Stochastic Differential Equations Explained: Brownian Motion & Finance

Stochastic Differential Equation (SDE) is a class of differential equations that include random perturbations, used to describe the dynamic behavior of stochastic processes. Unlike ordinary differential equations (ODE), SDE models contain one or more random terms, typically Brownian motion (Wiener process). SDEs have wide applications in finance, physics, and mathematical biology.

In simple terms, a stochastic differential equation adds random disturbance to a traditional differential equation. In mathematical notation, a typical SDE can be written as:

dX_t = a(X_t, t) dt + b(X_t, t) dW_t

Here, X_t is the stochastic process of interest, a(X_t, t) is the deterministic part describing the system's trend, and b(X_t, t) is the stochastic part reflecting random fluctuations. W_t denotes Brownian motion, also known as a Wiener process, the classic tool for modeling random disturbances.

Brownian motion has important properties: it starts at zero, its increments are independent and normally distributed, and its paths are continuous but nowhere differentiable. These properties make Brownian motion an ideal choice for characterizing random phenomena.

A classic SDE model in finance is the geometric Brownian motion, used to describe the random evolution of stock prices. Its mathematical form is:

dS_t = μ S_t dt + σ S_t dW_t

In this model, μ is the drift rate representing the average growth of the stock price, and σ is the volatility reflecting the degree of price fluctuations. The model assumes that the logarithmic returns of the stock price follow a normal distribution, capturing the stochastic nature of price movements.

Assuming an initial stock price S_0 , drift μ , volatility σ , a simulation over one year with a time step of 0.01 year yields the following stock price trajectory:

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