Ornstein-Uhlenbeck Processes of Bounded Variation

Abstract

Ornstein-Uhlenbeck process of bounded variation is introduced as a solution of an analogue of the Langevin equation with an integrated telegraph process replacing a Brownian motion. There is an interval I such that the process starting from the internal point of I always remains within I. Starting outside, this process a. s. reaches this interval in a finite time. The distribution of the time for which the process falls into this interval is obtained explicitly. The certain formulae for the mean and the variance of this process are obtained on the basis of the joint distribution of the telegraph process and its integrated copy. Under Kac’s rescaling, the limit process is identified as the classical Ornstein-Uhlenbeck process.

Appendix: The Telegraph Process

Let $({\varOmega }, \mathcal F, \mathcal F_{t}, \mathbb {P})$ be the complete filtered probability space. Consider the adapted telegraph process $\mathbb {T}(t), t\geq 0,$ with two alternating symmetric velocities a and − a,a > 0, switching with positive intensities λ₀ and λ₁.

The joint distribution of $\mathbb {T}(t)$ and ε(t) can be expressed by means of the (generalised) density functions

$$ {p_{i}^{j}}(x, t):=\mathbb{P}\{\mathbb{T}(t)\in\mathrm{d} x, {\varepsilon}(t)=j~|~{\varepsilon}(0)=i\}/\mathrm{d} x,\qquad i, j\in\{0, 1\},\quad t>0. $$

The following formulae seem to be generally known, but for the best of my belief, they have never been published.

Theorem A.1

The density functions ${p_{i}^{j}}(x, t), i, j\in \{0, 1\},$ are given by

$$ \begin{aligned} {p_{0}^{0}}(x, t)&=\mathrm{e}^{-{\lambda}_{0}t}\delta(x-a_{0}t) +\frac{\sqrt{{\lambda}_{0}{\lambda}_{1}}}{a_{0}-a_{1}}\sqrt{\frac{\xi}{t-\xi}} \mathrm{e}^{-{\lambda}_{0}\xi-{\lambda}_{1}(t-\xi)}I_{1}(2\sqrt{{\lambda}_{0}{\lambda}_{1}\xi(t-\xi)})\mathbbm{1}_{\{0<\xi<t\}}, \\ {p_{1}^{1}}(x, t)&= \mathrm{e}^{-{\lambda}_{1}t}\delta(x-a_{1}t) +\frac{\sqrt{{\lambda}_{0}{\lambda}_{1}}}{a_{0}-a_{1}}\sqrt{\frac{t-\xi}{\xi}}\mathrm{e}^{-{\lambda}_{0}\xi-{\lambda}_{1}(t-\xi)}I_{1}(2\sqrt{{\lambda}_{0}{\lambda}_{1}\xi(t-\xi)})\mathbbm{1}_{\{0<\xi<t\}},\\ {p_{0}^{1}}(x, t)&=\frac{\lambda_{0}}{a_{0}-a_{1}}\mathrm{e}^{-\lambda_{0}\xi-\lambda_{1}(t-\xi)}I_{0}(2\sqrt{\lambda_{0}\lambda_{1}\xi(t-\xi)})\mathbbm{1}_{\{0<\xi<t\}},\\ {p_{1}^{0}}(x, t)&=\frac{{\lambda}_{1}}{a_{0}-a_{1}}\mathrm{e}^{-{\lambda}_{0}\xi-{\lambda}_{1}(t-\xi)}I_{0}(2\sqrt{{\lambda}_{0}{\lambda}_{1}\xi(t-\xi)})\mathbbm{1}_{\{0<\xi<t\}}, \end{aligned} $$

(A.1)

where ξ = (x − a₁t)/(a₀ − a₁),t − ξ = (a₀t − x)/(a₀ − a₁),a₁t < x < a₀t.

Here I₀ and I₁ are the modified Bessel functions,

$$ I_{0}(z)=1+\sum\limits_{n=1}^{\infty}\frac{(z/2)^{2n}}{(n!)^{2}},\qquad I_{1}(z)=I_{0}^{\prime}(z)=\sum\limits_{n=1}^{\infty}\frac{(z/2)^{2n-1}}{(n-1)!n!}. $$

Proof

Let N(t) be the number of velocity switchings in the time interval [0,t).

