First-passage probabilities and invariant distributions of Kac-Ornstein-Uhlenbeck processes
aa r X i v : . [ m a t h . P R ] F e b First-passage probabilities and invariant distributions of Kac-Ornstein-Uhlenbeckprocesses
Nikita Ratanov
Chelyabinsk State University, Russia (Dated:)In this paper, we study Ornstein-Uhlenbeck processes with Markov modulation, whose parametersdepend on an external underlying two-state Markov process ε . Conditional mean and variance ofsuch processes under given modulation are investigated from the point of view of the first passageprobabilities and invariant measures. It is also studied the limiting behaviour under scaling conditionssimilar to Kac’s scaling. PACS numbers: 05.40.Fb,05.40.Jc,02.50.Ey
INTRODUCTION
The Ornstein-Uhlenbeck process X can be de-fined as the solution of the stochastic equationd X OU ( t ) = ( a − γ X OU ( t )) d t + b d W ( t ) , t > , (1)with the initial condition X OU ( ) = x , where a , b and γ are constants, b ≥ W = W ( t ) is a standardWiener process (Brownian motion).It is widely accepted that Ornstein-Uhlenbeckprocesses can be used as an alternative model ofBrownian motion that better matches the physicaldata than the pure Wiener process.The solution of (1) can be written explicitly, X OU ( t ) = a γ + (cid:18) x − a γ (cid:19) e − γ t + σ Z t e − γ ( t − s ) d W ( s ) . (2)Basic properties of X OU can be derived from (1)-(2).In particular, X ( t ) has a Gaussian distribution withmean E [ X ( t )] = a γ + (cid:18) x − a γ (cid:19) e − γ t , and covatianceCov ( X ( s ) , X ( t )) = σ γ e − γ s ( e γ t − e − γ t ) , s ≥ t ≥ . Having originated from physics, this model is ex-ploited in various application fields as an alternativeto Wiener process with an average tendency to re-turn, see [6, 19]. For instance, the Vaˇs´ıˇcek interestrate model, [32], gave rise to widespread financialapplication of this process. The same processes arealso intensively used for neuronal modelling, see e.g.[5, 23, 27].Processes X OU are studied since the seminal pa-per by Uhlenbeck and Ornstein, [31]. Subsequently,similar processes were constructed on the basis of afractional Wiener process, [3, 14], or, in general, bya L´evy process, [2], and a fractional L´evy process,[7]. See also [12] for rationales and nonstandard in-terpretations, and [14, 16] for statistics.Recently, some results have appeared on theMarkov-modulated Ornstein-Uhlenbeck processes,see [11, 26–28, 36]. This approach assumes thatequation (1) is modified by Markov oscillation of allparameters. The motivation to study the model inthis context arises from the following observations.First, by allowing alternation of the coefficients ofthe Langevin equation (1), we open up new good op-portunities for applications. The second idea is basedon the fact that the Brownian motion has some prop-erties that are contrary to physical intuition, such asinfinite total variation of paths and infinite propaga-tion speeds. Since a source of stochasticity such asMarkov modulation does not possess such disadvan-tages, such modification of the model could be fruit-ful.Replacing Brownian motion with a so-calledKac’s telegraph process T ( t ) , [13], is the most pop-ular approach.The telegraph process T ( t ) , t ≥ , describes asteady state motion of a particle with alternating atrandom times velocities, T ( t ) = Z t c ε ( s ) d s , (3)where ε = ε ( s ) ∈ { , } is a two-state Markov pro-cess with switching intensity λ , λ > . This math-ematical