Stochastic processes model systems that evolve randomly over time. Think of the daily fluctuations of a stock price, the weather patterns in a city, or the sequence of words a large language model generates. A key challenge in modelling these systems is complexity. If you needed to know the entire history of a system to predict its future, the calculations would quickly become unmanageable.

This is where a simplifying assumption, known as the Markov Property, becomes incredibly useful. It states that for a certain class of processes, the future is conditionally independent of the past, given the present. In simpler terms, to predict where the system will go next, you only need to know where it is right now. All the information about its past journey is irrelevant.

A process that follows this rule is called a Markov chain. Before we formalise this, let's define two key terms.

The Markov property can be expressed precisely using the language of conditional probability. For a discrete-time stochastic process, where we look at the system at distinct steps ($t=0, 1, 2, ...$), the property is defined as follows.

This simplification is what gives Markov chains their power. Instead of tracking an ever-growing history of events, we only need the system's present configuration. This drastically reduces the computational load required for modelling.

Consider a customer subscription model. A customer's state might be 'Active', 'At Risk', or 'Churned'. The Markov assumption implies that the probability of a customer churning next month depends only on their current status ('At Risk'), not on whether they have been a loyal customer for five years or just two months. While this is a simplification, it provides a powerful and often accurate way to model customer behaviour without getting bogged down in individual histories.

Similarly, when a large language model (LLM) predicts the next word in a sentence, it's acting like a Markov process. The probability of the next word depends most strongly on the last few words (the current state), not the entire text that came before. This is why techniques like attention mechanisms, which allow the model to 'look back' further, were such a breakthrough, as they relax the strict memoryless assumption.

Of course, not all processes are memoryless. A system is history-dependent if its past states continue to influence its future, even when the present state is known. Think about a game of chess. Knowing the current positions of the pieces is essential, but a player's strategy is also shaped by the sequence of moves that led to that position. The history reveals the opponent's tactics and intentions.

Another example is a loan application. The bank doesn't just look at your current bank balance. It assesses your entire credit history—your past financial behaviour—to predict your future reliability. In these cases, a Markov chain would be an inappropriate model because the past carries significant predictive information that is not captured by the present state alone.

The core takeaway is that the Markov property is a modelling choice. It provides a powerful simplification for complex systems where the immediate present is the most crucial factor for predicting the near future. By ignoring the full weight of history, we can create models that are both tractable and remarkably effective.