By virtue of Kolesnik and Ratanov (2013, (4.1.10)-(4.1.11)), ${p_{0}^{0}},$ can be represented as

$$ \begin{aligned} {p_{0}^{0}}(x, t) =&\sum\limits_{n=0}^{\infty}\mathbb{P}\{\mathbb{T}(t)\in\mathrm{d} x, N(t)=2n~|~{\varepsilon}(0)=0\}/\mathrm{d} x\\ =\mathrm{e}^{-{\lambda}_{0}t}\delta(x-a_{0}t)& +\frac{\exp(-{\lambda}_{0}\xi-{\lambda}_{1}(t-\xi))}{a_{0}-a_{1}} \sum\limits_{n=1}^{\infty}\frac{{{\lambda}_{0}^{n}}{{\lambda}_{1}^{n}}}{(n-1)!n!}\xi^{n}(t-\xi)^{n-1}\mathbbm{1}_{\{0<\xi<t\}}\\ =&\mathrm{e}^{-{\lambda}_{0}t}\delta(x-a_{0}t) +\frac{\sqrt{{\lambda}_{0}{\lambda}_{1}}}{a_{0}-a_{1}}\sqrt{\frac{\xi}{t-\xi}}I_{1}(2\sqrt{{\lambda}_{0}{\lambda}_{1}\xi(t-\xi)})\mathbbm{1}_{\{0<\xi<t\}}. \end{aligned}$$

see Gradshteyn and Ryzhik (1994, formula 8.445). The remaining equalities of Eq. A.1 are obtained in the same manner. □

The well-known formulae for the (conditional) distribution of $\mathbb {T}(t)$ follow from Eq. A.1:

$$ \begin{aligned} p_{0}(x, t)= \mathbb{P}\{\mathbb{T}(t)\in\mathrm{d} x~|~{\varepsilon}(0)=0\}/\mathrm{d} x &= {p_{0}^{0}}(x, t)+{p_{0}^{1}}(x, t), \\ p_{1}(x, t)= \mathbb{P}\{\mathbb{T}(t)\in\mathrm{d} x~|~{\varepsilon}(0)=1\}/\mathrm{d} x & = {p_{1}^{0}}(x, t)+{p_{1}^{1}}(x, t), \end{aligned} $$

cf Beghin et al. (2001) and López and Ratanov (2014) or see in the book by Kolesnik and Ratanov, (2013, (4.1.15)).

Similarly, $f(x, t)={p_{0}^{0}}(x, t)+{p_{1}^{0}}(x, t)$ (and $b(x, t)={p_{0}^{1}}(x, t)+{p_{1}^{1}}(x, t)$) are the distribution density functions of the moving forward (and backward) particles, cf Orsingher (1995), where these formulae were presented in the symmetric case, λ₀ = λ₁.

The rest of this section is devoted to a description of the first and the second moments of $\mathbb {T}(t)$ and $\mathbb {T}(t)\mathbbm {1}_{\{{\varepsilon }(t)=j\}}, j\in \{0, 1\}.$

We will use the following notations

$$ \mathbb{E}_{i}[g(\mathbb{T}(t))]=\mathbb{E}[g(\mathbb{T}(t))~|~{\varepsilon}(0)=i]={\int}_{-\infty}^{\infty} g(x)p_{i}(x, t)\mathrm{d} x $$

and

$$ {\mathbb{E}_{i}^{j}}[g(\mathbb{T}(t))]=\mathbb{E}[g(\mathbb{T}(t))\cdot\mathbbm{1}_{\{{\varepsilon}(t)=j\}}~|~{\varepsilon}(0)=i]={\int}_{-\infty}^{\infty} g(x){p_{i}^{j}}(x, t)\mathrm{d} x, \qquad i, j\in\{0, 1\}. $$

Theorem A.2

Let a₀ = −a₁ = a > 0.