construct is useful on its own for applica-tions, for example, for physically oriented applica-tions, such as the description of photon transport in ahighly scattering medium, or of neutron transport ina reactor, [20, 34, 35]. This model has recently beenapplied to studies of cosmic microwave backgroundradiation studies based on the hyperbolic heat equa-tion, [4]. Application of telegraph processes in mod-elling of financial markets and related mathematicalinnovations are presented in [15, 25].In this paper, we study Markov-modulated Ornstein-Uhlenbeck processes where the Wienerprocess is replaced by Kac’s telegraph process. Wecall the result of such a replacement Kac-Ornstein-Uhlenbeck processes.This paper concerns the following principal top-ics. First, we are interested in first passage proba-bilities of the Kac-Ornstein-Uhlenbeck process, Sec-tions and . This subject is related to applicationsof persistent random walks which are still of inter-est, [17, 20, 21, 27, 33]. Second, we study invari-ant measures for Markov process which is formed bythe Kac-Ornstein-Uhlenbeck process X and the un-derlying state process ε , h X ( t ) , ε ( t ) i , Sections and. Third, the limit behaviour of X under some tradi-tional parameter scalings is also analysed, see Sec-tion . MODEL AND MAIN OBJECTIVES
On a given probability space ( Ω , F , P ) , considerthe standard Brownian motion W ( t ) , t ≥ , an irre-ducible continuous-time Markov chain ε = ε ( t ) , t ≥ , with a finite state space { , , . . . , d } and a ran-dom variable x , independent of each other.We define Markov-modulated Ornstein-Uhlenbeck process M = M ( t ) , assuming thatthe parameters a , b and γ of the Ornstein-Uhlenbeckprocess undergo synchronous switching driven bythe underlying process ε .Precisely, let a i , b i and γ i , i ∈ { , , . . . , d } , bearbitrary constants, b i ≥ . The process M = M ( t ) follows the stochastic equationd M ( t ) = (cid:0) a ε ( t ) − γ ε ( t ) (cid:1) d t + b ε ( t ) d W ( t ) , t > , (4)with the initial condition M ( ) = x . This initial valueproblem is equivalent to the integral equation M ( t ) = x + Z t (cid:0) a ε ( s ) − γ ε ( s ) M ( s ) (cid:1) d s + Z t b ε ( s ) d W ( s ) , (5) t ≥ . The study of Markov-modulated Ornstein-Uhlenbeck processes has recently begun, first in[11, 36], dealing with the transient behaviour of mo-ments and some specific scaling of parameters, andthen in [26–28] in terms of first passage distributionsand with neural modelling applications.The solution to the integral equation (5) can beexpressed by means of piecewise deterministic pro-cesses. Let Γ ( t ) = R t γ ε ( s ) d s and A ( t ) = R t a ε ( s ) d s be two piecewise linear processes based on the com-mon underlying Markov process ε . We define alsothe time reversal process e Γ ( s , t ) by setting e Γ ( s , t ) = Γ ( t ) − Γ ( s ) = R ts γ ε ( u ) d u , ≤ s ≤ t . The unique solu-tion to (5) is given by M ( t ) = x e − Γ ( t ) + Z t e − e Γ ( s , t ) d A ( s )+ Z t b ε ( s ) e − e Γ ( s , t ) d W ( s ) . (6)Conditionally (for given { ε ( s ) } s ∈ [ , t ] ) the randomvariable M ( t ) is Gaussian with (random) mean X ( t ) = E (cid:0) M ( t ) | { ε ( s ) } s ∈ [ , t ] (cid:1) and (random) vari-ance V ( t ) = Var (cid:0) M ( t ) | { ε ( s ) } s ∈ [ , t ] (cid:1) , X ( t ) = x e − Γ ( t ) + Z t a ε ( s ) e − e Γ ( s , t ) d s , (7) V ( t ) = Z t b ε ( s ) e − e Γ ( s , t ) d s . (8)See [11, Theorem 2.1].In what follows, we assume that the underlyingprocess