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Alternative way to derive the distribution of the multivariate Ornstein–Uhlenbeck process
Advances in Difference Equations volume 2019, Article number: 276 (2019)
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 crosscovariance matrix. Moreover, we obtain its meanreverting condition and the longterm distribution.
Introduction
For decades, stochastic processes have become more popular as a model for fluctuations over time. Including the noise term is the main advantage of the stochastic model. The Ornstein–Uhlenbeck process is one of the most wellknown stochastic processes used in many research areas such as mathematical finance [1], physics [2], and biology [3]. It was introduced by L. Ornstein and G. Eugene Uhlenbeck (1930). This process is defined as the solution of stochastic differential equation
where \(\theta \neq 0\), μ, and \(\sigma >0\) are constant parameters, and \(W(t)\) is the Wiener process. The parameter μ is the longterm mean, θ is the velocity, and σ is the friction coefficient. Its analytic solution, a function of mean, variance, and covariance functions over time t were derived. An important feature of this process (with positive θ) is the mean reversion, which means that it tends to its longterm mean μ as t tends to infinity. So, at any moment in time, if the value of the yield is greater than the longterm mean, then the drift becomes negative, so that the yield is pulled down in the direction of the longterm mean. Similarly, if the value of the yield is smaller than the longterm mean, then the drift becomes positive, so that the yield is pushed up to the longterm mean.
The multivariate Ornstein–Uhlenbeck process is a generalization to multiple dimensions of the Ornstein–Uhlenbeck process. It is defined as the solution of the multivariate stochastic differential equation
where θ is an \(n\times n\) invertible real matrix, μ is an ndimensional real vector, σ is an \(n\times m\) positive real matrix, and \(\boldsymbol{W}(t)\) is an mdimensional standard Wiener process. The idea of this generalization arises when we simultaneously deal with more than one quantity. The univariate Ornstein–Uhlenbeck process forces us to model \(X(t)\) independently, which is not a realistic assumption. It certainly does not work when all quantities are related in some sense. Consequently, many researchers apply this process to their interesting situations with limitations [4,5,6], and [7].
In most research in the past, this process was considered as a solution of stochastic differential equation (2). It was solved for its solution, and then its distribution, mean, covariance, and crosscovariance function matrix was computed. This research is different in that we derive its distribution and parameters without solving it analytically: we consider the probability density function as a solution of the Fokker–Plank equation.
Preliminaries
In this section, we introduce some wellknown definitions and results, which can be found in [8,9,10,11].
Proposition 1
Let \(\boldsymbol{X}(t)\) be a multivariate Itô process defined by the stochastic differential equation
where \(\boldsymbol{\mu }(\boldsymbol{X}(t),t)\) is an ndimensional vector, \(\boldsymbol{\sigma }(\boldsymbol{X}(t),t)\) is an \(n\times m\) matrix, and \(\boldsymbol{W}(t)\) is an mdimensional standard Wiener process. The probability density function \(p(\boldsymbol{x},t)\) of \(\boldsymbol{X}(t)\) satisfies the Fokker–Planck equation
where \(\boldsymbol{D}(\boldsymbol{x}(t),t)=\boldsymbol{\sigma }(\boldsymbol{X}(t),t)\boldsymbol{\sigma }^{T}( \boldsymbol{X}(t),t)\). This equation is also known as the Kolmogorov forward equation.
Let \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) be continuous. The ndimensional Fourier transform of f is the function \(\mathfrak{F}(f):\mathbb{R}^{n}\rightarrow \mathbb{R}\) defined by
where i is the imaginary unit.
Lemma 1
Let \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) be a continuously differentiable function such that \(\lim_{\\boldsymbol{x}\\rightarrow \infty }f(\boldsymbol{x})=0\). For any \(n\times n\) real matrix A and ndimensional real vector c, the following properties hold:

1
\(\mathfrak{F}(\frac{\partial }{\partial \boldsymbol{x}}\cdot \boldsymbol{c}f( \boldsymbol{x}))=i\boldsymbol{u}^{T}\boldsymbol{c}\mathfrak{F}(f)(\boldsymbol{u})\),

