**Geometric Brownian Motion (GBM):**

Geometric Brownian Motion is a mathematical model commonly used to describe the stochastic movements of financial instruments, such as stock prices, over time. It is characterized by a continuous-time process where the logarithm of the variable follows a Brownian Motion (a random walk with normally distributed increments) with a drift and volatility. The formula for GBM is often expressed as:

[ dS_t = \mu S_t dt + \sigma S_t dW_t ]

where:

- (S_t) is the price of the financial instrument at time (t).
- (\mu) is the drift, representing the average rate of return.
- (\sigma) is the volatility, indicating the degree of randomness or variability.
- (dW_t) is a Brownian Motion, representing random increments.

### Applications to Mean Reversion Models:

**1. Ornstein-Uhlenbeck Process:**

The Ornstein-Uhlenbeck process is a mean-reverting extension of the GBM. It is used to model the behavior of a variable that tends to revert to a long-term mean over time. The process is governed by a mean-reversion parameter (\theta) and a speed of mean reversion parameter (\kappa). The equation for the Ornstein-Uhlenbeck process is:

[ dX_t = \kappa(\theta – X_t) dt + \sigma dW_t ]

where (X_t) is the variable being modeled.

**Applications:**

- Used in finance to model interest rates, where rates tend to revert to a long-term average.
- Applied in statistical physics to model the motion of particles under the influence of friction.

**2. Stochastic Volatility Models:**

GBM is also used in stochastic volatility models, where the volatility itself follows a stochastic process. In this case, the volatility term (\sigma) in the GBM equation becomes a function of time and possibly the current state.

**Applications:**

- Used in options pricing to account for changes in volatility over time.
- Applied in financial econometrics to model time-varying volatility in financial markets.

**3. Mean-Reverting Jump Diffusion:**

Mean-reverting jump diffusion models combine elements of mean reversion with jumps in asset prices. These models introduce additional jumps to the GBM process, representing sudden, discontinuous changes in the asset’s value.

**Applications:**

- Used in finance to capture both the mean-reverting behavior and occasional large price movements.
- Applied in modeling commodity prices, where jumps may be influenced by external events.

### Conclusion:

Geometric Brownian Motion is a versatile mathematical framework widely applied in finance and related fields. Its extension to mean reversion models, such as the Ornstein-Uhlenbeck process, allows for the modeling of processes that exhibit a tendency to revert to a long-term mean. These models are essential in understanding and predicting the dynamics of financial markets, interest rates, and other phenomena characterized by both random and deterministic components.