ε = ε ( t ) ∈ { , } , t ≥ , is the two-state continuous-time Markov chain with transition inten-sities λ and λ , λ , λ > . Let Λ = − λ λ λ − λ and Π = Π ( t ) = ( P { ε ( t ) = j | ε ( ) = i } ) i , j ∈{ , } , t ≥ , be the matrix of transition probabilities. It isknown that, see e. g. [30], Π ( t ) = exp ( t Λ )=( λ ) − λ + λ e − λ t λ (cid:0) − e − λ t (cid:1) λ (cid:0) − e − λ t (cid:1) λ + λ e − λ t , where 2 λ = λ + λ . For arbitrary distribution ~ π ofthe initial state ε ( ) , the distribution of ε ( t ) is givenby ~ πΠ ( t ) and the limit ~ π ∗ = ~ π ∗ ( λ , λ ) = lim t → ∞ ~ πΠ ( t ) is given by π ∗ ( λ , λ ) = ( λ ) − ( λ , λ ) . (9)Let γ , γ = . The conditional mean X = X ( t ) ofthe Markov-modulated process M obeys the integralequation X ( t ) = x + Z t (cid:0) a ε ( s ) − γ ε ( s ) X ( s ) (cid:1) d s , t ≥ , (10)see (5). The process X = X ( t ) , t ≥ , (10), canbe viewed as a (non-Gaussian) Ornstein-Uhlenbeckprocess, controlled by two Kac’s telegraph pro-cesses, Γ ( t ) and A ( t ) = R t a ε ( s ) d s , t ≥ , insteadof Brownian motion. We call it the Kac-Ornstein-Uhlenbeck process.By virtue of (7), the process X = X ( t ) sequen-tially follows the two deterministic patterns, φ and φ , switching from one to another randomly after ex-ponentially distributed holding times. These patternsare defined by φ ( t , x ) = e − γ t (cid:18) x + a Z t e γ s d s (cid:19) = a γ + (cid:18) x − a γ (cid:19) e − γ t , (11)and, similarly, φ ( t , x ) = a γ + (cid:18) x − a γ (cid:19) e − γ t , t ≥ . (12)In short, the patterns φ and φ are determined by thefunction φ ( t , x ) = ρ + ( x − ρ ) e − γ t with two pairs ofparameters, h ρ , γ i and h ρ , γ i , alternating at ran-dom times when the underlying process ε switches.Here ρ = a / γ , ρ = a / γ . Both patterns, t → φ ( t , x ) and t → φ ( t , x ) , satisfy semigroup property.If a / γ = a / γ = : ρ , then the solution of (10)comes down to the exponential telegraph process.Namely, one can see that in this case, formula (7) issimplified as X ( t ) = e − Γ ( t ) (cid:18) x + ρ Z t γ ε ( s ) e Γ ( s ) d s (cid:19) = ρ + ( x − ρ ) exp ( − Γ ( t )) . (13)The distribution of such a process is well studied,see [15, 18, 25, 29]. In this case, process X , (13), istime-homogeneous in the sense of [29, (2.13)] withrectifying diffeomorphism Φ ( x ) = log | x − ρ | . We assume that a / γ = a / γ . To be specific, let ρ = a / γ < a / γ = ρ . We are interested in theprobabilities of the first passage of a fixed level y byprocess X .In the case when the parameters ρ , γ and the vari-ables x , y satisfy the conditions γ > , x − ρ y − ρ > γ < , < x − ρ y − ρ < , (14) the inverse function t ( x , y ) = φ ( · , x ) − ( y ) is positive, t ( x , y ) = φ ( · , x ) − ( y ) = γ log x − ρ y − ρ > . (15)If for the pair h ρ , γ i condition (14) is not satisfied,we set t ( x , y ) = + ∞ . By t ( x , y ) and t ( x , y ) we de-note the inverse functions to φ ( · , x ) and φ ( · , x ) , re-spectively, which are defined above, (15). The values t ( x , y ) and t ( x , y ) coincide with the shortest time forprocess X to reach level y , starting with x and withoutswitching states. First passage time
