In financial modelling, information isn't revealed all at once. It arrives over time, piece by piece. To model this mathematically, we use a concept called a filtration. Imagine you're watching a football match. At any given moment, you know everything that's happened up to that point: the score, the fouls, the substitutions. You don't know what will happen next. A filtration, denoted as $\mathcal{F}_t$, represents this accumulating body of knowledge at time $t$. It's a sequence of information sets, where each set contains all the information of the previous ones, plus any new information revealed in the interim.

A process is said to be 'adapted' to a filtration if its value at any time $t$ can be determined solely by the information available at that time, $\mathcal{F}_t$. A stock price is an adapted process because its current value is known today; you don't need to see into the future to know its price right now. This concept is crucial because it ensures our models don't cheat by using information that isn't yet available.

Now, let's introduce one of the most important concepts in quantitative finance: the martingale. A martingale is a stochastic process that models a 'fair game.' In simple terms, if a process is a martingale, your best prediction for its future value is simply its current value, regardless of its past behaviour. It has no predictable drift upwards or downwards.

This property is the mathematical expression of an efficient, arbitrage-free market. If the discounted price of an asset were not a martingale, it would have a predictable drift. For instance, if $E[X_t | \mathcal{F}_s] > X_s$, it would mean we expect the price to rise on average. This would present a clear arbitrage opportunity: buy the asset and hold it for a guaranteed profit, risk-free. In a real market, such opportunities are quickly eliminated. The goal of risk-neutral pricing, which we'll explore later, is to find a probability measure under which all discounted asset prices become martingales.

Of course, not all processes are fair games. Some have a built-in tendency to drift in one direction. A process that is expected to increase or stay the same is called a submartingale, while one expected to decrease or stay the same is a supermartingale. A powerful result known as the Doob-Meyer decomposition allows us to break down any submartingale into more understandable parts.

The theorem states that any submartingale $X_t$ can be uniquely split into the sum of a martingale, $M_t$, and an increasing, predictable process, $A_t$. Think of $M_t$ as the 'fair game' or random noise component, and $A_t$ as the underlying, non-random drift or trend. This decomposition is fundamental because it allows us to isolate and analyse the predictable part of a process from its random fluctuations. In finance, this helps in identifying the 'risk premium' or expected return that compensates an investor for bearing risk.

In trading and investment, deciding when to act is as important as deciding what to do. The concept of a stopping time formalises this decision point. A stopping time is a random variable representing a time at which a decision is made, with the crucial condition that the decision to stop must be based only on information available up to that time, not on future events. For example, 'Sell the stock the first time its price reaches £100' defines a stopping time. You don't need to know the future to execute this rule. In contrast, 'Sell the stock the day before it hits its peak' is not a valid stopping time, because you can't know it's the peak until after it has passed.

These concepts—filtrations, martingales, decompositions, and stopping times—form the theoretical bedrock upon which modern option pricing is built. They provide the language to describe information flow and arbitrage-free conditions, which are essential for deriving the Black-Scholes-Merton equation.