Define the term "stochastic process." Can you give an example of a stochastic process in the context of finance?

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 {X_t : t ∈ T}, where X_t 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 S_t at time t, the continuous-time stochastic differential equation (SDE) describing the GBM is given by:

dS_t = μS_tdt + σS_tdW_t,

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,

- W_t 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.