Let T ( x , y ) be the time when the process X = X ( t ) first passes through y , starting from x = X ( ) , T ( x , y ) = inf { t > X ( t ) = y | X ( ) = x } , x = y . (16)Random variable T ( x , y ) has an atomic value t ( x , y ) , t ( x , y ) > , if the particle starting at x reaches y with-out switching. Further, the distribution of T ( x , y ) isrenewal after each state switch.The following distribution identities hold: (cid:2) T ( x , y ) | ε ( ) = (cid:3) D = t ( x , y ) { τ ( ) > t ( x , y ) } + (cid:0) τ ( ) + h T ( φ ( τ ( ) , x ) , y ) | ε ( ) = i (cid:1) { τ ( ) < t ( x , y ) } , [ T ( x , y ) | ε ( ) = ] D = t ( x , y ) { τ ( ) > t ( x , y ) } + (cid:0) τ ( ) + h T ( φ ( τ ( ) , x ) , y ) | ε ( ) = i (cid:1) { τ ( ) < t ( x , y ) } . (17)Here, the exponentially distributed random variables τ ( ) and τ ( ) , τ ( ) ∼ Exp ( λ ) , τ ( ) ∼ Exp ( λ ) , do notdepend on further dynamics; [ T | ε ( ) = i ] denotesthe conditional distribution of T under the given ini-tial state ε ( ) = i . The first terms on the right-handsides of equations (17) are set to zero if the corre-sponding t i ( x , y ) becomes equal to + ∞ . The Laplace transforms ℓ ( q , x , y ) : = E [ exp ( − qT ( x , y ))] ,ℓ ( q , x , y ) : = E [ exp ( − qT ( x , y ))] serves cumulative distribution function for the run-ning minimum X e q : = min ≤ t ≤ e q X ( t ) if x > y , and com-plementary cumulative distribution function for therunning maximum X e q : = max ≤ t ≤ e q X ( t ) if x < y . In-deed, integrating by parts one can see, i ∈ { , } ,ℓ i ( q , x , y ) = Z ∞ e − qt d P { T ( x , y ) < t | ε ( ) = i } = Z ∞ q e − qt P { T ( x , y ) < t | ε ( ) = i } d t = P { T ( x , y ) < e q | ε ( ) = i } = P { X e q < y | ε ( ) = i } x > y , P { X e q > y | ε ( ) = i } x < y , Due to (17), functions ℓ and ℓ obey the coupledintegral equations, ℓ ( q , x , y ) = e − ( q + λ ) t ( x , y ) + Z t ( x , y ) λ e − ( q + λ ) τ ℓ ( q , φ ( τ , x ) , y ) d τ ,ℓ ( q , x , y ) = e − ( q + λ ) t ( x , y ) + Z t ( x , y ) λ e − ( q + λ ) τ ℓ ( q , φ ( τ , x ) , y ) d τ . (18)If condition (14) is not satisfied for a set of parame-ters x , y , ρ i , γ i , then t i ( x , y ) = + ∞ ; the first term onthe right-hand side of the corresponding equation of(18) vanishes, and the integral is taken over the entirehalf-line [ , + ∞ ) . Differentiating (18) with respect to x , and thenintegrating by parts we get the coupled differentialequations: ( x − ρ ) ∂ ℓ ∂ x ( q , x , y ) = − β ( q ) ℓ ( q , x , y )+ β ( ) ℓ ( q , x , y ) , ( x − ρ ) ∂ ℓ ∂ x ( q , x , y ) = β ( ) ℓ ( q , x , y ) − β ( q ) ℓ ( q , x , y ) . (19)Here, we use the identities ( x − ρ ) ∂φ ∂ x ( τ , x ) ≡ − γ ∂φ ∂τ ( τ , x ) , ( x − ρ ) ∂φ ∂ x ( τ , x ) ≡ − γ ∂φ ∂τ ( τ , x ) , For various combinations of parameters, system(19) should be considered in different domains withdifferent boundary conditions. Below, we give ex-plicit formulae for ℓ ( q , x , y ) and ℓ ( q , x , y ) with dif-ferent parameters.In the case of non-strict attraction/repulsion,equations (18) and (19) can be written similarly.For example, let γ = , a = + , γ > x < y . Hence, equations (18) hold with t ( x , y ) = + ∞ and φ ( τ , x ) = x + a τ . Equivalently, the first equation ofsystem (19) don’t change, while the second equationturns into ∂ ℓ ∂ x ( q , x , y ) = − λ ℓ ( q , x , y ) + ( q + λ ) ℓ ( q , x , y ) . (20) Invariant measures
