events (subsets of Ω). Example: "odd numbers" = {1, 3, 5}
- Probability Measure (P): Assigns probabilities to events in the sigma algebra
A solid understanding of Introduction to Probability and Statistics: Key Concepts and Terminology can provide a helpful foundation for grasping these fundamental building blocks.
What Is a Stochastic Process?
Think of a stochastic process as a continuous sequence where:
- Input: Time (t ≥ 0) + an outcome (ω) from the sample space
- Output: A vector X(t, ω) that can be 1D, 2D, or 3D
Key Construction
Input: (t, ω) → Stochastic Process → Output: X(t, ω) in R^d
- Time is continuous (takes all positive real numbers)
- Output uses the Borel sigma algebra to enable probability measurement
Four Essential Properties of Stochastic Processes
1. Stationary Process
If you shift the process horizontally by time h, the statistical properties remain identical. The motion pattern is consistent, past behavior doesn't influence future behavior.
2. Stationary Increments
For any fixed time t and small interval h: X(t + h) − X(t) depends only on h, not on t. The distribution of increments is time-invariant.
3. Independent Increments
For any sequence of times t₁ < t₂ < t₃ < t₄:
- Increment X(t₂) − X(t₁) is independent of increment X(t₄) − X(t₃)
- Only requires non-overlapping time intervals
- A change from t₁ to t₂ doesn't affect changes from t₃ to t₄
4. Measurability
A function f: E → F is measurable if for any subset A in F's sigma algebra, its inverse image f⁻¹(A) belongs to E's sigma algebra. Applied to stochastic processes, this ensures we can trace outcomes back to the original probability space.
Paths (Sample Functions)
By fixing a specific outcome ω, the stochastic process becomes:
- A function of time only: X(t)
- Shows exactly how the process evolves for that particular outcome
Filtration: Capturing Information Over Time
Filtration = A growing family of sigma algebras F₀ ⊂ F₁ ⊂ F₂ ⊂ ... where each contains more information than the previous.
Coin Flip Example (3 flips)
F₀ (no information):
- Only knows "something will happen"
- Contains empty set and full sample space
F₁ (after first flip):
- Two new events: H (first flip heads) and T (first flip tails)
- Knows outcome of first flip
F₂ (after two flips):
- Events for all first-two-flip combinations: HH, HT, TH, TT
- Knows outcomes of first two flips
F₃ (after three flips):
- Events for all sequences: HTH, TTH, etc.
- Complete information about all three flips
Filtered Probability Space
Combining the original probability space with filtration creates a filtered probability space that tracks information evolution.
The concept of an Introduction to Probability: Foundations for AI & Bayes' Nets is central to building the initial probability space from which these filtrations evolve.
Adapted Process
A process X is F-adapted if:
- For every time t, X(t) is F(t)-measurable
- You have complete information about X up to time t
Example: Stock price S(t) is F-adapted means:
- At time t, you know the stock price at t and all previous prices
- You can make decisions based on past information only
Illustration
Filtration: F₀ → → F₁ → → F₂ → → F₃
Information: None → Flip 1 → Flips 1&2 → All 3 flips
Process: S₀ → S₁ → S₂ → S₃
(adapted at each step)
Practical Applications
Stochastic processes with these properties are used in:
- Option pricing (Black-Scholes model)
- Particle movement (Brownian motion)
- Financial modeling (stock prices)
- Physics (diffusion processes)
For a broader view of how probability models uncertainty in these applications, the Probability & Statistics: The Ultimate Guide to Modeling Uncertainty (Course Intro) provides an excellent overview. Additionally, understanding Continuous Random Variables: PDF, CDF, and Key Concepts Explained is crucial for modeling the continuous time and space often used in these practical examples.
[Music] hi welcome on stochastic in this video we will explore stochastic processes
these are actual tools for understanding option pricing and movement of particles okay so before we start we need to
introduce the concept of a property space a property space include three main components the sample space Omega
the sigma algebra F and the protty measure P first let's look at the sample space this consist of all possible
outcomes a simple example is rolling a dice the sample space in this case include 1 2 3 4 5 and
six next we have this Sigma algebra the sigma algebra is a set containing subset of the sample space to say it more
simply it include events for example we could have the event getting an order number 1 3 or five or we could also have
getting an even number 2 4 6 the sigma algebra would include these events the empty set and the entire sub
space and finally we have the property measure this assigns prob ities to the events in the sample space okay so now
let's dive into the stochastic process think of it like sequence however instead of being discrete like U of zero
U of one U of two it is continuous and takes all positive real numbers as inputs and also the output is just not
one dimension it can be any type of Vector in D Dimensions let's look at this more
complex formulation on the left side you take as input an outcome from the sample space which is
equipped with a sigma algebra on the right side you can see the output which is a stochastic process this process is
a three dimensional Vector equipped with the B Sigma algebra we won't dive into all the complex details of the Bor Sigma
algebra but the main idea is to ensure that we can use the prity measure on the stochastic process now let's look at
some example first let's start with one dimension next let's move to two dimensions and finally let's move to
three dimensions also I would like to emphasize that while time is continuous
the stochastic process itself don't need to be I didn't show gems in these examples but they all possible scenarios
okay now we look at some properties that a stochastic process can have the first property a stochastic process can have
is the property of stationary process a simple illustration of this is to take a process running for some time and then
slide it to the left until H reach the axis if you restart the process you will see
that the motion is the same as before it's choic but in the same way not moving faster or slower essentially the
pattern it follows remain consistent demonstrating that past Behavior doesn't influence future
Behavior then let's look at the property of stationary increments and independent increments a stationary increment is
simpler it mean that if you fix any positive time t and take a small interval t + H above T the difference
between the process at t plus h and T doesn't depend on T independent increments on the other end are a bit
more General this property mean that if you take any sequence of times T1 to TN the increments between these times are
all independent of each other okay let's illustrate this with an example imagine we take four times T1 to T4 where each
times is earlier than the next now consider the difference of X between T2 and T1 and T4 and T3 this
