Student OU-Type Processes

Abstract

The classical Ornstein-Uhlenbeck process $\{X_t,t\ge 0\}$, starting from $x\in R^d$, is a solution of linear equation

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The classical Ornstein–Uhlenbeck process $\{X_t,t\ge 0\}$, starting from $x\in R^d$, is a solution of linear equation.

$$\begin{aligned} X_t=x+B_t-c\int \limits ^{t}_{0}X_s\text{d}s, \quad t\ge 0, \end{aligned}$$

(5.1)

where $c>0$ and $\{B_t,t\ge 0\}$ is a standard d-dimensional Brownian motion. It is uniquely solved by

$$\begin{aligned} X_t=e^{-ct}x+\int \limits ^{t}_{0}e^{-c(t-s)}\text{d}B_s, \quad t\ge 0, \end{aligned}$$

where the last integral is a Wiener stochastic integral. We easily find that

$$\begin{aligned} \fancyscript{L}(X_t)=G_{e^{-ct}x,\frac{1}{2c}\left(1-e^{-2ct}\right)I_{d}} \Rightarrow G_{0,\frac{1}{2c}I_d}, \end{aligned}$$

as $t\rightarrow \infty$, where $I_d$ is an identity $d\times d$ matrix.

If we replace $\{B_t,t\ge 0\}$ in (5.1) by an arbitrary Lévy process $\{Z_t,t\ge 0\}$ with the triplet $(a,A,\Pi )$ of Lévy characteristics and the characteristic exponent $\varphi (z)=-\log Ee^{i\langle z,X_1\rangle }$, the solution

$$\begin{aligned} X_t=e^{-ct}x+\int \limits ^{t}_{0}e^{-c(t-s)}\text{d}Z_s, \quad t\ge 0 \end{aligned}$$

(5.2)

is called the starting from $x\in R^d$ Ornstein–Uhlenbeck type process generated by $(a,A,\Pi ,c)$.

The integral in (5.2) is defined analogously to the Wiener integral through converging in probability integral sums (see, e.g., [1]).

If we write

$$\begin{aligned} P_t(x,B)=\mathrm P \left\{ X_t\in B\right\} , \quad x\in R^d, \quad B\in \fancyscript{B}(R^d), \quad t\ge 0, \end{aligned}$$

it can be proved (see [2]) that

$$\begin{aligned} \int \limits _{R^d}e^{i\langle z,y\rangle }P_t(x,\text{d}y)=\exp \left\{ ie^{-ct}\langle x,z\rangle -\int \limits ^{t}_{0}\varphi (e^{-cs}z)\text{d}s\right\} , \quad x,z\in R^d, \quad t\ge 0, \end{aligned}$$

implying that $P_t(x,\cdot )$ is an infinitely divisible probability measure with the triplet $(a_{t,x},A_t,\Pi _t)$ of Lévy characteristics given by the formulas:

$$\begin{aligned} A_t=\int \limits ^{t}_{0}e^{-2cs}\text{d}sA, \quad t\ge 0 \end{aligned}$$

$$\begin{aligned} \Pi _t(B)=\int \limits _{R^d_0}\int \limits ^{t}_{0}1_B(e^{-cs}y)\text{d}s\Pi (\text{d}y), \quad B\in \fancyscript{B}(R^d_0), \quad t\ge 0, \end{aligned}$$

and

$$\begin{aligned} \begin{array}{ll} a_{t,x}&=e^{-ct}x+\int \limits ^{t}_{0}e^{-cs}\text{d}s\\&\quad +\int \limits _{R^d_0}\int ^{t}_{0}e^{-cs}y\left(1_{\{e^{-cs}|y|\le 1\}}-1_{\{|y|\le 1\}}\right)\text{d}s\Pi (\text{d}y), \quad t\ge 0, \quad x\in R^d. \end{array} \end{aligned}$$

Because

$$\begin{aligned}&\int \limits _{R^d}\int \limits _{R^d}e^{i\langle z,w\rangle }P_s(y,\text{d}w)P_t(x,\text{d}y)\\&\qquad =\int \limits _{R^d}\exp \left\{ i\langle y,e^{-cs}z\rangle -\int \limits ^{s}_{0}\varphi (e^{-cr}z)\text{d}r\right\} P_t(x,\text{d}y)\\&\qquad =\exp \left\{ i\langle x, e^{-c(t+s)}z\rangle -\int \limits ^{s}_{0}\varphi (e^{-c(r+s)}z)\text{d}r-\int \limits ^{t}_{0}\varphi (e^{-cr}z)\text{d}r\right\} \\&\qquad =\int \limits _{R^d}e^{i\langle z,w\rangle }P_{t+s}(x,\text{d}w), \end{aligned}$$

$P_t(x,B)$, $t\ge 0$, $x\in R^d$, $B\in \fancyscript{B}(R^d)$, satisfies the Chapman–Kolmogorov identity

$$\begin{aligned} \int \limits _{R^d}P_t(x,\text{d}y)P_s(y,B)=P_{t+s}(x,B) \end{aligned}$$

as the transition probability function of the time homogeneous Markov process $X$.

