A stochastic process is a mathematical representation of a collection of random variables indexed by either discrete or continuous time. The main characteristic of a stochastic process is that it describes the probabilistic behavior of a system evolving over time, where the future state of the system depends on its present state and possibly on the past states as well. More formally, a stochastic process is a collection {Xt : t ∈ T}, where Xt is a random variable and t belongs to an index set T, which can be either discrete (e.g., T = {0, 1, 2, ...} representing time in discrete steps) or continuous (e.g., T = [0, ∞) representing continuous time).
A well-known example of a stochastic process in finance is the Geometric Brownian Motion (GBM), which is used to model the dynamics of stock prices. Given a stock price St at time t, the continuous-time stochastic differential equation (SDE) describing the GBM is given by:
dSt = μStdt + σStdWt,
where:
- t ∈ [0, ∞) is the continuous time index,
- μ ∈ ℝ is the drift term representing the average rate of return on the stock,
- σ > 0 is the volatility term representing the uncertainty or risk associated with the stock’s return,
- Wt is a standard Brownian Motion (or Wiener process) representing a continuous-time stochastic process with independent and normally distributed increments.
In the context of finance, the GBM is often used in option pricing models, such as the Black-Scholes-Merton model, as it displays many characteristics of real-world stock prices, like their non-negativity and the fact that they exhibit continuous compounding of returns.