For t ≥ 0

$$ \begin{array}{@{}rcl@{}} {\mathbb{E}_{0}^{0}}\mathbb{T}(t)&=&a\mathrm{e}^{-{\lambda}_{0}t}\sum\limits_{n=0}^{\infty}\frac{{{\lambda}_{0}^{n}}{{\lambda}_{1}^{n}}}{(2n)!}t^{2n+1}G_{n}^{(1)}(t), \end{array} $$

(A.2)

$$ \begin{array}{@{}rcl@{}} {\mathbb{E}_{0}^{1}}\mathbb{T}(t) & =&a\mathrm{e}^{-{\lambda}_{0}t}\sum\limits_{n=0}^{\infty}\frac{{\lambda}_{0}^{n+1}{{\lambda}_{1}^{n}}}{(2n+1)!}t^{2n+2}H_{n}^{(1)}(t), \end{array} $$

(A.3)

$$ \begin{array}{@{}rcl@{}} {\mathbb{E}_{1}^{0}}\mathbb{T}(t)& =&-a\mathrm{e}^{-{\lambda}_{1}t}\sum\limits_{n=0}^{\infty}\frac{{{\lambda}_{0}^{n}}{\lambda}_{1}^{n+1}}{(2n+1)!}t^{2n+2}H_{n}^{(1)}(-t), \end{array} $$

(A.4)

$$ \begin{array}{@{}rcl@{}} {\mathbb{E}_{1}^{1}}\mathbb{T}(t)&=&-a\mathrm{e}^{-{\lambda}_{1}t}\sum\limits_{n=0}^{\infty}\frac{{{\lambda}_{0}^{n}}{{\lambda}_{1}^{n}}}{(2n)!}t^{2n+1}G_{n}^{(1)}(-t), \end{array} $$

(A.5)

and

$$ \begin{array}{@{}rcl@{}} {\mathbb{E}_{0}^{0}}\mathbb{T}(t)^{2}&=&a^{2}\exp(-{\lambda}_{0}t) \sum\limits_{n=0}^{\infty}\frac{{{\lambda}_{0}^{n}}{{\lambda}_{1}^{n}}}{(2n)!}t^{2n+2}G_{n}^{(2)}(t), \end{array} $$

(A.6)

$$ \begin{array}{@{}rcl@{}} {\mathbb{E}_{0}^{1}} \mathbb{T}(t)^{2}&=&a^{2}\exp(-{\lambda}_{0}t) \sum\limits_{n=0}^{\infty}\frac{{\lambda}_{0}^{n+1}{{\lambda}_{1}^{n}}}{(2n+1)!}t^{2n+3}H_{n}^{(2)}(t), \end{array} $$

(A.7)

$$ \begin{array}{@{}rcl@{}} {\mathbb{E}_{1}^{0}}\mathbb{T}(t)^{2}&=&a^{2}\exp(-{\lambda}_{1}t) \sum\limits_{n=0}^{\infty}\frac{{{\lambda}_{0}^{n}}{\lambda}_{1}^{n+1}}{(2n+1)!}t^{2n+3}H_{n}^{(2)}(-t), \end{array} $$

(A.8)

$$ \begin{array}{@{}rcl@{}} {\mathbb{E}_{1}^{1}} \mathbb{T}(t)^{2}&=&a^{2}\exp(-{\lambda}_{1}t) \sum\limits_{n=0}^{\infty}\frac{{{\lambda}_{0}^{n}}{{\lambda}_{1}^{n}}}{(2n)!}t^{2n+2}G_{n}^{(2)}(-t), \end{array} $$

(A.9)

where

$$ G_{n}^{(1)}(t)=-\frac{2n}{2n+1}{\varPhi}(n+1, 2n+2; 2\upbeta t)+{\varPhi}(n, 2n+1; 2\upbeta t), $$

(A.10)

$$ H_{n}^{(1)}(t)=-{\varPhi}(n+2, 2n+3; 2\upbeta t)+{\varPhi}(n+1, 2n+2; 2\upbeta t) $$

(A.11)

and

$$ G_{n}^{(2)}(t)=\frac{2n}{2n+1}{\varPhi}(n+2, 2n+3; 2\upbeta t) -\frac{4n}{2n+1}{\varPhi}(n+1, 2n+2; 2\upbeta t)+{\varPhi}(n, 2n+1; 2\upbeta t), $$

(A.12)

$$ H_{n}^{(2)}(t)=\frac{2n+4}{2n+3}{\varPhi}(n+3, 2n+4; 2\upbeta t) -2{\varPhi}(n+2, 2n+3; 2\upbeta t)+{\varPhi}(n+1, 2n+2; 2\upbeta t), $$

(A.13)

2β = λ₀ − λ₁.