It is known (see [2–6]) that, as $t\rightarrow \infty$, for each $x\in R^d$

$$\begin{aligned} P_t(x,\cdot )\Rightarrow \tilde{\mu }_c \end{aligned}$$

if and only if

$$\begin{aligned} \int \limits _{\{|y|>2\}}\log |y|\Pi (\text{d}y)<\infty , \end{aligned}$$

(5.3)

where the limit distribution $\tilde{\mu }_c$ satisfies

$$\begin{aligned} \int \limits _{R^d}e^{i\langle z,y\rangle }\tilde{\mu }_c(\text{d}y)=\exp \left\{ -\int \limits ^{\infty }_{0}\varphi (e^{-cs}z)\text{d}s\right\} , \quad z\in R^d. \end{aligned}$$

(5.4)

The distribution $\tilde{\mu }_c$ is self-decomposable with the triplet of Lévy characteristics $(\tilde{a}_c,\tilde{A}_c,\tilde{\Pi }_c)$, where

$$\begin{aligned} \tilde{a}_c=\frac{1}{c}a+\frac{1}{c}\int \limits _{\{|y|>1\}}\frac{y}{|y|}\Pi (\text{d}y), \end{aligned}$$

$$\begin{aligned} \tilde{A}_c=\frac{1}{2c}A \end{aligned}$$

and

$$\begin{aligned} \tilde{\Pi }_c(B)=\frac{1}{c}\int \limits _{R^d}\int \limits ^{\infty }_{0}1_B\left(e^{-s}y\right)\text{d}s\Pi (\text{d}y), \quad B\in \fancyscript{B}(R^d_0). \end{aligned}$$

There is one-to-one continuous in the topology of weak convergence correspondence between the class $ID_{\log }(R^d)$ of infinitely divisible distributions, satisfying the integrability assumption (5.3), and the class of self-decomposable distributions $L(R^d)$. It is given by the mapping

$$\begin{aligned} ID_{\log }(R^d)\ni \mu =\fancyscript{L}(Z_1)\leftrightarrow \fancyscript{L}\left(\int \limits ^{\infty }_{0}e^{-t}\text{d}Z_t\right)=\tilde{\mu }\in L(R^d). \end{aligned}$$

(5.5)

The correspondence (5.5) imply that for the triplet $(\tilde{a},\tilde{A},\tilde{\Pi })$ of Lévy characteristics for $\tilde{\mu }$ the following equalities hold true:

$$\begin{aligned} \tilde{a}=a+\int \limits _{\{|y|>1\}}\frac{y}{|y|}\Pi (\text{d}y) \end{aligned}$$

$$\begin{aligned} \tilde{A}=\frac{1}{2}A \end{aligned}$$

and

$$\begin{aligned} \tilde{\Pi }(B)=\int \limits _{R^d}\int \limits ^{\infty }_{0}1_B(e^{-s}y)\text{d}s\Pi (\text{d}y), \quad B\in \fancyscript{B}(R^d_0). \end{aligned}$$

Vice versa, if

$$\begin{aligned} \tilde{\Pi }(B)=\int \limits _{S^{d-1}}\lambda (\text{d}\xi )\int \limits ^{\infty }_{0}1_B(r\xi )\frac{k_{\xi }(r)}{r}\text{d}r, \quad B\in \fancyscript{B}(R_0), \end{aligned}$$

then

$$\begin{aligned} a=\tilde{a}-\int \limits _{\{|y|>1\}}\frac{y}{|y|}\Pi (\text{d}y), \end{aligned}$$

$$\begin{aligned} A=2\tilde{A} \end{aligned}$$

(5.6)

and

$$\begin{aligned} \Pi (B)=-\int \limits _{S^{d-1}}\lambda (\text{d}\xi )\int \limits ^{\infty }_{0}1_B(r\xi )\text{d}k_{\xi }(r). \end{aligned}$$

The process $\{Z_t,t\ge 0\}$, is called the background driving Lévy process (BDLP for short).

FormalPara Definition 5.1

The subclass of the Ornstein–Uhlenbeck type processes, obtained by the correspondence (5.5) with the Student $t$-distribution $\tilde{\mu }$, is called the class of the Ornstein–Uhlenbeck type Student processes (Student OU-type processes for short).

