Vasicek Process
Chapter 19: Stochastic Differential Equations — Stochastic Differential Equations
From the book
Chapter 19: Stochastic Differential Equations. In the chapter mind map this icon labels Stochastic Differential Equations. The discussion below is excerpted and lightly edited from § Stochastic Differential Equations in Mathematics for AI and Machine Learning.
- $\mathbf{X}_t \in \mathbb{R}^d$ is the state vector at time $t$
- $\mathbf{f}: \mathbb{R}^d \times \mathbb{R} \to \mathbb{R}^d$ is the drift function (deterministic component)
- $\mathbf{G}: \mathbb{R}^d \times \mathbb{R} \to \mathbb{R}^{d \times m}$ is the diffusion matrix (stochastic component)
- $\mathbf{W}_t \in \mathbb{R}^m$ is an $m$-dimensional Brownian motion
- $d\mathbf{W}_t$ is the infinitesimal increment of Brownian motion
What this drawing shows
What you see. Vasicek short-rate path oscillates around dashed long-run mean $\mu$ as red $r_t$ mean-reverts in time.
In the mind map. Chapter 19 — Stochastic Differential Equations. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 19: Stochastic Differential Equations — see the mind-map node Stochastic Differential Equations.