Stochastic Processes and Filtrations

Imagine flipping a coin over and over. With each flip, you learn a little more about what is happening. This chapter turns that simple idea into precise math: stochastic processes model how randomness unfolds over time, and filtrations track what you know at each step. These ideas are the first step toward martingales, Brownian motion, and stochastic calculus.

The Big Picture#

A stochastic process is just a list of random variables ordered by time. The tricky part is: at any moment, what does an observer actually know? A filtration answers that. It is a growing family of events that become measurable as time goes on. Together, a process and a filtration let us check if the process respects the flow of time—meaning its future is still uncertain based on what we know now. That property, called adaptedness, is the quiet foundation for all later dynamic models.

Sequences of Random Variables: What Is a Stochastic Process?#

You already know a random variable is a number that depends on chance. A discrete-time stochastic process is just a list of random variables in time order. Picture a movie: frame 0 shows one random outcome, frame 1 shows another, frame 2 another, and so on.

More formally, we work on a probability space . is the set of all possible outcomes. is a sigma-algebra—a collection of subsets of that tells us which events we can measure. gives each event a probability. A discrete-time stochastic process is a family

where each is a random variable: a function that is -measurable. (Roughly, the set of outcomes where falls into an interval is an event in .) The index stands for time, usually . We often write for short.

Discrete-time stochastic process: A sequence of random variables all defined on the same probability space and indexed by non‑negative integer time steps.

You can also view the process as a function of two arguments: . Fix the outcome to see a sample path (the actual numbers that unfold), or fix the time to see the random variable at that instant.

Example 1 – Repeated fair coin tosses. Let be the set of all infinite sequences of heads and tails. Define

