**Geometric Brownian Motion (GBM):**

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**Approach to Geometric Brownian Motion:**

Geometric Brownian Motion is a continuous-time stochastic process that is widely used in finance to model the evolution of asset prices. It is characterized by constant drift and volatility. The equation for GBM is given by:

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

where:

- ( S_t ) is the asset price at time ( t ).
- ( \mu ) is the constant drift (expected return).
- ( \sigma ) is the constant volatility.
- ( dW_t ) is the Wiener process (Brownian motion), representing random shocks.

**Applications to Mean Reversion Models:**

**GBM and Mean Reversion:**

- While GBM is commonly associated with trends and exponential growth, it can be adapted for mean-reverting processes by introducing a mean-reverting component.

**Mean-Reverting Drift (( \mu )):**

- In the context of mean reversion, ( \mu ) can be adjusted to incorporate a mean-reverting force. This means that the drift term (( \mu S_t dt )) can be adjusted to pull the process back towards a long-term mean.

**Ornstein-Uhlenbeck Process:**

- The Ornstein-Uhlenbeck process is an example of a mean-reverting extension of GBM. It introduces a mean-reverting term to the GBM equation, resulting in a process that tends to revert to a specified mean over time.

**Applications in Finance:**

- Mean reversion models based on GBM are used in financial modeling to capture the tendency of financial assets to revert to a long-term average. This is valuable in pricing options, forecasting future prices, and risk management.

**Calibration to Empirical Data:**

- Traders and analysts often calibrate mean-reverting GBM models to historical data to estimate parameters such as the mean-reversion speed and long-term mean.

In summary, the GBM can be adapted to incorporate mean-reverting behavior by adjusting the drift term, providing a flexible framework for modeling financial processes with both trend-following and mean-reverting characteristics. The Ornstein-Uhlenbeck process is a specific example of this adaptation in the context of mean reversion.