Our second goal is to study invariant measuresfor X . Notice that Ξ ( t ) = h X ( t ) , ε ( t ) i ∈ R × { , } , t ≥ , is the Markov process. Let P ( t , d y | x ) be thetransition function, P ( t , d y | x ) = ( p i j ( t , x ; d y )) i , j ∈{ , } , where p i j ( t , x ; d y ) = P { X ( t ) ∈ d y , ε ( t ) = j | X ( ) = x , ε ( ) = i } . Let P t be the corresponding Markovsemigroup, ~ f → P t ~ f , where ( P t ~ f )( x ) = E (cid:16) ~ f ( Ξ ( t )) (cid:12)(cid:12)(cid:12) Ξ ( ) = h x , i i (cid:17) = Z ∞ − ∞ P ( t , d y | x ) ~ f ( y ) , for any test-function ~ f = ( f , f ) . The infinitesimalgenerator for the semigroup P t is determined by L = − λ + ( a − γ x ) dd x λ λ − λ + ( a − γ x ) dd x . Indeed, let f , f be a pair of test function. By virtueof (11), we get E [ f ε ( t ) ( X x ( t ))] − f ( x ) t = ( − λ t ) f ( φ ( t , x )) + λ t f ( x ) − f ( x ) t + o ( t )= λ f ( x ) − λ f ( x ) + f ′ ( x ) · ( a − γ x ) + o ( t ) , t → , which give the first row of the matrix L . The secondrow is obtained similarly by using (12).We study invariant measures, which are definedas fixed points of the adjoint semigroup P ∗ t . Namely,the invariant measure ~ µ = ( µ , µ ) , supported on aset K ⊂ R , is defined by the equation ~ µ ( d y ) = Z K P ( t , d y | x ) ~ µ ( d x ) , y ∈ K . Here K is an invariant set with respect to thetime evolution of X . When the invariant measure ~ µ is determined by the probability density function ~ π = ~ π ( x ) = ( π ( x ) , π ( x )) , this is equivalent to theboundary value problem for the ordinary differentialequation, see e.g. [22], L ∗ ~ π ( x ) = , x ∈ K . (21)Here L ∗ is the adjoint operator to the generator L , and the following assumptions hold: π ( x ) ≥ , π ( x ) ≥ , ∀ x ∈ K and Z K ( π ( x ) + π ( x )) d x = . The existence of the invariant distribution for theprocess X and its shape depends on signs of the pa-rameters γ , γ , which determine the boundary con-ditions to equation (21).The explicit form of the adjoint operator L ∗ andthe boundary conditions can be obtained by integrat-ing by parts in (cid:16) L ~ f ( x ) ,~ π ( x ) (cid:17) = Z K (cid:16) − λ f ( x ) + ( a − γ x ) f ′ ( x ) + λ f ( x ) (cid:17) π ( x ) d x + Z K (cid:16) λ f ( x ) − λ f ( x ) + ( a − γ x ) f ′ ( x ) (cid:17) π ( x ) d x for any test function ~ f = ( f ( x ) , f ( x )) . We have (cid:16) L ~ f ( x ) ,~ π ( x ) (cid:17) = (cid:2) ( a − γ x ) f ( x ) π ( x ) + ( a − γ x ) f ( x ) π ( x ) (cid:3) | x ∈ ∂ K + Z ∞ − ∞ f ( x ) (cid:2) − λ π ( x )+ λ π ( x )+ γ π ( x )+( γ x − a ) π ′ ( x ) (cid:3) d x + Z ∞ − ∞ f ( x ) (cid:2) λ π ( x ) − λ π ( x )+ γ π ( x )+( γ x − a ) π ′ ( x ) (cid:3) d x . (22)Therefore, the adjoint operator L ∗ is defined by thematrix L ∗ = ( γ − λ ) + ( γ x − a ) dd x λ λ ( γ − λ ) + ( γ x − a ) dd x , and the boundary conditions for (21) are supplied bysetting the non-integral terms of (22) to be zero.Precisely, we have the following system: ( γ x − a ) d π ( x ) d x = ( λ − γ ) π ( x ) − λ π ( x ) , ( γ x − a ) d π ( x ) d x = − λ π ( x ) + ( λ − γ ) π ( x ) , (23) x ∈ K , with the boundary conditions ( a − γ x ) π ( x ) | ∂ K = , ( a − γ x ) π ( x ) | ∂ K = . (24)Below, Section , we study the invariant measuresunder different combinations of parameters of themodel. We distinguish two main cases: the attrac-ting-only dynamics when both γ are positive and themixed attraction-repulsion case.The first passage probabilities are explored in thenext section. FIRST PASSAGE PROBABILITIES FOR X In this section, we obtain some explicit formu-lae for the Laplace transforms ℓ and ℓ of the firstpassage time T ( x , y ) , (16). We will consider two dif-ferent models, when the paths of X are attracted to both points ρ and ρ , γ , γ > , and the case whenone level attracts and the other repels, γ · γ < ξ ( x ) = x − ρ ρ − ρ , ξ ( x ) = − ξ ( x ) = ρ − x ρ − ρ and b , = (cid:18) β + β ± q ( β − β ) + λ λ / ( γ γ ) (cid:19) , (25)where