driftsfunksjon

Driftsfunksjon, or drift function, describes the deterministic trend or expected rate of change of a stochastic process. In many continuous-time models, it is denoted a(x,t) and specifies the instantaneous drift of the process X_t given its current state X_t = x.

In Itô diffusion models, the process X_t satisfies an equation of the form dX_t = a(X_t,t) dt + b(X_t,t) dW_t, where W_t is a Wiener process and b is the diffusion coefficient. The drift term a governs the average trajectory, while the diffusion term b introduces random fluctuations. In a time-homogeneous setting, the drift depends only on the state x.

The drift function influences the evolution of the probability density p(x,t) through the Fokker-Planck equation ∂p/∂t = -∂(a(x,t)p)/∂x + (1/2) ∂^2((b(x,t))^2 p)/∂x^2, linking the deterministic trend to the probabilistic description of the process.

In discrete-time models, the drift can be interpreted as the conditional mean increment: E[X_{t+Δ} − X_t | X_t = x] ≈ a(x,t) Δ for small Δ.

Examples help illustrate the concept. For the Ornstein-Uhlenbeck process, dX_t = -θ(X_t − μ) dt + σ dW_t, the drift is a(x) = -θ(x − μ). For geometric Brownian motion, dX_t = μ X_t dt + σ X_t dW_t, the drift is a(x) = μ x.

Driftsfunksjonen is central in fields such as finance, physics, biology, and engineering, where it models systematic trends within stochastic dynamics and is often estimated together with the diffusion term from data.