Geometric brownian motion in stochastic differential equations? - Answers

Geometric Brownian motion (GBM) is a mathematical model used to describe the evolution of asset prices in finance, characterized by its stochastic differential equation (SDE) of the form ( dS_t = \mu S_t dt + \sigma S_t dW_t ). Here, ( S_t ) represents the asset price, ( \mu ) is the drift term (representing the expected return), ( \sigma ) is the volatility, and ( dW_t ) is a Wiener process or Brownian motion. GBM captures the continuous compounding of returns and the random fluctuations in asset prices, making it a fundamental model for option pricing and risk management. The solution to this SDE leads to a log-normal distribution of prices, emphasizing the multiplicative nature of returns over time.