# Integral of "white noise" multiplied by exponential term (multivariate Ornstein-Uhlenbeck)

Consider a matrix differential equation for the vector $$\mathbf{y}$$ of the form $$$$\dot{\mathbf{y}}(t) = A \, \mathbf{y}(t) + \mathbf{b}(t) \, ,$$$$ where $$A$$ is a constant square matrix and $$\mathbf{b}(t)$$ a given time-dependent vector of the same dimension of $$\mathbf{y}$$. Now, assume that $$\mathbf{b}(t)$$ is a "white noise" term such that $$\langle b_i(t) \rangle=0 \qquad \langle b_i(t) b_j(t') \rangle = 2 D_{ij} \, \delta(t-t')$$

where $$D$$ is a square, positive definite, "diffusion matrix". This process is a multivariate Ornstein–Uhlenbeck process. Is there some good reference for this kind of problem?

We can formally solve the problem by means of the standard matrix exponentiation, namely $$$${\mathbf{y}(t) } = e^{A t} \mathbf{y}(0) + e^{A t} \int_0^t e^{-A z} \mathbf{b}(z) \, dz$$$$ Clearly, if $$A=0$$, we just have the usual Wiener (aka "Brownian") process: $$$${\mathbf{y}(t) } = \mathbf{y}(0) + \int_0^t \mathbf{b}(z) \, dz$$$$ and the integral gives us a random variable at time $$t$$ that is Gaussian with a variance that grows linearly in time $$t$$. However, I do not know how to deal with the generic integral when $$A\neq 0$$. References/hints/ideas?

NOTE: I wrote the process in the "Langevin" form. It is immediate to interpret it in the Ito's way by saying that $$\mathbf{b}dt = d\mathbf{W}$$, where $$d\mathbf{W}$$ is the increment of the associated multivariate Wiener process (i.e. the "white noise" is the formal time derivative of the Wiener process).

Thanks to the comments I found many references: I added an answer to "close the case" and to list them for the interested ones. This tutorial on stochastic calculus ("Tutorial on Stochastic Differential Equations" by J. R. Movellan) has been proved to be very useful, clear and to the point.

• just to questions, what's $B_{ij}$ and the delta function is a dirac's delta? namely the variance of your white noise if infinite? Feb 16, 2021 at 17:30
• No it's the standard "physicist" notation, see e.g. en.wikipedia.org/wiki/Langevin_equation See e.g. the section "Itô calculus for physicists" here: en.wikipedia.org/wiki/It%C3%B4_calculus . In brief: $b$ is the "time derivative" of the standard Weiner process $W(t)$: $b(t)dt = dW(t)$ Feb 16, 2021 at 17:37
• It is a multivariate Ornstein-Uhlenbeck process. Feb 16, 2021 at 20:53
• And also you argument on the distribution doesn't change too much. In 1d case the Ito-Integral $\int_{0}^{t}e^{as}b(s)ds$ is zero-mean Gaussian with variance $\int_{0}^{t}e^{2as}ds$. In the multivariate case it's similar but tediuous. Feb 16, 2021 at 21:05
• I believe the multivariate ornstein uhlenbeck process will be discussed at length in Gardiner 1983 "Handbook of stochastic methods in the physical sciences". Jul 28, 2021 at 21:22