differences shown in red are completely independent of each other as long as the time intervals don't overlap this means
that the change from T1 to T2 doesn't affect the change from T3 to T4 okay so this property of independent
increments is very simple instead of dealing with two inputs T and Omega we are fixing Omega the outcome of the
experiment saying that the input is just time T once we have done that we just have a
classic function where time T is the input and the vector X is the output this function shows how the process
evolve over time for that particular fixed outcome Omega the final property is a little bit
more complex so before talking about it let's first recap what a measurable function is if you have a function that
goes from set e to set F and both sets have their own Sigma algebras the function is measurable if for any
element in the sigma algebra of the output the inverse image of that element is in the sigma algebra of the input in
simple terms if you can take any subset from the output space and trace it back to the the function to the input space
the subset you get should be part of the input space Sigma algebra if you are a little bit confused just pause the video
and read carefully the property now applying this to a stochastic process we get this
property okay so now that we have covered these four important properties of stochastic process let's switch to a
different but crucial topic filtration filtration involves taking a large Sigma algebra which contains many
elements and constructing a family of smaller sub Sigma algebras from it these sub Sigma algebras are indexed and each
one is contained within the next so you essentially have a family of Sigma algebras that grow
larger well this may seem theoretical the key ID of filtration is that it represents the amount of information you
have about a stochastic process at any given time as stochastic process evolves you don't know the future only
the past to illustrate this let's use the example of flipping a coin three times we'll see how the information
evolve with each flip and of course this is very specific and discret example unlike the continued stochastic process
we discussed earlier but it's easier to understand let's say we flip account three times at the top you can see the
the actual sequence of flips head Tails heads at Time Zero you have no information about the future flips the
simple Sigma algebra at this point is F0 where you essentially have no information you only know that some
outcome would happen Okay next we can create a slightly more complex Sigma algebra F1 which contains two more
elements a H represent the event where the first flip is heads and a where the first lip is
tailed F1 is also a signal algebra you can verify this because the union ah and a t gives you the entire sample space
satisfying all the properties of a sigma algebra at this point in the experiment after the first flip you will see that
ah is the actual event occurring if the first slip is heads this mean you now have enough information about the
outcome of the first slip okay now let's move to the next step and introduce
F2 this one is more complex because it includes more elements to satisfy all the properties of a sigma algebra the
key elements in F2 are the four events representing the outcomes of the first two flips you can see them after the
empty set and the entire sample space the many additional elements ensure that F2 is a sigma algebra at this stage you
have enough information to know the outcome of the first two flips okay now let's move to F3 F3 has
many elements but we'll focus on the one that matter in F3 you have the necessary information to know the outcomes of all
three flips this includes events like HTH tth and so on representing every possible sequence of three flips while
there are many other elements this one ensure you have complete information about the three
flips okay once we have this family of growing Sigma algebras we can integrate it into the proerty space this give us
what is called a filtered proerty space now that we know what a filtration is we can introduce adapted process basically
an adapted process is like I explained it's a way of mathematically measuring the the level of information you have so
if you say that X is f adapted it means that for any T XT is f measurable for instance if you take a
stock s and you say that it is f measurable it just mean that you have all the information about the stock
price at T here you can see that in blue because s is f measurable you have the information about everything that
happened at T and before all right that's all for this video thank you for watching and see you
in the next one
The four essential properties are: 1) Stationary Process, where shifting the process horizontally by time h leaves statistical properties unchanged; 2) Stationary Increments, meaning the distribution of increments X(t+h) − X(t) depends only on the interval length h; 3) Independent Increments, where increments over non-overlapping time intervals are independent; 4) Measurability, ensuring that the inverse image of any measurable set in the output space belongs to the sigma algebra of the original probability space.
A filtration is a nested sequence of sigma algebras F₀ ⊂ F₁ ⊂ F₂ ⊂ ... that represent growing information. In the three-coin-flip example: F₀ contains only the empty set and full sample space (no information). F₁ includes events for the first flip being heads or tails. F₂ includes all first-two-flip combinations (HH, HT, TH, TT). F₃ includes all complete three-flip sequences (e.g., HTH), representing full knowledge of all outcomes.
A filtered probability space combines the original probability space (Ω, F, P) with a filtration. A process X is F-adapted if for every time t, the value X(t) is measurable with respect to F(t), meaning you have complete information about the process up to time t. This allows you to make decisions based on past information only, such as knowing a stock price S(t) and all its previous values at that moment.
A stationary process has statistical properties that remain unchanged when shifted horizontally by any time h; the pattern of motion is consistent over time. Stationary increments, in contrast, focus on changes: the distribution of X(t+h) − X(t) depends only on the interval length h and not on the starting time t, meaning increment behavior is time-invariant.
Measurability ensures stochastic process outputs can be traced back to the probability space. For a process mapping sample space to real numbers (using the Borel sigma algebra), measurability means that for any Borel set A, its inverse image belongs to the sigma algebra of the probability space. This guarantees you can assign probabilities to events involving the process's values.
A stochastic process takes two inputs: time t and outcome ω, producing output X(t, ω). When you fix a specific outcome ω, the process becomes a function of time only, X(t), called a sample path. This reveals the exact evolution the process takes for that particular outcome, showing how the trajectory unfolds moment by moment.
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