Here Φ(a,b; z) denotes the confluent hypergeometric function,

$$ {\varPhi}(\alpha, \upbeta; z):={\sum}_{n=0}^{\infty}\frac{(\alpha)_{n}}{(\upbeta)_{n}}\frac{z^{n}}{n!}, $$

(⋅)_n is the Pochhammer symbol; (γ)_n = γ(γ + 1)…(γ + n − 1),n ≥ 1, (γ)₀ = 1.

Proof

Consider

$$ \begin{aligned} \psi_{i}(z, t)=&\mathbb{E}_{i}\exp(z\mathbb{T}(t))=\mathbb{E}\left[\exp(z\mathbb{T}(t))~|~{\varepsilon}(0)=i\right],\\ \psi_{i}(z, t; n)=&\mathbb{E}_{i}\left[\exp(z\mathbb{T}(t))\mathbbm{1}_{\{N(t)=n\}}\right],\qquad n\geq0, \end{aligned} \qquad i\in\{0, 1\},$$

and

$$ {\psi_{i}^{j}}(z, t)={\mathbb{E}_{i}^{j}}\left[\exp(z\mathbb{T}(t))\right]=\mathbb{E}_{i}\left[\exp(z\mathbb{T}(t))\mathbbm{1}_{\{{\varepsilon}(t)=j\}}\right],\qquad i, j\in\{0, 1\}, $$

corresponding to the moment generating function of $\mathbb {T}(t).$ Notice that $\psi _{i}(z, t)=\sum \limits _{n=0}^{\infty } \psi _{i}(z, t; n)$ and

$$ {\psi_{i}^{i}}(z, t)=\sum\limits_{n=0}^{\infty} \psi_{i}(z, t; 2n),\qquad \psi_{i}^{1-i}(z, t)=\sum\limits_{n=0}^{\infty} \psi_{i}(z, t; 2n+1), \quad i\in\{0, 1\}. $$

The explicit expressions for ψ₀(z,t; n) and ψ₁(z,t; n) can be written separately for even and odd n,n ≥ 0. Due to López and Ratanov (2014, Theorem 2.1), we have

$$ \begin{array}{@{}rcl@{}} \psi_{0}(z, t; 2n) &=\frac{{{\lambda}_{0}^{n}}{{\lambda}_{1}^{n}}}{(2n)!} t^{2n}{\varPhi}\left( n, 2n+1; 2(\upbeta-az)t\right)\exp(-({\lambda}_{0}-az)t), \end{array} $$

(A.14)

$$ \begin{array}{@{}rcl@{}} \psi_{1}(z, t; 2n) &=\frac{{{\lambda}_{0}^{n}}{{\lambda}_{1}^{n}}}{(2n)!} t^{2n}{\varPhi}(n, 2n+1; 2(az-\upbeta)t)\exp(-({\lambda}_{1}+az)t), \end{array} $$

(A.15)

$$ \begin{array}{@{}rcl@{}} \psi_{0}(z, t; 2n+1) &=\frac{{\lambda}_{0}^{n+1}{{\lambda}_{1}^{n}}}{(2n+1)!} t^{2n+1}{\varPhi}\left( n+1, 2n + 2; 2(\upbeta - az)t\right)\exp(-({\lambda}_{0} - az)t), \end{array} $$

(A.16)

$$ \begin{array}{@{}rcl@{}} \psi_{1}(z, t; 2n+1) \!&=\frac{{{\lambda}_{0}^{n}}{\lambda}_{1}^{n+1}}{(2n+1)!} t^{2n+1} {\varPhi}(n + 1, 2n+2; 2(az - \upbeta)t)\exp(-({\lambda}_{1} + az)t). \end{array} $$

(A.17)

