OORNSTEIN-UHLENBECK PROCESSES OF BOUNDED VARIATION
NIKITA RATANOVA
BSTRACT . Ornstein-Uhlenbeck process of bounded variation is introduced as a solutionof an analogue of the Langevin equation with an integrated telegraph process replacing aBrownian motion. There is an interval I such that the process starting from the internal pointof I always remains within I . Starting outside, this process a. s. reaches this interval ina finite time. The distribution of the time for which the process falls into this interval isobtained explicitly.The certain formulae for the mean and the variance of this process are obtained on thebasis 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-Uhlenbeckprocess. Keywords:
Ornstein-Uhlenbeck process; Langevin equation; telegraph process; Kac’sscaling
1. I
NTRODUCTION
For a long time by various reasons, different finite-velocity diffusion models have beenstudied as a substitute for classical diffusion, which is described by a parabolic equationwith infinitely fast propagation. The main model represents motions performed by a particlemoving along a line at a finite velocity and changing directions after exponentially distributedholding times, see [4]. The corresponding random process of particle’s position is called anintegrated telegraph process . The distribution of this process is described by the dampedwave equation (hyperbolic diffusion equation, the so-called telegraph equation).The one-dimensional version of the telegraph process T ( t ) , t ≥ , with two alternatingregimes is well studied, starting with the seminal works of M.Kac, see [10]. This theory hasa huge literature, see, for example, surveys in [12, 23].To introduce the telegraph process, we consider a two-state Markov process ε = ε ( t ) ∈{ , } , defined on the complete filtered probability space ( Ω , F , F t , P ) . Process ε is deter-mined by two positive switching parameters λ , λ : P { ε ( t + d t ) = i | ε ( t ) = i } = − λ i d t + o ( d t ) , d t → , i ∈ { , } . We define the (integrated) telegraph process by T ( t ) = (cid:90) t a ε ( s ) d s , where a , a are constants; T ( t ) is the position of a particle moving in a line with velocities a and a alternating at random times. Since ε is the time-homogeneous Markov process, Published in
Methodology and Computing in Applied Probability , (2020) DOI 10.1007/s11009-020-09794-x. a r X i v : . [ m a t h . P R ] J u l NIKITA RATANOV the (conditional) distribution of T ( t ) − T ( s ) = (cid:82) ts a ε ( s (cid:48) ) d s (cid:48) and T ( t − s ) = (cid:82) t − s a ε ( s (cid:48) ) d s (cid:48) areidentical for any s , t , ≤ s < t , precisely, the following identity in law holds:(1.1) (cid:20) T ( t ) − T ( s ) = (cid:90) ts a ε ( s (cid:48) ) d s (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) F s (cid:21) D = (cid:20) T ( t − s ) = (cid:90) t − s a ε ( s (cid:48) ) d s (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ε ( ) (cid:21) , see e.g. [12].The Gaussian Ornstein-Uhlenbeck process X OU is another class of processes we are inter-ested in. This process can be defined as the solution to the stochastic differential equation(1.2) d X OU ( t ) = − γ X OU ( t ) d t + σ d W ( t ) , t > , where W = W ( t ) is the standard Brownian motion and γ > . This model is used in variousapplication areas as an alternative to Brownian motion with an average tendency to return,see [5, 15]. Let me mention here only two of these application areas. The Vaˇs´ıˇcek interestrate model, [21], is the most famous financial application of this process. The same processesare also widely used for neuronal modelling, see e. g. [3]. Similar application areas havetelegraph processes: for financial applications see e. g. [6, 12, 17]; the first steps in theneuronal modelling based on the telegraph process are presented by [19] and [8, Section2.3.2].In this paper, we study the Ornstein-Uhlenbeck process of bounded variation, which is de-termined by the version of Langevin equation (1.2) when the Brownian motion W is replacedby telegraph process T . More precisely, let X = X ( t ) be a stochastic process defined by theequation d X ( t ) = − γ ε ( t ) X ( t ) d t + d T ( t ) , t > , where T ( t ) is the telegraph process based on the Markov process ε . Since Kac’s telegraphprocess T is used instead of the Wiener process in the usual Langevin equation, [5], thisequation can be called the Kac-Langevin equation. The latter stochastic equation is equiva-lent to an integral equation of the following form,(1.3) X ( t ) = x − (cid:90) t γ ε ( s ) X ( s ) d s + T ( t ) , t > . Here x = X ( ) is the starting point of the process X . To the best of my knowledge, such amodification of the Langevin equation has not been studied before.The detailed problem settings are presented by Section 2. Not surprisingly, the analysis ofthe distribution of X ( t ) is not as simple as for the Gaussian process X OU = X OU ( t ) . The firstpeculiarity is the following.If the starting point x is in the interval I = ( a / γ , a / γ ) , x ∈ I , the process X ( t ) remainsinside the band, that is X ( t ) ∈ ( a / γ , a / γ ) , t ≥ . In contrary, if the process X starts fromoutside of I , x / ∈ [ a / γ , a / γ ] , the process reaches I a. s. in finite time (here, we assumethat a / γ < a / γ ).The main goal of this paper (see Section 3) is to study the distribution of time over whichthe process X = X ( t ) , starting from the outside of interval I , falls into I . This problem isassociated with the first passage time of the telegraph process, which has been intensivelystudied recently, see, e. g. [7, 18, 19, 22, 23].