You can't use 'macro parameter character #' in math mode\begin{cases} 1 & \text{if the }(n+1)\text{th toss is H}, \\ 0 & \text{if it is T}. \end{cases}$$ $(X_n)$ is a stochastic process. $X_0$ is the first toss. **Example 2 – Symmetric random walk.** Start at $0$. At each step you add $+1$ with probability $1/2$ and $-1$ with probability $1/2$, independently. Let $S_0 = 0$ and $S_n = \sum_{k=1}^{n} Z_k$, where the $Z_k$ are the independent steps. $(S_n)_{n \ge 0}$ is a stochastic process. Its value at time $n$ depends on the first $n$ steps—not on any future ones. We always assume that all the random variables live on the same probability space, even if we don’t spell it out every time. This shared space is what lets us ask questions about how the variables relate to each other across time. > 📝 **Section Recap:** A discrete-time stochastic process is a time-ordered list of random variables on the same probability space. It captures randomness unfolding step by step. --- ## Information Over Time: The Idea of a Filtration Watching a process is like reading a diary—you only know what has been written up to the page you are on. A **filtration** makes this idea of growing information precise. Remember, a sigma-algebra $\mathcal{F}$ tells you which events are measurable—which questions you can answer with a probability. As time passes, you learn more, so your set of answerable questions grows. A **filtration** is an increasing family of sigma-algebras $$\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \mathcal{F}_2 \subseteq \cdots \subseteq \mathcal{F}$$ all inside the original sigma-algebra $\mathcal{F}$ of the probability space. We write $(\mathcal{F}_n)_{n\ge 0}$ and call it a filtration over discrete time. > **Filtration:** A nested sequence of sigma-algebras $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \mathcal{F}_2 \subseteq \cdots$ where each $\mathcal{F}_n$ represents the information known at time $n$. Intuitively, an event $A$ is in $\mathcal{F}_n$ if, by time $n$, you can say for sure whether the true outcome $\omega$ is in $A$ or not. As $n$ grows, $\mathcal{F}_n$ gets finer—you can answer more questions. **Example – coin tosses again.** Suppose you watch the first $n$ tosses of a fair coin. Before any toss ($n = 0$), you know nothing: the only events you can recognize are the empty set $\emptyset$ and the whole space $\Omega$. So $\mathcal{F}_0 = \{\emptyset, \Omega\}$ (the trivial sigma-algebra). After the first toss, you know whether it was H or T, so $\mathcal{F}_1$ contains all events that depend only on the first toss—like “first toss is H.” After two tosses, $\mathcal{F}_2$ contains events that depend on the first two tosses, like “both first two tosses are H,” and so on. Clearly $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \mathcal{F}_2 \subseteq \cdots$, giving a filtration. In general, a filtration can be any increasing family of sigma-algebras; it does not have to come from a specific process. But in practice, we almost always build it from the process we are studying. > 📝 **Section Recap:** A filtration is an expanding sequence of sigma-algebras that captures the growth of measurable information as time advances. Each $\mathcal{F}_n$ tells you which events you can already check with certainty at step $n$. --- ## The Filtration Generated by a Process The most natural way to build a filtration is to let the process itself define what is known. This is called the **natural filtration** of the process. Given a stochastic process $X = (X_n)_{n\ge 0}$, define $$\mathcal{F}_n^X = \sigma(X_0, X_1, \dots, X_n),$$ the smallest sigma-algebra that makes all the random variables $X_0, X_1, \dots, X_n$ measurable. In plain words, $\mathcal{F}_n^X$ contains exactly those events that can be described using $X_0$ through $X_n$—for example, “$X_0 \le 2$ and $X_1 > 0$” or “the maximum of $X_0,\dots,X_n$ exceeds $5$.” > **Natural filtration (or filtration generated by $X$):** The family $(\mathcal{F}_n^X)$ where $\mathcal{F}_n^X = \sigma(X_0, X_1, \dots, X_n)$ is the smallest sigma-algebra that makes all $X_0,\dots,X_n$ measurable. It represents exactly the information you get by observing $X$ up to time $n$. Because $X_{n+1}$ is not included in $\mathcal{F}_n^X$, future values stay genuinely random from the viewpoint of time $n$. This is key: the natural filtration captures the idea that you don’t peek at tomorrow’s price when deciding today. **Example – random walk’s natural filtration.** Recall the symmetric random walk $S_n = S_{n-1} + Z_n$ with $S_0=0$. At time $n$, you have observed $S_0, S_1, \dots, S_n$, which is the same as knowing the first $n$ steps $Z_1, \dots, Z_n$. So $\mathcal{F}_n^S$ is exactly $\sigma(Z_1, \dots, Z_n)$. The event $\{S_{n+1} = k\}$ is not in $\mathcal{F}_n^S$ because it depends on the unknown $Z_{n+1}$. We often drop the superscript and just write $\mathcal{F}_n$ when the process is clear. The natural filtration is the smallest filtration that makes $X$ **adapted**—our next topic. > 📝 **Section Recap:** The filtration generated by a process is the minimal one that carries all the information up to each time $n$ from the process itself. It keeps future values out of the current information set. --- ## Adaptedness: Processes That Respect the Flow of Information A process and a filtration are meant to work together. The property that makes this happen is **adaptedness**. A stochastic process $X = (X_n)_{n\ge 0}$ is **adapted** to a filtration $(\mathcal{F}_n)_{n\ge 0}$ if, for every $n$, the random variable $X_n$ is $\mathcal{F}_n$-measurable. > **Adapted process:** A process $(X_n)$ is adapted to the filtration $(\mathcal{F}_n)$ if, at each time $n$, the value $X_n$ is completely determined by the information in $\mathcal{F}_n$. In math language, $X_n$ is $\mathcal{F}_n$-measurable. What does $\mathcal{F}_n$-measurable mean? It means that any event like $\{X_n \in B\}$ (where $B$ is an interval, say) belongs to $\mathcal{F}_n$. So by time $n$, you can definitely check whether $X_n$ falls in $B$ because that event is already part of your knowledge $\mathcal{F}_n$. If $X_n$ were not $\mathcal{F}_n$-measurable, you would need future information to know its value—which breaks the idea of a real-time, cause‑and‑effect unfolding. Every process is automatically adapted to its own natural filtration: by construction, $\mathcal{F}_n^X$ contains all events that $X_0,\dots,X_n$ can generate, so $X_n$ is clearly $\mathcal{F}_n^X$-measurable. So the natural filtration is the default for a given process. But sometimes we work with a larger filtration—for example, one that also includes outside randomness or another process—and we then require $X$ to be adapted to that larger filtration, meaning $X$ must not use any information from that outside source ahead of time. **Example – adapted vs. non‑adapted.** Let $(\mathcal{F}_n)$ be the natural filtration of a coin toss process $X_0, X_1, X_2, \dots$. Define a new process $Y_n = X_{n+1}$. Is $Y$ adapted to $(\mathcal{F}_n)$? At time $n$, $Y_n$ equals the still‑unknown $(n+1)$st coin toss. That is not $\mathcal{F}_n$-measurable—it depends on the next toss, which is in the future relative to $n$. So $Y$ is **not** adapted to $(\mathcal{F}_n)$. In finance, this would be like a trading strategy that uses tomorrow’s stock price to decide today’s trade—impossible in a fair market. Adaptedness may seem like a small technical detail, but it is the gatekeeper for almost every important result that follows. For a process to be a martingale, it must be adapted; for a stochastic integral to make sense, the integrand must be adapted. In short, adaptedness makes sure the timeline is respected. **Think of a security camera recording a building.** The footage up to hour $n$ is $\mathcal{F}_n$. Your description $X_n$ of what is happening at hour $n$ must be based only on the footage you already have—not on what will be recorded later. That’s adaptedness. > 📝 **Section Recap:** A process is adapted to a filtration if, at every moment, its current value depends only on the information already available—no peeking into the future. Any process is automatically adapted to its own natural filtration. --- ## Summary We have taken the first step from static probability to the dynamic world where randomness changes over time. A stochastic process is a time-ordered list of random variables. Filtrations track what we know as time passes. The natural filtration, built from the process, makes it adapted. Adaptedness means no peeking into the future—each value depends only on past and present information. | Key idea | What it means (plain English) | Why it matters | |----------|-------------------------------|----------------| | **Discrete-time stochastic process** | A sequence of random variables $X_0, X_1, X_2, \dots$, all on the same probability space, indexed by time steps. | It gives us a mathematical model for anything that evolves randomly over discrete time—stock prices, queue lengths, weather states, etc. | | **Probability space** $(\Omega, \mathcal{F}, P)$ | The triple containing the set of outcomes $\Omega$, a sigma-algebra $\mathcal{F}$ of measurable events, and a probability measure $P$ that gives each event a number between $0$ and $1$. | It is the shared stage on which all random variables and processes

The Big Picture#

Sequences of Random Variables: What Is a Stochastic Process?#

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