Formulae (A.14)–(A.17) directly give the desired result (A.2)–(A.8). For instance, by differentiating in Eq. A.14 we have

$$ {\mathbb{E}_{0}^{0}}[\mathbb{T}(t)]=\sum\limits_{n=0}^{\infty}\frac{{\partial}\psi_{0}(z, t; 2n)}{{\partial} z}|_{z=0} $$

$$ =a\mathrm{e}^{-{\lambda}_{0}t}\sum\limits_{n=0}^{\infty}\frac{{{\lambda}_{0}^{n}}{{\lambda}_{1}^{n}}}{(2n)!}t^{2n+1} \Big[ -2{\varPhi}^{\prime}(n, 2n+1; 2\upbeta t)+{\varPhi}(n, 2n+1; 2\upbeta t) \Big] $$

and

$$ {\mathbb{E}_{0}^{0}}[\mathbb{T}(t)^{2}]=\sum\limits_{n=0}^{\infty}\frac{{\partial}^{2}\psi_{0}(z, t; 2n)}{{\partial} z^{2}}|_{z=0} $$

$$ =a^{2}\mathrm{e}^{-{\lambda}_{0}t}\sum\limits_{n=0}^{\infty}\frac{{{\lambda}_{0}^{n}}{{\lambda}_{1}^{n}}}{(2n)!}t^{2n+2} \Big[ 4{\varPhi}^{\prime\prime}(n, 2n+1; 2\upbeta t)-4{\varPhi}^{\prime}(n, 2n+1; 2\upbeta t)+{\varPhi}(n, 2n+1; 2\upbeta t) \Big]. $$

The following known identities,

$$ {\varPhi}^{\prime}(\alpha, \upbeta; z)= \frac{\mathrm{d}{\varPhi}}{d z}(\alpha, \upbeta; z)=\frac{\alpha}{\upbeta}{\varPhi}(\alpha+1, \upbeta+1;z)$$

and

$${\varPhi}^{\prime\prime}(\alpha, \upbeta; z)=\frac{\alpha(\alpha+1)}{\upbeta(\upbeta+1)}{\varPhi}(\alpha+2, \upbeta+2;z),$$

see Gradshteyn and Ryzhik (1994, [formula 9.213]), give the result, Eqs. A.2, A.10 and A.6, A.12. Formulae A.3 and A.7 can be obtained similarly from Eq. A.16.

The remaining formulae of the theorem can be derived from Eqs. A.2–A.3 and A.6–A.7 by symmetry: Eq. A.5 follows from Eq. A.2; Eq. A.4 follows from Eq. A.3; Eq. A.9 follows from Eq. A.6; Eq. A.8 follows from Eq. A.7 after replacements $a\rightarrow -a$ and ${\lambda }_{0}\leftrightarrow {\lambda }_{1}.$ □

Equations A.2–A.9 permit us to evaluate the covariance between $\mathbb {T}(t)$ and $\mathbb {T}(s).$

Theorem A.3

For t > s > 0

$$ \begin{array}{@{}rcl@{}} \mathbb{E}_{0}\mathbb{T}(t)\mathbb{T}(s)=& \mathbb{E}_{0}\mathbb{T}(t-s){\cdot\mathbb{E}_{0}^{0}}\mathbb{T}(s)+\mathbb{E}_{1}\mathbb{T}(t-s){\cdot\mathbb{E}_{0}^{1}}\mathbb{T}(s)+\mathbb{E}_{0}[\mathbb{T}(s)^{2}], \end{array} $$

(A.18)

$$ \begin{array}{@{}rcl@{}} \mathbb{E}_{1}\mathbb{T}(t)\mathbb{T}(s)=& \mathbb{E}_{0}\mathbb{T}(t-s){\cdot\mathbb{E}_{1}^{0}}\mathbb{T}(s)+\mathbb{E}_{1}\mathbb{T}(t-s){\cdot\mathbb{E}_{1}^{1}}\mathbb{T}(s)+\mathbb{E}_{1}[\mathbb{T}(s)^{2}], \end{array} $$

(A.19)

where ${\mathbb {E}_{i}^{j}}[\mathbb {T}(s)], \mathbb {E}_{i}[\mathbb {T}(t-s)]$ and $\mathbb {E}_{i}[\mathbb {T}(s)^{2}], i, j\in \{0, 1\},$ are given by Eqs. A.2–A.9.