RNSTEIN-UHLENBECK PROCESSES OF BOUNDED VARIATION 3
To analyse Ornstein-Uhlenbeck process of bounded variation, (1.3), we need to study theproperties of the stochastic integral(1.4) I ( t ) = (cid:90) t ϕ ( s ) d T ( s ) = (cid:90) t ϕ ( s ) a ε ( s ) d s , t > . Since ϕ ( · ) a ε ( · ) a. s. has a finite number of discontinuities on [ , t ] , the integral in (1.4)can be considered as a pathwise Riemann integral. The stochastic process I ( t ) , t > , canbe considered as a generalised telegraph process with two time-varying velocity patterns, a ϕ ( t ) and a ϕ ( t ) , alternating after exponentially distributed holding times. The rectifiableversion of such a process has been studied in detail by [20], but I = I ( t ) , defined by (1.4),does not belong to this class.The distribution of X ( t ) looks much more sophisticated than the Gaussian distribution ofthe Ornstein-Uhlenbeck process X OU ( t ) . Sections 4 and 5 take only a few simple first stepsfor this analysis.For completeness, in the Appendix we recall some modern results on telegraph processes,including explicit formulas for the joint distribution of X ( t ) and ε ( t ) , which have never beenpublished before. 2. T HE PROBLEM SETTING
We study the path-continuous random process X = X ( t ) , t ≥ , satisfying the stochasticequation (1.3), that is(2.1) d X ( t ) = (cid:0) a ε ( t ) − γ ε ( t ) X ( t ) (cid:1) d t , t > , with the initial condition X ( ) = x . Recall, that ε = ε ( t ) is the two-state Markov process, a , a ∈ ( − ∞ , ∞ ) and γ , γ > , a / γ > a / γ . After applying the usual integrating factor technique, one can see that (2.1) is equivalentto d (cid:16) e Γ ( t ) X ( t ) (cid:17) = e Γ ( t ) a ε ( t ) d t , where Γ ( t ) = (cid:82) t γ ε ( s ) d s is the integrated telegraph process based on the same underlyingprocess ε as the telegraph process T ( t ) = (cid:82) t a ε ( s ) d s . This yields the formula for the solutionof (2.1):(2.2) X ( t ) = e − Γ ( t ) (cid:18) x + (cid:90) t e Γ ( s ) a ε ( s ) d s (cid:19) = e − Γ ( t ) (cid:18) x + (cid:90) t e Γ ( s ) d T ( s ) (cid:19) , t ≥ . The process X = X ( t ) can be considered as a piecewise deterministic path-continuousprocess of bounded variation which follows the two patterns, φ ( x , t ) = e − γ t (cid:18) x + a (cid:90) t e γ s d s (cid:19) = a γ + (cid:18) x − a γ (cid:19) e − γ t , (2.3) φ ( x , t ) = a γ + (cid:18) x − a γ (cid:19) e − γ t , t ≥ , (2.4)alternating at Poisson times. Similarly defined generalisation of the telegraph process wererecently studied by [20]; however, here the process X does not satisfy the homogeneityproperty with a common rectifying mapping, see [20, (2.13)], which creates new difficulties. NIKITA RATANOV
Let τ = τ ( ) ( τ = τ ( ) ) be the first switching time if ε ( ) = ε ( ) = [ X ( t ) | ε ( ) = , X ( ) = x ] D = [ X ( t − τ ) | ε ( ) = , X ( ) = φ ( x , τ )] , (2.5) [ X ( t ) | ε ( ) = , X ( ) = x ] D = [ X ( t − τ ) | ε ( ) = , X ( ) = φ ( x , τ )] . (2.6)If there are no switching to the time horizon t , that is τ > t , we have [ X ( t ) | ε ( ) = , X ( ) = x ] = φ ( x , t ) , [ X ( t ) | ε ( ) = , X ( ) = x ] = φ ( x , t ) , a.s.Note that the mappings t → φ ( x , t ) and t → φ ( x , t ) satisfy semigroup property,lim t → ∞ φ ( x , t ) = a γ , lim t → ∞ φ ( x , t ) = a γ . A sample path is shown in Fig. 1. τ τ τ τ tXxa / γ a / γ T ( x ) F IGURE
1. The sample path of X = X ( t ) . It follows from the definition that if the starting point x = X ( ) is in the interval, x ∈ I =( a / γ , a / γ ) , then(2.7) a / γ < φ ( x , t ) , φ ( x , t ) < a / γ , ∀ t ≥ . Further, if the starting point x = X ( ) is outside the interval I , the trajectory X = X ( t ) a. s.falls into I and remains there for all subsequent t . The distribution of this falling time is studied in the next section.3. T
HE FALLING TIME INTO THE INTERVAL I = ( a / γ , a / γ ) . Let x > a / γ , and T ( x ) be the time of first passage through the level a / γ by the process X ( t ) , which starts at point x , x > a / γ , (3.1) T ( x ) : = inf { t : X ( t ) < a / γ | X ( ) = x > a / γ } , see Fig.1.We denote by t ∗ ( x ) the shortest time for crossing the level a / γ by the process X = X ( t ) ,which starts at x , x > a / γ . This corresponds to movement only along the pattern φ ( x , t ) without switching. Thus, t ∗ ( x ) is determined by the formula(3.2) t ∗ ( x ) = γ log x − a / γ a / γ − a / γ > , RNSTEIN-UHLENBECK PROCESSES OF BOUNDED VARIATION 5 which is the root of the equation, φ ( x , t ) = a / γ , (2.4). A motion only with the pattern φ ( x , t ) , x > a / γ , (2.3), (without switching) never crosses the level a / γ .The distribution of T ( x ) is supported on { t : t ≥ t ∗ ( x ) } , and can be found in the form ofthe (generalised) density functions Q ( t , x ) and Q ( t , x ) , P { T ( x ) ∈ d t | ε ( ) = } = Q ( t , x ) d t , P { T ( x ) ∈ d t | ε ( ) = } = Q ( t , x ) d t , assuming that Q ( t , x ) | t < t ∗ ( x ) = , Q ( t , x ) | t < t ∗ ( x ) = . Due to identities (2.5)-(2.6), functions Q ( t , x ) and Q ( t , x ) follow the system of the inte-gral equations,(3.3) Q ( t , x ) = (cid:90) t λ e − λ τ Q ( t − τ , φ ( x , τ )) d τ , Q ( t , x ) = e − λ t ∗ ( x ) δ ( t − t ∗ ( x )) + (cid:90) t ∗ ( x ) λ e − λ τ Q ( t − τ , φ ( x , τ )) d τ , t > t ∗ ( x ) , x > a / γ . Here, δ = δ ( · ) is Dirac’s delta-function.By definition of T ( x ) , (3.1), equations (3.3) must be supplied with the boundary conditions(3.4) lim x ↓ a / γ Q ( t , x ) = λ e − λ t , lim x ↓ a / γ Q ( t , x ) = δ ( t ) . Since lim x ↓ a / γ t ∗ ( x ) = x ↓ a / γ φ ( x , τ ) ≡ a / γ (see (2.3)), the samefollows from the equations (3.3) themselves.Let L : = ∂∂ t + ( γ x − a ) ∂∂ x , L : = ∂∂ t + ( γ x − a ) ∂∂ x . Since L (cid:2) ( γ x − a ) e − γ t (cid:3) = , L (cid:2) ( γ x − a ) e − γ t (cid:3) = , and ( γ x − a ) dd x [ t ∗ ( x )] = , we have the following identities: L [ φ ( x , t )] = , L [ φ ( x , t )] = , L [ t ∗ ( x )] = , see (2.3)-(2.4) and (3.2). By applying L and L to (3.3) we obtain L [ Q ( t , x )] = λ e − λ t Q ( , φ ( x , t )) − (cid:90) t λ e − λ τ dd τ (cid:104) Q ( t − τ , φ ( x , τ )) (cid:105) d τ , L [ Q ( t , x )] = − λ e − λ t ∗ ( x ) δ ( t − t ∗ ( x )) + λ e − λ t ∗ ( x ) Q ( t − t ∗ ( x ) , a / γ ) − (cid:90) t ∗ ( x ) λ e − λ τ dd τ (cid:104) Q ( t − τ , φ ( x , τ )) (cid:105) d τ . Integrating by parts, one can see that system (3.3) of the integral equations is equivalent tothe system of the partial differential equations,(3.5) (cid:26) L [ Q ( t , x )] = − λ Q ( t , x ) + λ Q ( t , x ) , L [ Q ( t , x )] = λ Q ( t , x ) − λ Q ( t , x ) , t > t ∗ ( x ) , x > a / γ , with the boundary conditions (3.4). NIKITA RATANOV
