Welcome to our exploration of Stochastic Differential Equations. An SDE is a differential equation that includes random terms, unlike ordinary differential equations which are deterministic. SDEs are essential for modeling systems where randomness plays a crucial role, such as stock prices or particle motion in fluids.
The general form of a stochastic differential equation is dX_t equals f of X_t, t times dt plus g of X_t, t times dW_t. Here, f represents the drift term, which describes the deterministic trend of the process. The g term is the diffusion coefficient, controlling the magnitude of random fluctuations. dW_t represents increments of a Wiener process, introducing the stochastic component.
The Wiener process, also known as Brownian motion, is the fundamental building block of stochastic calculus. It's a continuous-time stochastic process with several key properties: it starts at zero, has independent increments that are normally distributed, follows continuous but nowhere differentiable paths. Each sample path looks erratic and unpredictable, representing pure randomness in continuous time.
Itô calculus is the mathematical framework for working with stochastic integrals and SDEs. The key difference from classical calculus is Itô's formula, which includes an additional correction term. This arises because the quadratic variation of Brownian motion is non-zero: dW squared equals dt, not zero as in classical calculus. This correction term is essential for properly transforming stochastic processes.
SDEs have widespread applications across many fields. In finance, the Black-Scholes model uses geometric Brownian motion to model stock prices. In physics, they describe particle motion in random environments. Biology uses SDEs for population dynamics with environmental noise. Engineering applies them in signal processing and control theory. These applications demonstrate the power of SDEs in modeling real-world systems where randomness plays a crucial role.