Proof

Notice that $\mathbb {E}_{0}\mathbb {T}(t)\mathbb {T}(s)=\mathbb {E}_{0}\left [(\mathbb {T}(t)-\mathbb {T}(s))\cdot \mathbb {T}(s)\right ]+\mathbb {E}_{0}[\mathbb {T}(s)^{2}].$

Due to persistence and time-homogeneity of the process $\mathbb {T},$ see Eq. 1.1,

$$ \begin{aligned} &\mathbb{E}_{0}\left[(\mathbb{T}(t)-\mathbb{T}(s))\cdot\mathbb{T}(s)\right]= \\ &\mathbb{E}[(\mathbb{T}(t)-\mathbb{T}(s)~|~{\varepsilon}(s)=0]{\cdot\mathbb{E}_{0}^{0}}\mathbb{T}(s) +\mathbb{E}[(\mathbb{T}(t)-\mathbb{T}(s)~|~{\varepsilon}(s)=1]{\cdot\mathbb{E}_{0}^{1}}\mathbb{T}(s) \\ &=\mathbb{E}_{0}\mathbb{T}(t-s){\cdot\mathbb{E}_{0}^{0}}\mathbb{T}(s)+\mathbb{E}_{1}\mathbb{T}(t-s){\cdot\mathbb{E}_{0}^{1}}\mathbb{T}(s), \end{aligned} $$

which gives Eq. A.18. Equation A.19 follows similarly. □

Remark 4

A.2 In the symmetric case λ₀ = λ₁ = λ > 0, the results of Theorem A.2 (Eqs. A.2–A.9) and Eqs. A.18–A.19 look much simpler, cf Kolesnik (2012).

Since β = 0 and Φ(⋅,⋅; 0) = 1, we have

$$ G_{n}^{(1)}(t)|_{\upbeta=0}=G_{n}^{(2)}(t)|_{\upbeta=0}\equiv\frac{1}{2n+1},\qquad H_{n}^{(1)}(t)|_{\upbeta=0}\equiv0,\qquad H_{n}^{(2)}(t)|_{\upbeta=0}\equiv\frac{1}{2n+3}. $$

Therefore, for the symmetric case, the first moments Eqs. A.2–A.4 are given by

$$ \begin{aligned} {\mathbb{E}_{0}^{0}}\mathbb{T}(t)&=- {\mathbb{E}_{1}^{1}}\mathbb{T}(t)=at\mathrm{e}^{-{\lambda} t}\sum\limits_{n=0}^{\infty}\frac{({\lambda} t)^{2n}}{(2n+1)!} =at\mathrm{e}^{-{\lambda} t}\frac{\sinh{{\lambda} t}}{{\lambda} t} =\frac{a}{2{\lambda}}\left( 1-\mathrm{e}^{-2{\lambda} t}\right), \\ {\mathbb{E}_{0}^{1}}\mathbb{T}(t)&={\mathbb{E}_{1}^{0}}\mathbb{T}(t)=0, \end{aligned} $$

(A.20)

and $\mathbb {E}_{0}\mathbb {T}(t)=-\mathbb {E}_{1}\mathbb {T}(t)=\frac {a}{2{\lambda }}\left (1-\mathrm {e}^{-2{\lambda } t}\right ).$

The second moments are given by

$$ \begin{aligned} {\mathbb{E}_{0}^{0}}\mathbb{T}(t)^{2} &={\mathbb{E}_{1}^{1}}\mathbb{T}(t)^{2}\\ &=(at)^{2}\mathrm{e}^{-{\lambda} t}\sum\limits_{n=0}^{\infty}\frac{({\lambda} t)^{2n}}{(2n+1)!}=(at)^{2}\mathrm{e}^{-{\lambda} t}\frac{\sinh{{\lambda} t}}{{\lambda} t} =\frac{a^{2}t}{2{\lambda}} \left( 1-\mathrm{e}^{-2{\lambda} t}\right), \\ {\mathbb{E}_{0}^{1}}\mathbb{T}(t)^{2} &={\mathbb{E}_{1}^{0}}\mathbb{T}(t)^{2}\\ &=(at)^{2}\mathrm{e}^{-{\lambda} t}\sum\limits_{n=0}^{\infty}\frac{({\lambda} t)^{2n+1}}{(2n+1)!(2n+3)} =(at)^{2}\mathrm{e}^{-{\lambda} t}\left( \frac{\sinh z-z}{z}\right)'|_{z={\lambda} t}\\ &=\frac{a^{2}t}{2{\lambda}}\left( 1+\mathrm{e}^{-2{\lambda} t}\right)-\frac{a^{2}}{2{\lambda}^{2}}\left( 1-\mathrm{e}^{-2{\lambda} t}\right), \end{aligned} $$

and, by summing we have

$$ \mathbb{E}_{0}\mathbb{T}(t)^{2}= {\mathbb{E}_{0}^{0}}\mathbb{T}(t)^{2}+ {\mathbb{E}_{0}^{1}}\mathbb{T}(t)^{2} =\frac{a^{2}}{2{\lambda}^{2}}\left( \mathrm{e}^{-2{\lambda} t}-1+2{\lambda} t\right)=\mathbb{E}_{1}\mathbb{T}(t)^{2}. $$