Consider the Laplace transforms(3.6) (cid:98) Q ( q , x ) = E [ exp ( − qT ( x ))] = (cid:90) ∞ e − qt Q ( t , x ) d t , (cid:98) Q ( q , x ) = E [ exp ( − qT ( x ))] = (cid:90) ∞ e − qt Q ( t , x ) d t , q > . Note that 0 ≤ (cid:98) Q i ( q , x ) ≤ , i ∈ { , } . Functions (cid:98) Q ( q , x ) and (cid:98) Q ( q , x ) have a sense of thecomplementary cumulative distribution function of X e q = max ≤ t ≤ e q X ( t ) , where e q is an expo-nentially distributed random variable, Exp ( q ) , independent of X . Indeed, integrating by partsin (3.6), one can see (cid:98) Q i ( q , x ) = (cid:90) ∞ e − qt d [ P { T ( x ) < t | ε ( ) = i } ] = (cid:90) ∞ q e − qt P { T ( x ) < t | ε ( ) = i } d t = P { T ( x ) < e q | ε ( ) = i } = P { X e q > x | ε ( ) = i } . System (3.5) corresponds to the system of the ordinary equations,(3.7) ( x − a / γ ) d (cid:98) Q d x ( q , x ) = − β ( q ) (cid:98) Q ( q , x ) + β ( ) (cid:98) Q ( q , x ) , ( x − a / γ ) d (cid:98) Q d x ( q , x ) = β ( ) (cid:98) Q ( q , x ) − β ( q ) (cid:98) Q ( q , x ) , x > a / γ , where(3.8) β ( q ) = λ + q γ , β ( q ) = λ + q γ . Due to (3.4), system (3.7) is supplied with the boundary conditions (cid:98) Q ( q , a / γ +) = λ / ( λ + q ) , (cid:98) Q ( q , a / γ +) = . Consider the series representations: (cid:98) Q ( q , x ) = ∞ ∑ n = A n ( x − a / γ ) n , (cid:98) Q ( q , x ) = ∞ ∑ n = B n ( x − a / γ ) n , x > a / γ . The boundary conditions give A = λ / ( λ + q ) , B = , and by the system (3.7) we havethe sequence of coupled equations for coefficients A n and B n , n ≥ (cid:26) nA n = − β ( q ) A n + β ( ) B n , nB n + ( n + )( a / γ − a / γ ) B n + = β ( ) A n − β ( q ) B n , which is equivalent to(3.9) A n = β ( ) β ( q ) + n B n , B n + = β ( ) A n − ( β ( q ) + n ) B n ( n + )( a / γ − a / γ ) = β ( ) β ( ) − ( β ( q ) + n )( β ( q ) + n )( n + )( a / γ − a / γ )( β ( q ) + n ) B n . The second equation can be rewritten as B n + = − ( b ( q ) + n ) ( b ( q ) + n ) β ( q ) + n · B n ( n + )( a / γ − a / γ ) , RNSTEIN-UHLENBECK PROCESSES OF BOUNDED VARIATION 7 where(3.10) b , = (cid:18) β ( q ) + β ( q ) ± (cid:113) ( β ( q ) − β ( q )) + β ( ) β ( ) (cid:19) . Due to the boundary conditions, B = , B = b b β · · − a / γ − a / γ , B = b ( b + ) · b ( b + ) β ( β + ) · · (cid:18) − a / γ − a / γ (cid:19) ,. . . , B n = ( b ) n ( b ) n ( β ) n · n ! · (cid:18) − a / γ − a / γ (cid:19) n , and, by the first equation of (3.9), A n = β ( ) β ( q ) + n · ( b ) n ( b ) n ( β ( q )) n n ! · (cid:18) − a / γ − a / γ (cid:19) n = λ λ + q · ( b ) n ( b ) n ( β + ) n · n ! · (cid:18) − a / γ − a / γ (cid:19) n , n ≥ , where β = β ( q ) , β = β ( q ) and b = b ( q ) , b = b ( q ) are defined by (3.8) and (3.10); ( b ) n = Γ ( b + n ) / Γ ( b ) = b ( b + ) . . . ( b + n − ) is the Pochhammer symbol.As a result, functions (cid:98) Q and (cid:98) Q are expressed by(3.11) (cid:98) Q ( q , x ) = λ λ + q F (cid:18) b ( q ) , b ( q ) ; β ( q ) + a / γ − xa / γ − a / γ (cid:19) , (cid:98) Q ( q , x ) = F (cid:18) b ( q ) , b ( q ) ; β ( q ) ; a / γ − xa / γ − a / γ (cid:19) . Here F is the Gaussian hypergeometric function, defined by the series(3.12) F ( b , b ; β ; z ) = + ∞ ∑ n = ( b ) n ( b ) n ( β ) n · n ! z n , if one of the following conditions holds:(1) | z | < | z | = β − b − b > | z | = , z (cid:54) = , and − < β − b − b ≤ . Function F is also defined by analytic continuation everywhere in z , z ≤ − . see [9, Chap.9.1] and [1].Therefore, functions (cid:98) Q and (cid:98) Q are defined by formulae (3.11) and by series (3.12), ifthe starting point x satisfies a / γ ≤ x < a / γ − a / γ . If x is far from a / γ , analyticcontinuation is applied. Theorem 3.1. If λ > , then a.s.T ( x ) < ∞ , x > a / γ . Proof.
Since b ( ) = b ( ) = λ / γ + λ / γ , β ( ) = λ / γ , we have P { T ( x ) < ∞ | ε ( ) = } = (cid:98) Q ( , x ) = F ( b ( ) ; β ( ) + z ) ≡ , P { T ( x ) < ∞ | ε ( ) = } = (cid:98) Q ( , x ) = F ( b ( ) ; β ( ) ; z ) ≡ , NIKITA RATANOV which give the proof. (cid:3) (cid:3)
From (3.11) one can obtain the moments of the falling time T ( x ) . For simplicity, we givethe explicit formulae for the mean values of T ( x ) , when the initial point x is not so far fromthe attractive band. Theorem 3.2.
Let T ( x ) , x ≥ a / γ , be defined by (3.1) .In the following two cases (1) a / γ ≤ x < a / γ − a / γ ;(2) x = a / γ − a / γ and λ < γ ; the mean values of T ( x ) are given by the series E { T ( x ) | ε ( ) = } = − b (cid:48) ( ) ∞ ∑ n = ( λ / γ + λ / γ ) n ( + λ / γ ) n · z n n + λ < ∞ , (3.13) E { T ( x ) | ε ( ) = } = − b (cid:48) ( ) ∞ ∑ n = ( λ / γ + λ / γ ) n ( λ / γ ) n · z n n < ∞ . (3.14) Here b (cid:48) ( ) = λ + λ λ γ + λ γ > is the derivative of the minor root, b ( q ) , (3.10), and z = z ( x ) = a / γ − xa / γ − a / γ .If x = a / γ − a / γ and γ ≤ λ < γ , then only the series (3.14) for E { T ( x ) | ε ( ) = } is finite.In all other cases, the expectations E { T ( x ) | ε ( ) = } and E { T ( x ) | ε ( ) = } followafter analytic continuation of (3.11) .Proof. Since b ( ) = (cid:18) β ( ) + β ( ) − (cid:113) ( β ( ) − β ( )) + β ( ) β ( ) (cid:19) = , b ( ) = (cid:18) β ( ) + β ( ) + (cid:113) ( β ( ) − β ( )) + β ( ) β ( ) (cid:19) = λ γ + λ γ and by (3.12), F (cid:48) b ( b , b ; β + z ) | q = = , F (cid:48) β ( b , b ; β + z ) | q = = , we have E [ T ( x ) | ε ( ) = ] = − ∂∂ q (cid:104) (cid:98) Q ( q , x ) (cid:105) | q = = λ ( λ + q ) | q = − λ λ + q · (cid:16) b (cid:48) ( ) F (cid:48) b + b (cid:48) ( ) F (cid:48) b + β (cid:48) ( ) F (cid:48) β (cid:17) ( b , b ; 1 + β ; z ) | q = = λ − b (cid:48) ( ) F (cid:48) b ( , λ / γ + λ / γ ; 1 + λ / γ ; z ) RNSTEIN-UHLENBECK PROCESSES OF BOUNDED VARIATION 9 with z = z ( x ) = a / γ − xa / γ − a / γ . Further, F (cid:48) b ( b , b ; 1 + β ; z ) | q = = ∞ ∑ n = ( n − ) ! ( b ( )) n ( + β ( )) n · z n n ! = ∞ ∑ n = ( λ / γ + λ / γ ) n ( + λ / γ ) n · z n n , if the series converges.Formula (3.13) follows from b (cid:48) ( q ) | q = = (cid:18) γ + γ − ( λ / γ − λ / γ )( / γ − / γ ) λ / γ + λ / γ (cid:19) =
12 2 ( λ + λ ) / ( γ γ ) λ / γ + λ / γ = λ + λ γ λ + γ λ Similarly, one can obtain (3.14). (cid:3) (cid:3) (a) (b)(c) (d) F IGURE
2. The expectation E = E [ T ( x ) | ε ( ) = ] in the case a = − a = a and γ = γ = γ , as function of (a): x , ≤ x ≤ , with λ = λ = a = γ =
1; 2 .