2
\(\mathfrak{F}(\frac{\partial }{\partial \boldsymbol{x}}\frac{\partial }{\partial \boldsymbol{x}}:\boldsymbol{A}f(\boldsymbol{x}))=\boldsymbol{u}^{T}\boldsymbol{A}\boldsymbol{u} \mathfrak{F}(f)(\boldsymbol{u})\),

3
\(\mathfrak{F}(\frac{\partial }{\partial \boldsymbol{x}}\cdot \boldsymbol{Ax}f( \boldsymbol{x}))=  (\frac{\partial \mathfrak{F}(f)(\boldsymbol{u})}{\partial \boldsymbol{u}} )^{T}\boldsymbol{A}^{T}\boldsymbol{u}\).
For any square matrix A, we define the exponential of A, denoted \(e^{\boldsymbol{A}}\), as \(\sum_{k=0}^{\infty }\frac{\boldsymbol{A} ^{k}}{k!}\), where \(\boldsymbol{A}^{0}\) is the identity matrix I. Note that this series always converges, so the exponential is welldefined.
Lemma 2
For any square matrix A, the following properties hold:

1
\(\boldsymbol{A}e^{\boldsymbol{A}}=e^{\boldsymbol{A}}\boldsymbol{A}\),

2
\((e^{\boldsymbol{A}})^{T}=e^{\boldsymbol{A}^{T}}\),

3
\(e^{\boldsymbol{A}}\) is invertible with \(e^{\boldsymbol{A}}\) as its inverse,