(A.21)

Equations A.20 and A.21 are consistent with known results, see e.g. Kolesnik and Ratanov (2013, (4.2.24)).

By Eqs. A.18–A.19, A.20 and A.21,

$$\begin{aligned} &\mathbb{E}_{0}[\mathbb{T}(t)\mathbb{T}(s)]=\mathbb{E}_{1}[\mathbb{T}(t)\mathbb{T}(s)]\\ &=\frac{a}{2{\lambda}}\left( 1-\mathrm{e}^{-2{\lambda}(t-s)}\right)\cdot\frac{a}{2{\lambda}}\left( 1-\mathrm{e}^{-2{\lambda} s}\right) +0+\frac{a^{2}}{2{\lambda}^{2}}\left( \mathrm{e}^{-2{\lambda} s}-1+2{\lambda} s\right) \end{aligned}$$

$$ =\frac{a^{2}}{4{\lambda}^{2}}\left[4{\lambda} s-(1+\mathrm{e}^{-2{\lambda}(t-s)})(1-\mathrm{e}^{-2{\lambda} s})\right], $$

(A.22)

and the covariance becomes

$$\begin{aligned} \text{cov}(\mathbb{T}(t), \mathbb{T}(s))=&\mathbb{E}[\mathbb{T}(t)\cdot\mathbb{T}(s)]-\mathbb{E}[\mathbb{T}(t)]\cdot\mathbb{E}[\mathbb{T}(s)]\\ =&\frac{a^{2}}{2{\lambda}^{2}} \left[2{\lambda} s-1+\mathrm{e}^{-2{\lambda} s}+\mathrm{e}^{-2{\lambda} t}-\frac12\left( \mathrm{e}^{-2{\lambda}(t-s)}+\mathrm{e}^{-2{\lambda}(t+s)}\right)\right]. \end{aligned}$$

It is known that under Kac’s scaling, $a, {\lambda }\to \infty , a^{2}/{\lambda }\to \sigma ^{2},$ see Kac (1974), Kolesnik and Ratanov (2013), and López and Ratanov (2012), the symmetric telegraph process $\mathbb {T}(t)$ converges to Brownian motion σW(t). Equations A.20, A.21 and A.22 consist with this convergence: under this scaling we have

• by Eq. A.20,

  $$ \lim\mathbb{E}_{0}[\mathbb{T}(t)]=\lim\mathbb{E}_{1}[\mathbb{T}(t)]=0; $$

• by Eq. A.21,

  $$ \lim\mathbb{E}_{0}[\mathbb{T}(t)^{2}]=\lim\mathbb{E}_{1}[\mathbb{T}(t)^{2}]=\sigma^{2}t, $$

• by Eq. A.22,

  $$ \lim\mathbb{E}_{0}[\mathbb{T}(t)\mathbb{T}(s)]=\lim\mathbb{E}_{1}[\mathbb{T}(t)\mathbb{T}(s)]=\sigma^{2}s,\qquad s\leq t. $$

Remark 2

Notice that the “general” telegraph process $\mathbb {T}(t), t\geq 0,$ with two alternating velocities a₀ and a₁,a₀ > a₁, can be reduced to the symmetric case:

$$ \mathbb{T}(t)\stackrel{D}{=}(a_{0}+a_{1})t/2+\mathbb{T}_{\pm a}^{\text{sym}}(t), $$

where $\mathbb {T}_{\pm a}^{\mathrm sym}(t)$ is the telegraph process with symmetric velocities ± a,a = (a₀ − a₁)/2. Therefore, without loss of generality, only a “symmetric” process $\mathbb {T}(t), t\geq 0,$ can be studied.