5; 5 (from top to bottom); (b): a , ≤ a ≤ , with x = , γ = a , λ = λ = x = .
5; 2; 2 . (c): λ , . ≤ λ ≤ . , with x = , a = γ = λ = .
1; 0 .
25; 0 . (d): λ , ≤ λ ≤ . , with x = , a = γ = λ = .
1; 0 .
25; 0 . The subsequent moments, E [ T ( x ) n | ε ( ) = i ] , n ≥ , i ∈ { , } , can be obtained by se-quential differentiation.Some plots are presented in Fig. 2. Remark 3.1.
Let X = X ( t ) , starts from x = X ( ) , x < a / γ , andT − ( x ) = inf { t : X ( t ) > a / γ } , x < a / γ , be the first passage time through the level x = a / γ . The formulae for the expectations ofT − ( x ) can be easily written by symmetry. Remark 3.2.
Formulae (3.11) are consistent with some simple reasonable results.Let λ = .If ε ( ) = , then X ( t ) = φ ( x , t ) ∀ t > , a. s. and, hence, the process X never crosses thelevel a / γ . That is, T ( x ) = + ∞ .If ε ( ) = , then the process X = X ( t ) passes through a / γ if and only if there is noswitching up to the time t ∗ ( x ) , (3.2) . Therefore , conditionally (under ε ( ) = ) (cid:104) T ( x ) = (cid:40) t ∗ ( x ) , with probability e − λ t ∗ ( x ) , + ∞ , with probability − e − λ t ∗ ( x ) | ε ( ) = (cid:105) . In this case, (3.15) (cid:98) Q ( q , x ) ≡ , (cid:98) Q ( q , x ) = exp ( − qt ∗ ( x )) · e − λ t ∗ ( x ) = (cid:18) x − a / γ a / γ − a / γ (cid:19) − ( λ + q ) / γ . The same result is given by (3.11) : if λ = , then, due to (3.11) , we have (cid:98) Q ( q , x ) ≡ , and b ( q ) , b ( q ) coincide with β = q / λ , β = ( λ + q ) / γ . Hence, (cid:98) Q ( q , x ) = + ∞ ∑ n = ( β ( q )) n n ! z ( x ) n = ( − z ( x )) − λ + q γ = (cid:18) x − a / γ a / γ − a / γ (cid:19) − ( λ + q ) / γ , which coincides with (3.15) .Let λ = . If the particle begins to move from point x , x > a / γ , according to the pattern φ ( x , t ) , (2.4) , then it will arrive without switching to a / γ at time t ∗ ( x ) . It means that (3.16) Q ( t , x ) = δ ( t − t ∗ ( x )) . Thus, by (3.6) and (3.2) (cid:98) Q ( q , x ) = e − qt ∗ ( x ) = (cid:18) x − a / γ a / γ − a / γ (cid:19) − q / γ = ( − z ( x )) − q / γ = + ∞ ∑ n = ( b ) n n ! z ( x ) n with b = b ( q ) = q / γ . This is repeated by (3.11) with b = β = q / γ and b = β = ( λ + q ) / γ . On the other hand, if the particle begins with the pattern φ ( x , t ) , (2.3) , then it falls intoa / γ after a single switch (at time τ ) to the pattern φ . This means that (3.17) (cid:98) Q ( q , x ) = E [ exp ( − q ( τ + t ∗ ( φ ( x , τ )))] . RNSTEIN-UHLENBECK PROCESSES OF BOUNDED VARIATION 11
Since t ∗ ( φ ( x , τ )) = γ log φ ( x , τ ) − a / γ a / γ − a / γ = γ log a / γ − a / γ + ( x − a / γ ) e − γ τ a / γ − a / γ = γ log (cid:0) − z ( x ) e − γ τ (cid:1) , z = a / γ − xa / γ − a / γ < , equation (3.17) becomes (cid:98) Q ( q , x ) = (cid:90) ∞ λ e − ( λ + q ) τ (cid:0) − z ( x ) e − γ τ (cid:1) − q / γ d τ (3.18) = λ γ (cid:90) y − +( λ + q ) / γ ( − z ( x ) y ) − q / γ d y . Due to the integral representation of Gaussian hypergeometric function [9, formula 9.111] , (3.19) (cid:98) Q ( q , x ) = λ λ + q F ( q / γ , ( λ + q ) / γ ; 1 + ( λ + q ) / γ ; z ( x )) , which coincides with the first equation of (3.11) (with λ = ).
4. T
HE MEAN AND VARIANCE OF X ( t ) . The marginal distribution of X ( t ) , (2.2), can not be so easily written as the distribution ofthe Gaussian Ornstein-Uhlenbeck process. In this section we give only a few hints on thismatter.Let 0 = τ < τ < . . . < τ n < . . . be the sequence of switching times of the of the underlyingMarkov process ε . Let N ( t ) corresponds to the number switchings till time t , t > , N ( t ) = n , if τ n ≤ t < τ n + . Recalling the distribution of the inhomogeneous Poisson process N ( t ) , see [14, Theorem2.1], we have(4.1) π ( s ) = P { N ( s ) is even } = e − λ s (cid:104) + Ψ ( s , ( λ − λ ) s ) (cid:105) , π ( s ) = P { N ( s ) is even } = e − λ s (cid:104) + Ψ ( s , ( λ − λ ) s ) (cid:105) , π ( s ) = P { N ( s ) is odd } = λ e − λ s Ψ ( s , ( λ − λ ) s ) , π ( s ) = P { N ( s ) is odd } = λ e − λ s Ψ ( s , ( λ − λ ) s ) , where(4.2) Ψ ( t , z ) = ∞ ∑ n = λ n λ n ( n ) ! t n Φ ( n , n + z ) , Ψ ( t , z ) = ∞ ∑ n = λ n − λ n − ( n − ) ! t n − Φ ( n , n ; z ) ; Φ ( · , · ; · ) is the confluent hypergeometric function, [1]. Due to representation (2.2), the mean of X ( t ) is given by(4.3) E [ X ( t )] = E (cid:20) e − Γ ( t ) (cid:18) x + (cid:90) t e Γ ( s ) a ε ( s ) d s (cid:19)(cid:21) = x ψ Γ ( t )+ (cid:90) t (cid:104) a π ( s ) E (cid:16) e − ( Γ ( t ) − Γ ( s )) | ε ( s ) = (cid:17) + a π ( s ) E (cid:16) e − ( Γ ( t ) − Γ ( s )) | ε ( s ) = (cid:17)(cid:105) d s = x ψ Γ ( t ) + a (cid:90) t π ( s ) ψ Γ ( t − s ) d s + a (cid:90) t π ( s ) ψ Γ ( t − s ) d s . Similarly,(4.4) E [ X ( t )] = x ψ Γ ( t ) + a (cid:90) t π ( s ) ψ Γ ( t − s ) d s + a (cid:90) t π ( s ) ψ Γ ( t − s ) d s . Here π ik ( · ) are determined by (4.1)-(4.2), and the moment generating functions, ψ Γ k ( t ) = E k [ exp ( − (cid:90) t γ ε ( s ) d s )] , k ∈ { , } , of the telegraph process Γ ( t ) are also known, ψ Γ ( t ) = e − ( λ + γ ) t [ + Ψ ( t , ( λ − λ + γ − γ ) t ) + λ Ψ ( t , ( λ − λ + γ − γ ) t )] , ψ Γ ( t ) = e − ( λ + γ ) t [ + Ψ ( t , ( λ − λ + γ − γ ) t ) + λ Ψ ( t , ( λ − λ + γ − γ ) t )] , see e.g. [14, (2.21)]. Remark 4.1.