4
\(\frac{de^{\boldsymbol{A}t}}{dt}=\boldsymbol{A}e^{\boldsymbol{A}t}\), so if A is invertible, then \(\int e^{\boldsymbol{A}t}\,dt=\boldsymbol{A}^{1}e^{\boldsymbol{A}t}\).
Main results
Theorem 1
The characteristic function of the ndimensional Ornstein–Uhlenbeck process \(\boldsymbol{X}(t)\) satisfying (2) with the initial value \(\boldsymbol{X}(0)=\boldsymbol{x}_{0}\) is given by
Proof
The Fokker–Planck equation of (2) is given by
where \(\boldsymbol{D}=\boldsymbol{\sigma }\boldsymbol{\sigma }^{T}\), with initial condition
First, taking the ndimensional Fourier transform of equation (6), we get
where \(\hat{p}(\boldsymbol{u},t)\) is the ndimensional Fourier transform of \(p(\boldsymbol{x},t)\).
The initial condition (7) becomes
Note that equation (8) is a firstorder partial differential equation, so we will apply the method of characteristic.
Consider the system
with initial condition \(\boldsymbol{u}(0)=\boldsymbol{u}_{0}\). The solution of this system is
Consider the other equation
Substituting u from (10) into (11), we get
So
Then, substituting \(\hat{p}_{0}\) from (9) and \(\boldsymbol{u}_{0}\) by inverting (10) into (13), we get
Since the characteristic function is the Fourier transform with opposite sign in the complex exponential, we are done. □
Corollary 1
The ndimentianal Ornstein–Uhlenbeck process \(\boldsymbol{X}(t)\) satisfying (2) has an ndimensional normal distribution with mean vector
and covariance matrix
Moreover, the probability density function of \(\boldsymbol{X}(t)\) is given by
Proof
Comparing (5) with a characteristic function of multivariate normal distribution with mean M and covariance matrix Σ,
we obtain the result. □
Theorem 2
The crosscovariance function matrix of an ndimensional Ornstein– Uhlenbeck process \(\boldsymbol{X}(t)\) satisfying (2) is given by
Proof
Let \(\boldsymbol{\varGamma }(s,t)=\mathbb{E}[(\boldsymbol{X}(t)\boldsymbol{M}(t))(\boldsymbol{X}(s) \boldsymbol{M}(s))^{T}]\). From (15) we can see that \(\boldsymbol{M}'(t)= \boldsymbol{\theta }(\boldsymbol{M}(t)\mu )\). Then
where \(\boldsymbol{\xi }(t)\) is an ndimensional white noise, \(\boldsymbol{K}(s,t)= \mathbb{E}[(\boldsymbol{X}(s)\boldsymbol{M}(s))\boldsymbol{\xi }^{T}(t)]\), and \(\boldsymbol{L}(s,t)= \mathbb{E}[\boldsymbol{\xi }(s)(\boldsymbol{X}(t)\boldsymbol{M}(t))^{T}]\).
Taking the derivative of \(\boldsymbol{K}(s,t)\) with respect to t, we get
Since \(\mathbb{E}[\boldsymbol{\xi }(s)\boldsymbol{\xi }^{T}(t)]=\boldsymbol{\delta }^{m}(st)\) and \(\boldsymbol{K}(0,t)=0\), we get the solution
Similarly, we get
So, if \(t>s\), then
with initial condition \(C(0,t)=C(s,0)=0\). This equation has the solution
On the other hand, if \(s>t\), then we similarly obtain that
This completes the proof. □
From this result it follows that if we let \(s=t\), then the crosscovariance function matrix becomes the covariance matrix as in (16).
If the parameter θ of univariate Ornstein–Uhlenbeck process is positive, then the process is meanreverting. For the multivariate case, we also have a condition for meanreverting, which is stated in the following theorem.
Theorem 3
The ndimensional Ornstein–Uhlenbeck process \(\boldsymbol{X}(t)\) satisfying (2) is meanreverting if all eigenvalues of θ are positive.
Proof
Since \(e^{\boldsymbol{\theta } t}\) tends to the zero matrix as t tends to infinity if all eigenvalues of θ are positive, we can conclude from (15) that, with this condition, \(\boldsymbol{M}(t)\) tends to μ.
For \(\boldsymbol{\varSigma }(t)\), the situation is different, since we cannot take t in (16) to infinity directly as we do for \(\boldsymbol{M}(t)\). We apply the identity \(\operatorname{vec}(\boldsymbol{ABC})=(\boldsymbol{C}^{T}\otimes \boldsymbol{A}) \operatorname{vec}( \boldsymbol{B})\), where ⊗ is the Kronecker product defined in [12], and \(\operatorname{vec}(\boldsymbol{A})\) is defined as the column vector made of the columns of A stacked atop one another from left to right. Then
Now we use another identity \(e^{\boldsymbol{A}\otimes \boldsymbol{B}}=e^{\boldsymbol{A}} \oplus e^{\boldsymbol{B}}\) where ⊕ is the Kronecker sum. Then we obtain
Since all eigenvalues of \(\boldsymbol{\theta }\oplus \boldsymbol{\theta }\) are still positive, the covariance matrix converts to a constant matrix Σ such that \(\operatorname{vec}(\boldsymbol{\varSigma })= (\boldsymbol{\theta }\oplus \boldsymbol{\theta })^{1} \operatorname{vec}(\boldsymbol{\sigma }\boldsymbol{\sigma }^{T})\). □
Conclusion
In this paper, we propose a new method to derive the distribution of the multivariate Ornstein–Uhlenbeck process by solving its forward equation. We apply the characteristic method and Fourier transform to solve the equation. We obtain the characteristic function of the multivariate Ornstein–Uhlenbeck process and also its density function. Our explicit result shows that the multivariate Ornstein–Uhlenbeck process, at any time, is a multivariate normal random variable. We also derive the mean vector, covariance matrix, and cross covariance matrix and obtain its meanreverting condition, which is an extension of the univariate case. It is well known that the univariate Ornstein–Uhlenbeck process has a meanreverting property when the parameter θ is positive. In our study, for the multivariate case, we have found that the process is meanreverting as t increases and all eigenvalues of the matrix θ are positive.
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Both authors contributed equally to this work. All authors read and approved the final manuscript.
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Correspondence to N. Phewchean.
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MSC
 93E03
 60H10
Keywords
 Multivariate Ornstein–Uhlenbeck process
 Multivariate normal distribution
 Fokker–Planck equation
 ndimensional Fourier transform