FormalPara Definition 5.2

The subclass of the Ornstein–Uhlenbeck type processes, obtained by the correspondence (5.5) with the noncentral Student $t$-distributions $\tilde{\mu }$, satisfying self-decomposability condition (iii) of Proposition 4.6, is called the class of the noncentral Ornstein–Uhlenbeck type Student processes (noncentral Student OU-type processes for short).

We shall describe the BGDP, generating the Student OU-type processes.

FormalPara Proposition 5.3

• (i) The Student OU-type processes are generated by the BDLP${\rm Z}=\{{\rm Z}_t,t\ge 0\}$with the triplets of Lévy characteristics$(\gamma _0,0,\Pi _0)$, where

  $$\begin{aligned}&\gamma _0=\int \limits _{\{|x|\le 1\}}x\pi _0(x)\text{d}x+\alpha , \quad \alpha \in R^d,\\&\Pi _0(B)=\int \limits _{B}\pi _0(x)\text{d}x, \quad B\in \fancyscript{B}(R^d_0),\\&\pi _0(x)=-\frac{\text{d}}{\text{d}r}\left(r^dl_0(r\xi )\right)|_{r\xi =x} \end{aligned}$$

  and

  $$\begin{aligned} l_0(x)=\frac{\nu 2^{\frac{\text{d}}{4}+1}\left(\langle x\Sigma ^{-1},x\rangle \right)^{-\frac{\text{d}}{4}}}{\sqrt{|\Sigma |}(2\pi )^{\frac{\text{d}}{2}}}\int \limits ^{\infty }_{0}u^{\frac{\text{d}}{4}}K_{\frac{\text{d}}{2}}\left((2t\langle x\Sigma ^{-1},x\rangle )^{\frac{1}{2}}\right)g_{\frac{\nu }{2}}(2\nu t)\text{d}t, \end{aligned}$$

  $\nu >0$, $\Sigma$is a symmetric positive definite$d\times d$matrix.

• (ii) The Student OU-type process$X$, generated by the BDLP$\rm Z$with the triplet of Lévy characteristics$(\gamma _0,0,\Pi _0)$and$\mathcal L (X_0)=T_d(\nu ,\Sigma ,\alpha )$is strictly stationary Markov process.

FormalPara Proof

• (i) Follows directly from the Definition 5.1, the above stated properties of BGDP and the Proposition 4.5.

• (ii) It is well-known property of time homogeneous Markov processes.$\square$

FormalPara Proposition 5.4

• (i) The noncentral Student OU-type processes are generated by the BDLP${\rm Z}=\{{\rm Z}_t,t\ge 0\}$with the triples of Lévy characteristics$(\gamma _a,0,\Pi _a)$, where

  $$\begin{aligned}&\gamma _a=\int \limits _{\{|x|\le 1\}}x\pi _a(x)\text{d}x+\alpha , \quad \alpha ,a\in R^d, \\&\Pi _a(B)=\int \limits _{B}\pi _a(x)dx, \quad B\in \fancyscript{B}(R^d_0),\\&\pi _a(x)=-\frac{\text{d}}{\text{d}r}\left(r^dl_a(r\xi )\right)|_{r\xi =x} \end{aligned}$$

  and

  $$\begin{aligned} l_a(x)=&\frac{2\nu \exp \left\{ \langle a\Sigma ^{-1},x\rangle \right\} }{\sqrt{|\Sigma |}(2\pi )^{\frac{\text{d}}{2}}\left(\langle x\Sigma ^{-1},x\rangle \right)^{\frac{\text{d}}{4}}}\int \limits ^{\infty }_{0}\left(\langle a\Sigma ^{-1},a\rangle +2t\right)^{\frac{\text{d}}{4}}\\&\times K_{\frac{\text{d}}{2}}\left(\left((\langle a\Sigma ^{-1},a\rangle +2t)\langle x\Sigma ^{-1},x\rangle \right)^{\frac{1}{2}}\right)g_{\frac{\nu }{2}}(2\nu t). \end{aligned}$$

• (ii) The noncentral Student OU-type process$X^{(a)}$, generated by the BDLP$\rm Z$with the triplet of Lévy characteristics$(\gamma _a,0,\Pi _a)$and$\mathcal L (X_0)=T_d(\nu ,\Sigma ,\alpha ,a)$is strictly stationary Markov process.

FormalPara Proof

• (i) Follows directly from the Definition 5.2, the above stated properties of BDLP and the Proposition 4.5.

• (ii) It is well-known property of time homogeneous Markov processes.$\square$