In the symmetric case, λ = λ = λ , γ = γ = γ and a = − a = a , formulae (4.3) - (4.4) can be simplified.Since, ψ Γ ( t ) = ψ Γ ( t ) = e − γ t and π ( s ) = π ( s ) = ( + e − λ s ) / , π ( s ) = π ( s ) =( − e − λ s ) / , by (4.3) - (4.4) we have (4.5) E [ X ( t )] = x e − γ t + a e − λ t − e − γ t γ − λ , if γ (cid:54) = λ , t e − γ t , if γ = λ , and (4.6) E [ X ( t )] = x e − γ t − a e − λ t − e − γ t γ − λ , if γ (cid:54) = λ , t e − γ t , if γ = λ . Further, notice that in the symmetric case, E [ a ε ( s ) a ε ( s ) ] = E [ a ε ( s ) a ε ( s ) ] = a + e − λ | s − s | − a − e − λ | s − s | = a exp ( − λ | s − s | ) . Hence, E (cid:20) (cid:90) t e − γ ( t − s ) d T ( t ) (cid:21) = a e − γ t (cid:90) t (cid:90) t exp ( γ ( s + s ) − λ | s − s | ) d s d s (4.7) = a γ + λ γ − γ − λ e − ( γ + λ ) t + γ + λγ ( γ − λ ) e − γ t , if γ (cid:54) = λ , − e − γ t − γ t e − γ t γ , if γ = λ , which gives the expression for the variance of X ( t ) , (4.8)Var [ X ( t )] = E (cid:20) (cid:90) t e − γ ( t − s ) d T ( t ) (cid:21) − (cid:18) E (cid:20) (cid:90) t e − γ ( t − s ) d T ( t ) (cid:21)(cid:19) = a γ ( γ + λ ) − e − γ t ( γ − λ ) (cid:20) e ( γ − λ ) t − λγ + λ e ( γ − λ ) t + λγ (cid:21) , if γ (cid:54) = λ , γ (cid:2) − e − γ t (cid:0) + γ t + γ t (cid:1)(cid:3) , if γ = λ . The limiting behaviour of X ( t ) is consistent with known results.As t → ∞ , the limits are given by lim t → ∞ E [ X ( t )] = lim t → ∞ E [ X ( t )] = , lim t → ∞ Var [ X ( t )] = a γ ( γ + λ ) . On the other hand, under Kac’s scaling, a , λ → ∞ , a / λ → σ , the limits of the expectation (4.9) lim E [ X ( t )] = x e − γ t ± lim a e − λ t − e − γ t γ − λ = x e − γ t , see (4.3) - (4.4) , and the variance (4.10)lim Var [ X ( t )] = lim a (cid:26) γ ( γ + λ ) − e − γ t ( γ − λ ) (cid:20) e ( γ − λ ) t − λγ + λ e ( γ − λ ) t + λγ (cid:21)(cid:27) = lim a γ ( γ + λ ) − e − γ t γ lim 2 a λ ( γ − λ ) = σ γ (cid:0) − e − γ t (cid:1) , Formulae (4.9) - (4.10) coincide with the known results for the classical Ornstein-Uhlenbeckprocess, see e.g. [15, (4)-(5)] .
5. O
N THE JOINT DISTRIBUTION OF X ( t ) AND N ( t ) . Due to technical difficulties, the distribution of the Ornstein-Uhlenbeck process withbounded variation cannot be presented explicitly. However, let’s sketch it out.Consider the Ornstein-Uhlenbeck process of bounded variation X = X ( t ) based on thecompletely symmetric telegraph process T : the velocities are ± a the switching intensitiesare identical, λ = λ = λ , and γ = γ = γ . Let f i ( y , t ; n | x ) , n ≥ , i ∈ { , } , be the densityfunctions characterising the joint distribution of the particle position X ( t ) and the number ofthe patterns switchings N ( t ) , f i ( y , t ; n | x ) = P { X ( t ) ∈ d y , N ( t ) = n | X ( ) = x , ε ( ) = i } / d y . By definition, we have(5.1) f ( y , t ; 0 | x ) = e − λ t δ ( y − φ ( x , t )) , f ( y , t ; 0 | x ) = e − λ t δ ( y − φ ( x , t )) , φ ( x , t ) = a / γ + ( x − a / γ ) e − γ t , φ ( x , t ) = − a / γ + ( x + a / γ ) e − γ t . Further, by virtue of (2.5)-(2.6), functions f ( y , t ; n | x ) and f ( y , t ; n | x ) satisfy the sequence of coupled integral equa-tions, n ≥ , f ( y , t ; n | x ) = λ (cid:90) t e − λ τ f ( y , t − τ ; n − | φ ( x , τ )) d τ , (5.2) f ( y , t ; n | x ) = λ (cid:90) t e − λ τ f ( y , t − τ ; n − | φ ( x , τ )) d τ . (5.3)Due to the total symmetry of the underlying process T , we have the identity in law: [ X ( t ) | ε ( ) = , X ( ) = x ] D = [ − X ( t ) | ε ( ) = , X ( ) = − x ] , t > . Moreover, by induction, one can verify the following identities: for all n , n ≥ , (5.4) f ( y , t ; n | x ) ≡ f ( − y , t ; n | − x ) , t > . Since φ ( − x , t ) ≡ − φ ( x , t ) , for n = n −
1. Equations (5.2)-(5.3) give f ( − y , t ; n | − x ) = λ (cid:90) t e − λ τ f ( − y , t − τ ; n − | φ ( − x , τ )) d τ = λ (cid:90) t e − λ τ f ( − y , t − τ ; n − | − φ ( x , τ )) d τ = λ (cid:90) t e − λ τ f ( y , t − τ ; n − | φ ( x , τ )) d τ = f ( y , t ; n | x ) , which proves the result (5.4).In order to determine the explicit expressions of the density functions f ( y , t ; n | x ) and f ( y , t ; n | x ) , consider first (5.2)-(5.3) with n = . By (5.1) we have f ( y , t ; 1 | x ) = λ e − λ t (cid:90) t δ ( y − φ ( φ ( x , τ ) , t − τ )) d τ , (5.5) f ( y , t ; 1 | x ) = λ e − λ t (cid:90) t δ ( y − φ ( φ ( x , τ ) , t − τ )) d τ . (5.6)Notice that the equations y − φ ( φ ( x , τ ) , t − τ ) = , (5.7) y − φ ( φ ( x , τ ) , t − τ ) = , (5.8)have the solutions, τ , ≤ τ ≤ t , if and only if y ∈ I ( x , t ) : = [ φ ( x , t ) , φ ( x , t )] , that is(5.9) − a γ + (cid:18) x + a γ (cid:19) e − γ t = φ ( x , t ) ≤ y ≤ φ ( x , t ) = a γ + (cid:18) x − a γ (cid:19) e − γ t . RNSTEIN-UHLENBECK PROCESSES OF BOUNDED VARIATION 15
Since φ ( φ ( x , τ ) , t − τ ) ≡ − a γ + a γ e − γ ( t − τ ) + (cid:18) x − a γ (cid:19) e − γ t , (5.10) φ ( φ ( x , τ ) , t − τ ) ≡ a γ − a γ e − γ ( t − τ ) + (cid:18) x + a γ (cid:19) e − γ t , (5.11)see (2.3)-(2.4), the solution of (5.7), τ = τ ( y , t | x ) , is given by(5.12) τ = τ ( y , t | x ) = t + γ log a + γ y + ( a − γ x ) e − γ t a . Similarly, the solution of (5.8), τ = τ ( y , t | x ) = τ ( − y , t | − x ) , is(5.13) τ = τ ( y , t | x ) = t + γ log a − γ y + ( a + γ x ) e − γ t a . Note thate − γ ( t − τ ( y , t | x )) + e − γ ( t − τ ( y , t | x )) ≡ a + γ y + ( a − γ x ) e − γ t a + a − γ y + ( a + γ x ) e − γ t a ≡ + e − γ t . Due to equations (5.5)-(5.6), (5.12)-(5.13) and the differential equalitiesd τ [ y − φ ( φ ( x , τ ) , t − τ )] = − a exp ( − γ ( t − τ )) d τ , d τ [ y − φ ( φ ( x , τ ) , t − τ )] = a exp ( − γ ( t − τ )) d τ , < τ < t , one can obtain the explicit expressions for the density functions with a single velocity switch-ing, f ( y , t ; 1 | x ) = λ e − λ t { y ∈ I ( x , t ) } a + γ y + ( a − γ x ) e − γ t = λ e − ( λ − γ ) t { y ∈ I ( x , t ) } a e − γτ ( y , t | x ) , f ( y , t ; 1 | x ) = λ e − λ t { y ∈ I ( x , t ) } a − γ y + ( a + γ x ) e − γ t = λ e − ( λ − γ ) t { y ∈ I ( x , t ) } a e − γτ ( y , t | x ) . We continue to solve equations (5.2)-(5.3) using the following lemma.
Lemma 5.1.
Let τ ( y , t | x ) and τ ( y , t | x ) be defined by (5.12) - (5.13) . We have the followingidentities :(5.14) τ ( y , t − τ | φ ( x , τ )) = τ ( y , t | x ) − τ , τ ( y , t − τ | φ ( x , τ )) = τ ( y , t | x ) − τ , < τ < t ; and (5.15) τ ( y , t − τ | φ ( x , τ )) = t − τ + γ log a + γ y − ( a + γ x ) e − γ t + a e − γ t e γτ a , τ ( y , t − τ | φ ( x , τ )) = t − τ + γ log a − γ y − ( a − γ x ) e − γ t + a e − γ t e γτ a , < τ < t . Furthermore, (5.16) y ∈ I ( φ ( x , τ ) , t − τ ) ⇔ < τ ≤ τ ( y , t | x ) , y ∈ I ( φ ( x , τ ) , t − τ ) ⇔ τ ( y , t | x ) ≤ τ < t . Proof.
Equalities (5.14)-(5.16) can be verified directly by definition. For instance, by (5.12)and (2.3)-(2.4) one can obtain the first identities of (5.14) and (5.15): τ ( y , t − τ | φ ( x , τ )) = t − τ + γ log a + γ y + ( a − γφ ( x , τ )) e − γ ( t − τ ) a = t − τ + γ log a + γ y + ( a − γ x ) e − γ t a = t − τ + τ ( y , t | x ) − t = τ ( y , t | x ) − τ and τ ( y , t − τ | φ ( x , τ )) = t − τ + γ log a + γ y + ( a − γφ ( x , τ )) e − γ ( t − τ ) a = t − τ + γ log a + γ y − ( a + γ x ) e − γ t + a e − γ t e γτ a . Further, y ∈ I ( φ ( x , τ ) , t − τ ) is equivalent to φ ( φ ( x , τ ) , t − τ ) ≤ y ≤ φ ( φ ( x , τ ) , t − τ ) ≡ φ ( x , t ) , see (5.9). Function τ → φ ( φ ( x , τ ) , t − τ ) , see (5.10), increases. Hence, y ∈ I ( φ ( x , τ ) , t − τ ) is equivalent to 0 < τ < τ ( y , t | x ) . Other equalities of the lemma are verified similarly. (cid:3) (cid:3) Due to Lemma 5.1 equations (5.2)-(5.3) give(5.17) f ( y , t ; 2 | x ) = λ e − λ t (cid:90) τ ( y , t | x ) d τ a − γ y + ( γ x − a ) e − γ t + a e − γ t e γτ , f ( y , t ; 2 | x ) = λ e − λ t (cid:90) t τ ( y , t | x ) d τ a + γ y − ( γ x + a ) e − γ t + a e − γ t e γτ , Since (cid:90) ba d τ A + B e γτ = γ A log (cid:20) A + B e γ a A + B e γ b e γ ( b − a ) (cid:21) , integrating in (5.17), we get f ( y , t ; 2 | x ) = λ e − λ t τ ( y , t | x ) + τ ( y , t | x ) − ta − γ y + ( γ x − a ) e − γ t , f ( y , t ; 2 | x ) = λ e − λ t τ ( y , t | x ) + τ ( y , t | x ) − ta + γ y − ( γ x + a ) e − γ t , where τ ( y , t | x ) and τ ( y , t | x ) are determined by (5.12)-(5.13).Applying Lemma 5.1 successively, one can obtain a sequence of the formulae for f i ( · , · ; n | · ) , which look more and more sophisticated.A PPENDIX : THE TELEGRAPH PROCESS
Let ( Ω , F , F t , P ) be the complete filtered probability space. Consider the adapted tele-graph process T ( t ) , t ≥ , with two alternating symmetric velocities a and − a , a > , switch-ing with positive intensities λ and λ . RNSTEIN-UHLENBECK PROCESSES OF BOUNDED VARIATION 17
The joint distribution of T ( t ) and ε ( t ) can be expressed by means of the (generalised)density functions p ji ( x , t ) : = P { T ( t ) ∈ d x , ε ( t ) = j | ε ( ) = i } / d x , i , j ∈ { , } , t > . The following formulae seem to be generally known, but for the best of my belief, they havenever been published.
