Alternative way to derive the distribution of the multivariate Ornstein–Uhlenbeck process

Abstract

Abstract In this paper, we solve the Fokker–Planck equation of the multivariate Ornstein–Uhlenbeck process to obtain its probability density function. This approach allows us to ascertain the distribution without solving it analytically. We find that, at any moment in time, the process has a multivariate normal distribution. We obtain explicit formulae of mean, covariance, and cross-covariance matrix. Moreover, we obtain its mean-reverting condition and the long-term distribution.

Keywords

• Multivariate Ornstein–Uhlenbeck process
• Multivariate normal distribution
• Fokker–Planck equation
• n-dimensional Fourier transform