β = β ( q ) = ( q + λ ) / γ , β = β ( q ) = ( q + λ ) / γ ;by F ( b , b ; b ; · ) we denote the Gaussian hypergeo-metric function, defined by the series F ( b , b ; b ; z ) = + ∞ ∑ n = ( b ) n ( b ) n ( b ) n n ! z n (26)if one of the following conditions holds: | z | < | z | = b − b − b > | z | = , z = − < b − b − b ≤ . (27)Here ( b ) n = b · ( b + ) · . . . · ( b + n − ) = Γ ( b + n ) / Γ ( b ) is the Pochhammer symbol. This function is definedby analytic continuation everywhere in z , z < − , see [1]. Attracting-only case, γ , γ > . In this case, both parameters γ and γ are re-garded as positive revertive rates, and both patterns, φ and φ , defined by (11)-(12), converge as t → ∞ , lim t → ∞ φ ( t , x ) = a γ = : ρ , lim t → ∞ φ ( t , x ) = a γ = : ρ . The interval [ ρ , ρ ] serves as an attractor for thepaths of X : if the process starts at point x outsidethis interval, x / ∈ [ ρ , ρ ] , it falls into [ ρ , ρ ] a.s.in a finite time. Moreover, once caught, the processremains there forever, see [26]. In this regard, westudy the first passage through the threshold y , ρ < y < ρ . A sample path is shown in Fig. 1. τ τ τ τ Xx ρ ρ y T ( x , y ) FIG. 1. A sample path of X = X ( t ) , γ , γ > . The first passage time T ( x , y ) , (16), is finite a.s. ∀ x . The distribution of T ( x , y ) can be studied sepa-rately for x < y and x > y . In both these cases, func-tions ℓ and ℓ corresponding to the Laplace trans-form obey the boundary value problems for equa-tions (19) on the half-lines x < y and x > y , respec-tively.First, let x < y . Since both levels, ρ and ρ , at-tract, then t ( x , y ) = + ∞ , t ( x , y ) = γ log x − ρ y − ρ < ∞ . If the initial state is 1 = ε ( ) , then T ( x , y ) → a . s . as x ↑ y . The latter gives the boundary condition ℓ ( q , x , y ) | x ↑ y = x < y with boundary condition (28), writing the solution in the form ℓ ( q , x , y ) = ∞ ∑ n = A n ( q , y ) ξ ( x ) n ,ℓ ( q , x , y ) = ∞ ∑ n = B n ( q , y ) ξ ( x ) n .. (29)Substituting functions ℓ and ℓ , defined by series(29), into (19) and using the identities ( x − ρ ) d ξ ( x ) d x ≡ ξ ( x ) , ( ρ − ρ ) d ξ ( x ) d x ≡ , we obtain nA n = − β ( q ) A n + λ γ B n , nB n − ( n + ) B n + = λ γ A n − β ( q ) B n . (30)After a simple algebra, see e.g. [26], we find thesolution of system (30): A n = λ q + λ · ( b ) n ( b ) n ( + β ) n n ! B , B n = ( b ) n ( b ) n ( β ) n n ! B , n ≥ . (31)Here, recall, β = β ( q ) = ( q + λ ) / γ and b , b aredefined by (25).Due to (27), series (29) with coefficients A n and B n determined by (31) converge if | ξ ( x ) | < , thatis, if 2 ρ − ρ < x < ρ . (32)Therefore, the Laplace transform of the first passagetime T ( x , y ) , y ∈ ( ρ , ρ ) , ρ − ρ < x < y , is ex-pressed through the Gaussian hypergeometric series,(26), ℓ ( q , x , y ) = λ q + λ F ( b , b ; 1 + β ; ξ ( x )) · B ,ℓ ( q , x , y ) = F ( b , b ; β ; ξ ( x )) · B . (33)The indefinite parameter B follows from theboundary condition (28). Thus, we finally obtain theexplicit formulae for ℓ and ℓ in the case ρ ≤ y < ρ , ρ − ρ < x < y ,ℓ ( q , x , y ) = λ q + λ · F ( b , b ; 1 + β ( q ) ; ξ ( x )) F ( b , b ; β ( q ) ; ξ ( y )) ,ℓ ( q , x , y ) = F ( b , b ; β ( q ) ; ξ ( x )) F ( b , b ; β ( q ) ; ξ ( y )) . (34)In the case ρ < y ≤ ρ , x > y formulae for ℓ ( q , x , y ) and ℓ ( q , x , y ) can be obtained by sym-metry in the form of series with ξ ( x ) . We have ℓ ( q , x , y ) | x ↓ y = ℓ ( q , x , y ) = F ( b , b ; β ( q ) ; ξ ( x )) F ( b , b ; β ( q ) ; ξ ( y )) ,ℓ ( q , x , y ) = λ q + λ · F ( b , b ; 1 + β ( q ) ; ξ ( x )) F ( b , b ; β ( q ) ; ξ ( y )) , (35) ρ < y < x < ρ − ρ , y ≤ ρ . Note that for y ∈ ( ρ , ρ ) we have 0 < ξ ( y ) , ξ ( y ) < , therefore the denominators in (34)and (35) are also determined by Gaussian hypergeo-metric series (26).In the case of non-strict attraction, the Laplacetransforms ℓ and ℓ can be obtained similarly to (34)and (35). For example, let γ = , a = + , γ > x < y . We have: ℓ ( q , x , y ) = λ / γ β ( q ) Φ ( δ ; 1 + β ( q ) ; ( x − ρ )( q + λ )) Φ ( δ ; β ( q ) ; ( y − ρ )( q + λ ))) ,ℓ ( q , x , y ) = Φ ( δ ; β ( q ) ; ( x − ρ )( q + λ )) Φ ( δ ; β ( q ) ; ( y − ρ )( q + λ ))) , where δ = ( q + λ )( q + λ ) − λ λ γ ( q + λ ) and Φ ( · ; · ; · ) isthe confluent hypergeometric function. Attraction-repulsion: γ and γ have opposite signs,“raznotyk” To be specific, assume that γ > > γ , thatis, the pattern φ is repelled from the threshold ρ , while φ is attracted to ρ . In this case, process X = X ( t ) , t ≥ , a.s. falls under ρ , into the half-line { z | z ≤ ρ } , in a finite time and, once falling,remains there forever, see Fig. 2. Let the threshold y belong to the attractor, y < ρ . Similarly to the caseof two attractive levels, we obtain the boundary con-ditions in dependence of the starting point x . τ τ τ X ρ ρ y T ( x , y ) FIG. 2. The sample path of X = X ( t ) , γ > > γ . If the process begins with 0-state from below ofthreshold y , x < y , then T ( x , y ) → , a . s . as x ↑ y . For x < y , we must consider the system (19) with theboundary condition ℓ ( q , x , y ) | x ↑ y = , (36)which gives the solution of the form (33), 2 ρ − ρ < x < y < ρ . By the boundary condition (36) we ob-tain B = (cid:18) λ q + λ F ( b , b ; 1 + β ; ξ ( x )) (cid:19) − ℓ ( q , x , y ) = F ( b , b ; 1 + β ; ξ ( x )) F ( b , b ; 1 + β ; ξ ( y )) ,ℓ ( q , x , y ) = q + λ λ · F ( b , b ; β ; ξ ( x )) F ( b , b ; 1 + β ; ξ ( y )) , ρ − ρ < x < y < ρ . Acting in the similar way, we find that in the case2 ρ − ρ < y < x < ρ , y < ρ we obtain, ℓ ( q , x , y ) = q + λ λ · F ( b , b ; β ; ξ ( x )) F ( b , b ; 1 + β ; ξ ( y )) ,ℓ ( q , x , y ) = F ( b , b ; 1 + β ; ξ ( x )) F ( b , b ; 1 + β ; ξ ( y )) . In the case of γ < < γ , the formulae for thedistribution of T ( x , y ) , y > ρ , are symmetric. INVARIANT MEASURES
Like the distribution of the first passage time, theform of the invariant measure also differs in the caseof positive values of γ , γ and in the case of oppositesigns. Attracting-only case, γ , γ > Since the paths of X remain inside the inter-val ( ρ , ρ ) after an almost surely finite transi-tion time, the invariant measure ~ µ is supported on [ ρ , ρ ] , ρ = a / γ , ρ = a / γ . As it was shown in Section , the invariant prob-ability density function ~ π = ~ π ( x ) = ( π ( x ) , π ( x )) obeys the system (23) of the ordinary differentialequations, ρ < x < ρ . This system is supplied with the boundary conditions, see (24), π ( x ) | x = ρ − = , ( x − ρ ) π ( x ) | x = ρ + = , ( ρ − x ) π ( x ) | x = ρ − = , π ( x ) | x = ρ + = . (37)By substituting π ( x ) = C ( x − ρ ) k ( ρ − x ) k { ρ < x < ρ } , π ( x ) = C ( x − ρ ) k ( ρ − x ) k { ρ < x < ρ } , into equations (23) and taking into account theboundary conditions (37), we obtain π ( x ) = c γ − ( x − ρ ) − + α ( ρ − x ) α { ρ < x < ρ } , π ( x ) = c γ − ( x − ρ ) α ( ρ − x ) − + α { ρ < x < ρ } , where α = λ / γ , α = λ / γ , α , α > . The nor-malising constant c can be found from the equality Z ρ ρ [ π ( x ) + π ( x )] d x = . Due to [10, 3.196] we obtain c − =( ρ − ρ ) α + α h γ − B ( α , α + ))+ γ − B ( α + , α )) i , (38)where B ( · , · ) is Euler’s beta-function. Example .1