Theorem A.1.
The density functions p ji ( x , t ) , i , j ∈ { , } , are given by (A.1) p ( x , t ) = e − λ t δ ( x − a t ) + (cid:112) λ λ a − a (cid:115) ξ t − ξ e − λ ξ − λ ( t − ξ ) I ( (cid:112) λ λ ξ ( t − ξ )) { < ξ < t } , p ( x , t ) = e − λ t δ ( x − a t ) + (cid:112) λ λ a − a (cid:115) t − ξξ e − λ ξ − λ ( t − ξ ) I ( (cid:112) λ λ ξ ( t − ξ )) { < ξ < t } , p ( x , t ) = λ a − a e − λ ξ − λ ( t − ξ ) I ( (cid:112) λ λ ξ ( t − ξ )) { < ξ < t } , p ( x , t ) = λ a − a e − λ ξ − λ ( t − ξ ) I ( (cid:112) λ λ ξ ( t − ξ )) { < ξ < t } , where ξ = ( x − a t ) / ( a − a ) , t − ξ = ( a t − x ) / ( a − a ) , a t < x < a t . Here I and I are the modified Bessel functions , I ( z ) = + ∞ ∑ n = ( z / ) n ( n ! ) , I ( z ) = I (cid:48) ( z ) = ∞ ∑ n = ( z / ) n − ( n − ) ! n ! . Proof.
Let N ( t ) be the number of velocity switchings in the time interval [ , t ) .By virtue of [12, (4.1.10)-(4.1.11)], p , can be represented as p ( x , t ) = ∞ ∑ n = P { T ( t ) ∈ d x , N ( t ) = n | ε ( ) = } / d x = e − λ t δ ( x − a t ) + exp ( − λ ξ − λ ( t − ξ )) a − a ∞ ∑ n = λ n λ n ( n − ) ! n ! ξ n ( t − ξ ) n − { < ξ < t } = e − λ t δ ( x − a t ) + (cid:112) λ λ a − a (cid:115) ξ t − ξ I ( (cid:112) λ λ ξ ( t − ξ )) { < ξ < t } . see [9, formula 8.445]. The remaining equalities of (A.1) are obtained in the same manner. (cid:3) (cid:3) The well-known formulae for the (conditional) distribution of T ( t ) follow from (A.1): p ( x , t ) = P { T ( t ) ∈ d x | ε ( ) = } / d x = p ( x , t ) + p ( x , t ) , p ( x , t ) = P { T ( t ) ∈ d x | ε ( ) = } / d x = p ( x , t ) + p ( x , t ) , cf [2], [14] or see in the book by Kolesnik and Ratanov, [12, (4.1.15)].Similarly, f ( x , t ) = p ( x , t ) + p ( x , t ) (and b ( x , t ) = p ( x , t ) + p ( x , t ) ) are the distribu-tion density functions of the moving forward (and backward) particles, cf [16], where theseformulae were presented in the symmetric case, λ = λ . The rest of this section is devoted to a description of the first and the second moments of T ( t ) and T ( t ) { ε ( t )= j } , j ∈ { , } . We will use the following notations E i [ g ( T ( t ))] = E [ g ( T ( t )) | ε ( ) = i ] = (cid:90) ∞ − ∞ g ( x ) p i ( x , t ) d x and E ji [ g ( T ( t ))] = E [ g ( T ( t )) · { ε ( t )= j } | ε ( ) = i ] = (cid:90) ∞ − ∞ g ( x ) p ji ( x , t ) d x , i , j ∈ { , } . Theorem A.2.
Let a = − a = a > . For t ≥ E T ( t ) = a e − λ t ∞ ∑ n = λ n λ n ( n ) ! t n + G ( ) n ( t ) , (A.2) E T ( t ) = a e − λ t ∞ ∑ n = λ n + λ n ( n + ) ! t n + H ( ) n ( t ) , (A.3) E T ( t ) = − a e − λ t ∞ ∑ n = λ n λ n + ( n + ) ! t n + H ( ) n ( − t ) (A.4) E T ( t ) = − a e − λ t ∞ ∑ n = λ n λ n ( n ) ! t n + G ( ) n ( − t ) , (A.5) and E T ( t ) = a exp ( − λ t ) ∞ ∑ n = λ n λ n ( n ) ! t n + G ( ) n ( t ) , (A.6) E T ( t ) = a exp ( − λ t ) ∞ ∑ n = λ n + λ n ( n + ) ! t n + H ( ) n ( t ) , (A.7) E T ( t ) = a exp ( − λ t ) ∞ ∑ n = λ n λ n + ( n + ) ! t n + H ( ) n ( − t ) , (A.8) E T ( t ) = a exp ( − λ t ) ∞ ∑ n = λ n λ n ( n ) ! t n + G ( ) n ( − t ) , (A.9) where (A.10) G ( ) n ( t ) = − n n + Φ ( n + , n +
2; 2 β t ) + Φ ( n , n +
1; 2 β t ) , (A.11) H ( ) n ( t ) = − Φ ( n + , n +
3; 2 β t ) + Φ ( n + , n +
2; 2 β t ) and (A.12) G ( ) n ( t ) = n n + Φ ( n + , n +
3; 2 β t ) − n n + Φ ( n + , n +
2; 2 β t ) + Φ ( n , n +
1; 2 β t ) , (A.13) H ( ) n ( t ) = n + n + Φ ( n + , n +
4; 2 β t ) − Φ ( n + , n +
3; 2 β t ) + Φ ( n + , n +
2; 2 β t ) , RNSTEIN-UHLENBECK PROCESSES OF BOUNDED VARIATION 19 β = λ − λ . Here Φ ( a , b ; z ) denotes the confluent hypergeometric function, Φ ( α , β ; z ) : = ∞ ∑ n = ( α ) n ( β ) n z n n ! , ( · ) n is the Pochhammer symbol ; ( γ ) n = γ ( γ + ) . . . ( γ + n − ) , n ≥ , ( γ ) = . Proof.
