In finance, we often look at charts showing an asset's price at the close of each day. This gives us a series of snapshots, discrete points in time. But what happens between those points? The price doesn't just teleport from one day's close to the next. It moves continuously, influenced by a torrent of news, trades, and random events.

To model this reality, we shift from discrete data points to continuous-time stochastic processes This is a fancy term for a simple idea: a variable that evolves randomly over time. Think of it like a story where the plot is written moment by moment, with each new twist being unpredictable based on what came before. In finance, that story is the asset's price path.

The foundational model for randomness in continuous time is the Wiener Process, more commonly known as Brownian Motion. Originally used to describe the erratic movement of pollen grains suspended in water, it provides a powerful mathematical framework for modeling unpredictable paths.

A process $W_t$ is a standard Wiener Process if it satisfies three key properties:

1. It starts at zero: $W_0 = 0$.
2. Its paths are continuous. There are no sudden jumps.
3. Its increments are independent and normally distributed. The change over any time interval, $W_t - W_s$, is a random variable drawn from a normal distribution with a mean of 0 and a variance equal to the time elapsed, $t-s$. This means that future movements don't depend on past movements.

The independence of increments is what makes the process a "random walk." Knowing the entire history of the path up to today gives you no edge in predicting whether it will go up or down in the next second. This concept is central to the Efficient Market Hypothesis, which suggests that all available information is already reflected in an asset's price.

While standard Brownian Motion captures randomness, it has two problems for modeling stocks. First, it can become negative, but stock prices can't fall below zero (thanks to limited liability). Second, investors think in terms of percentage returns, not absolute dollar changes. A $1 move is huge for a $10 stock but trivial for a $1000 stock.

To fix this, we use Geometric Brownian Motion (GBM). Instead of the price itself following a random walk, GBM models the logarithm of the price as a random walk. This ensures the price stays positive and that changes are proportional to the current price level. It's the standard model for stock price behavior in quantitative finance.

The behavior of GBM is described by a Stochastic Differential Equation (SDE). An SDE is like a regular differential equation but with an added term to represent randomness.

How do we formally represent the flow of information in a market? We use a concept called a filtration A filtration, denoted $\mathcal{F}_t$, is a mathematical object that represents all the information available up to time $t$. As time moves forward, the filtration grows, because we accumulate more information. Think of it as an ever-expanding library of past price movements and news.

A process is said to be "adapted" to a filtration if its value at any time $t$ can be determined from the information available at time $t$. In simple terms, you can't know the future price of a stock today.

This leads us to the idea of a martingale. A stochastic process is a martingale if its expected future value, given all the information we have today, is simply its current value. It represents a fair game. If you're playing a game where your wealth follows a martingale process, your best guess for your wealth tomorrow is what you have today. Neither you nor the house has an edge.

In finance, this concept is deeply linked to the Efficient Market Hypothesis. If a discounted stock price were not a martingale, it would imply that its future expected value is predictably higher or lower than its current price. This would create an arbitrage opportunity, allowing traders to make risk-free profits. The theory suggests such opportunities are quickly eliminated, forcing the price process back into a martingale.

These building blocks—stochastic processes, Brownian motion, and martingales—form the language of modern quantitative finance. They allow us to move beyond simple historical analysis and build models that can price complex derivatives and manage risk in a world defined by uncertainty.