Stochastic Processes and Filtrations: A Practical Guide for Quants

Stochastic Processes Explained Simply

Think of a stochastic process as a sequence of random variables that evolves over time. Unlike a discrete sequence, a stochastic process is continuous and can take any positive real number as input. The output isn't limited to one dimension; it can be a vector in multiple dimensions.

• Simplified: Imagine tracking the price of a stock. The price changes continuously over time, and each price point is a random variable. This entire sequence of price movements can be modeled as a stochastic process.

• Technical Definition: A stochastic process takes an outcome from a sample space (equipped with a sigma algebra) as input and produces a multi-dimensional vector (equipped with a Borel Sigma algebra) as output. This ensures we can apply probability measures to the process.

Properties of Stochastic Processes

• Stationary Process: If you observe a process for a while and then restart it, the motion will statistically look the same. The past behavior doesn't influence the future.

  • Quant Example: Consider a mean-reverting trading strategy. If the market conditions remain the same, the strategy's performance should be statistically similar each time you deploy it.

• Stationary Increments: The difference between the process at time t + h and time t doesn't depend on t.

• Independent Increments: Changes in the process over non-overlapping time intervals are independent. If you take any sequence of times T1 to TN, the increments between these times are all independent of each other as long as the time intervals don't overlap.

  • Quant Example: The change in a stock price from Monday to Tuesday is independent of the change from Wednesday to Thursday.

Filtrations: Capturing the Flow of Information

A filtration represents the amount of information you have about a stochastic process at any given time.

• As the process evolves, you only know the past, not the future.

• Filtration involves taking a large Sigma algebra and constructing a family of smaller sub Sigma algebras from it.

• These sub-Sigma algebras are indexed, with each one contained within the next, creating a growing family of Sigma algebras.

• Simplified: Imagine flipping a coin three times.

  • At time zero, you have no information.

  • After the first flip, you know the outcome of that flip.

  • After the second flip, you know the outcomes of the first two flips.

  • A filtration formalizes this increasing knowledge.

• Technical Definition: A filtration is a family of sub-sigma algebras, each contained within the next, representing the evolution of information over time.

Adapted Process: Measuring Your Information

An adapted process is a way to mathematically measure the level of information you possess.

• If a process X is F adapted, it means that for any time T, XT is F measurable.

  • Real-World Example: If a stock price S is F measurable, it means you have all the information about the stock price at time T and everything that happened before.

Why Are These Concepts Important?

• Modeling: Stochastic processes are used to model a wide range of phenomena in finance, from stock prices to interest rates.

• Risk Management: Understanding the properties of stochastic processes and filtrations is crucial for managing risk.

• Pricing Derivatives: Many derivative pricing models rely on stochastic calculus.

Looking Ahead

This blog post provided an introduction to stochastic processes and filtrations. In future posts, we'll delve deeper into these concepts and explore their applications in quantitative finance.