Consider ψ i ( z , t ) = E i exp ( z T ( t )) = E [ exp ( z T ( t )) | ε ( ) = i ] , ψ i ( z , t ; n ) = E i (cid:2) exp ( z T ( t )) { N ( t )= n } (cid:3) , n ≥ , i ∈ { , } , and ψ ji ( z , t ) = E ji [ exp ( z T ( t ))] = E i (cid:2) exp ( z T ( t )) { ε ( t )= j } (cid:3) , i , j ∈ { , } , corresponding to the moment generating function of T ( t ) . Notice that ψ i ( z , t ) = ∞ ∑ n = ψ i ( z , t ; n ) and ψ ii ( z , t ) = ∞ ∑ n = ψ i ( z , t ; 2 n ) , ψ − ii ( z , t ) = ∞ ∑ n = ψ i ( z , t ; 2 n + ) , i ∈ { , } . The explicit expressions for ψ ( z , t ; n ) and ψ ( z , t ; n ) can be written separately for evenand odd n , n ≥ . Due to [14, Theorem 2.1], we have ψ ( z , t ; 2 n ) = λ n λ n ( n ) ! t n Φ ( n , n +
1; 2 ( β − az ) t ) exp ( − ( λ − az ) t ) , (A.14) ψ ( z , t ; 2 n ) = λ n λ n ( n ) ! t n Φ ( n , n +
1; 2 ( az − β ) t ) exp ( − ( λ + az ) t ) , (A.15) ψ ( z , t ; 2 n + ) = λ n + λ n ( n + ) ! t n + Φ ( n + , n +
2; 2 ( β − az ) t ) exp ( − ( λ − az ) t ) , (A.16) ψ ( z , t ; 2 n + ) = λ n λ n + ( n + ) ! t n + Φ ( n + , n +
2; 2 ( az − β ) t ) exp ( − ( λ + az ) t ) . (A.17)Formulae (A.14)-(A.17) directly give the desired result (A.2)-(A.8). For instance, by dif-ferentiating in (A.14) we have E [ T ( t )] = ∞ ∑ n = ∂ ψ ( z , t ; 2 n ) ∂ z | z = = a e − λ t ∞ ∑ n = λ n λ n ( n ) ! t n + (cid:104) − Φ (cid:48) ( n , n +
1; 2 β t ) + Φ ( n , n +
1; 2 β t ) (cid:105) and E [ T ( t ) ] = ∞ ∑ n = ∂ ψ ( z , t ; 2 n ) ∂ z | z = = a e − λ t ∞ ∑ n = λ n λ n ( n ) ! t n + (cid:104) Φ (cid:48)(cid:48) ( n , n +
1; 2 β t ) − Φ (cid:48) ( n , n +
1; 2 β t ) + Φ ( n , n +
1; 2 β t ) (cid:105) . The following known identities, Φ (cid:48) ( α , β ; z ) = d Φ d z ( α , β ; z ) = αβ Φ ( α + , β + z ) and Φ (cid:48)(cid:48) ( α , β ; z ) = α ( α + ) β ( β + ) Φ ( α + , β + z ) , see [9, formula 9.213], give the result, (A.2), (A.10) and (A.6), (A.12). Formulae (A.3) and(A.7) can be obtained similarly from (A.16).The remaining formulae of the theorem can be derived from (A.2)-(A.3) and (A.6)-(A.7)by symmetry: formula (A.5) follows from (A.2); (A.4) follows from (A.3); (A.9) followsfrom (A.6); (A.8) follows from (A.7) after replacements a → − a and λ ↔ λ . (cid:3) (cid:3) Formulae (A.2)-(A.9) permit us to evaluate the covariance between T ( t ) and T ( s ) . Theorem A.3.
For t > s > E T ( t ) T ( s ) = E T ( t − s ) · E T ( s ) + E T ( t − s ) · E T ( s ) + E [ T ( s ) ] , (A.18) E T ( t ) T ( s ) = E T ( t − s ) · E T ( s ) + E T ( t − s ) · E T ( s ) + E [ T ( s ) ] , (A.19) where E ji [ T ( s )] , E i [ T ( t − s )] and E i [ T ( s ) ] , i , j ∈ { , } , are given by (A.2) - (A.9) .Proof. Notice that E T ( t ) T ( s ) = E [( T ( t ) − T ( s )) · T ( s )] + E [ T ( s ) ] . Due to persistence and time-homogeneity of the process T , see (1.1), E [( T ( t ) − T ( s )) · T ( s )] = E [( T ( t ) − T ( s ) | ε ( s ) = ] · E T ( s ) + E [( T ( t ) − T ( s ) | ε ( s ) = ] · E T ( s )= E T ( t − s ) · E T ( s ) + E T ( t − s ) · E T ( s ) , which gives (A.18). Formula (A.19) follows similarly. (cid:3) (cid:3) Remark 5.1.
In the symmetric case λ = λ = λ > , the results of Theorem A.2 (formulae (A.2) - (A.9) ) and (A.18) - (A.19) look much simpler, cf [11] .Since β = and Φ ( · , · ; 0 ) = , we haveG ( ) n ( t ) | β = = G ( ) n ( t ) | β = ≡ n + , H ( ) n ( t ) | β = ≡ , H ( ) n ( t ) | β = ≡ n + . Therefore , for the symmetric case , the first moments (A.2) - (A.4) are given by (A.20) E T ( t ) = − E T ( t ) = at e − λ t ∞ ∑ n = ( λ t ) n ( n + ) ! = at e − λ t sinh λ t λ t = a λ (cid:16) − e − λ t (cid:17) , E T ( t ) = E T ( t ) = , and E T ( t ) = − E T ( t ) = a λ (cid:16) − e − λ t (cid:17) . RNSTEIN-UHLENBECK PROCESSES OF BOUNDED VARIATION 21
The second moments are given by E T ( t ) = E T ( t ) = ( at ) e − λ t ∞ ∑ n = ( λ t ) n ( n + ) ! = ( at ) e − λ t sinh λ t λ t = a t λ (cid:16) − e − λ t (cid:17) , E T ( t ) = E T ( t ) = ( at ) e − λ t ∞ ∑ n = ( λ t ) n + ( n + ) ! ( n + ) = ( at ) e − λ t (cid:18) sinh z − zz (cid:19) (cid:48) | z = λ t = a t λ (cid:16) + e − λ t (cid:17) − a λ (cid:16) − e − λ t (cid:17) , and, by summing we have (A.21) E T ( t ) = E T ( t ) + E T ( t ) = a λ (cid:16) e − λ t − + λ t (cid:17) = E T ( t ) . Formulae (A.20) and (A.21) are consistent with known results, see e.g. [12, (4.2.24)] .By (A.18) - (A.19) , (A.20) and (A.21) , E [ T ( t ) T ( s )] = E [ T ( t ) T ( s )]= a λ (cid:16) − e − λ ( t − s ) (cid:17) · a λ (cid:16) − e − λ s (cid:17) + + a λ (cid:16) e − λ s − + λ s (cid:17) (A.22) = a λ (cid:104) λ s − ( + e − λ ( t − s ) )( − e − λ s ) (cid:105) , and the covariance becomes cov ( T ( t ) , T ( s )) = E [ T ( t ) · T ( s )] − E [ T ( t )] · E [ T ( s )]= a λ (cid:20) λ s − + e − λ s + e − λ t − (cid:16) e − λ ( t − s ) + e − λ ( t + s ) (cid:17)(cid:21) . It is known that under Kac ’ s scaling , a , λ → ∞ , a / λ → σ , see [10, 12, 13], the symmet-ric telegraph process T ( t ) converges to Brownian motion σ W ( t ) . Formulae (A.20) , (A.21) and (A.22) consist with this convergence: under this scaling we have • by (A.20) , lim E [ T ( t )] = lim E [ T ( t )] = • by (A.21) , lim E [ T ( t ) ] = lim E [ T ( t ) ] = σ t , • by (A.22) , lim E [ T ( t ) T ( s )] = lim E [ T ( t ) T ( s )] = σ s , s ≤ t . Remark 5.2.
Notice that the “ general ” telegraph process T ( t ) , t ≥ , with two alternatingvelocities a and a , a > a , can be reduced to the symmetric case : T ( t ) D = ( a + a ) t / + T sym ± a ( t ) , where T s ym ± a ( t ) is the telegraph process with symmetric velocities ± a , a = ( a − a ) / . There-fore , without loss of generality, only a “symmetric” process T ( t ) , t ≥ , can be studied.
6. C
ONCLUSION
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NIVERSIDAD DEL R OSARIO , C L . 12 C , N O . 4-69, B OGOT ´ A , C OLOMBIA , NIKITA . RATANOV @ UROSARIO . EDU ..