Let γ = γ = γ > . In this case , by (38) c − = ( ρ − ρ ) γ − and π ( x ) =( ρ − ρ ) − ( ρ − x ) , π ( x ) =( ρ − ρ ) − ( x − ρ ) , ρ < x < ρ . It’s curious that when the process X begins withstates 0 or 1 with equal probability , the invariant dis-tribution is uniform on [ ρ , ρ ] . Attraction-repulsion: γ and γ have opposite signs,“raznotyk” First, let γ > > γ . In this case, after an almostsurely finite transition period, process X falls into thehalf-line { x < ρ } , see Fig. 2. The invariant distribu-tions are defined by the probability density functions ~ π = ( π ( x ) , π ( x )) , satisfying system (23), x < ρ , with the boundary conditions ( ρ − x ) π ( x ) | x = ρ = , ( ρ − x ) π ( x ) | x = ρ = . The solution is given by π ( x ) = C ( ρ − x ) − + α ( ρ − x ) α { x < ρ } , π ( x ) = C ( ρ − x ) α ( ρ − x ) − + α { x < ρ } , (39)where C = c γ − , C = − c γ − , C , C > , and α = λ / γ , α = λ / γ , α > > α . In the dual case, γ < < γ , process X is cap-tured by the upper half-line, K = { x > ρ } . The in-variant probability density function is determined by π ( x ) = C ( x − ρ ) − + α ( x − ρ ) α { x > ρ } , π ( x ) = C ( x − ρ ) α ( x − ρ ) − + α { x > ρ } , (40)where C = − c γ − , C = c γ − . In both cases, we assume that α + α < . (41)The normalising constants c and c are determinedby the condition Z K [ π ( x ) + π ( x )] d x = . (42)The integral in (42) converges if (41) holds. In thecase γ > > γ , the normalising constant c for (39) is determined by c − = ( ρ − ρ ) α + α (cid:0) γ − B ( − α − α , α ) − γ − B ( − α − α , + α ) (cid:1) , α + α < , α > . In the symmetric case γ < < γ , the normalisingconstant c for (40) is determined by c − = ( ρ − ρ ) α + α (cid:0) − γ − B ( − α − α , + α )+ γ − B ( − α − α , α ) (cid:1) . α + α < , α > . If, on the contrary, (41) is not met, the invariantprobability distribution does not exist.Repulsion-only case, γ , γ < , corresponds toa subordinator, that is, all paths of X ( t ) , t ≥ − X ( t ) , t ≥
0) are strictly monotonically increasing.Therefore, there are no invariant probability distribu-tions.
Non-strict attraction
If one of attraction rates is zero, say γ = , and γ > , then the pattern φ defined by (11) is attrac-tive, and φ ( t , x ) ≡ x + a t . τ τ X ρ ρ yx τ FIG. 3. The sample path of X = X ( t ) , γ > = γ , a = . a = a is positive), a = − a is negative). Regardless of the initial point x , the trajectories of X (possibly, after a finite transi-tion period) remain above (or below) the threshold ρ = ρ = a / γ in the case a = + a = − π ( x ) , π ( x ) follow system (23) on x < ρ , if a = − x > ρ ,if a = + ( x − ρ ) d π ( x ) d x = (cid:18) λ γ − (cid:19) π ( x ) − λ γ π ( x ) , − a d π ( x ) d x = − λ π ( x ) + λ π ( x ) . In both cases, the boundary condition (24) turns into ( ρ − x ) π ( x ) | x = ρ = , π ( x ) | x = ρ = . (43)The solutions of these two boundary value problemsare given by π ± ( x ) = C | x − ρ | α − e − λ | x − ρ | θ ± ( x ) , π ± ( x ) = γ C | x − ρ | α e − λ | x − ρ | θ ± ( x ) , (44)where α = λ / γ , ρ = a / γ ; C is a normalisingconstant. Functions θ − ( x ) = { x < ρ } and θ + ( x ) = { x > ρ } specify the measure support below and, re-spectively, above the level ρ . The explicit value ofnormalising constant C follows from the equalities1 = Z ∞ρ [ π + ( x ) + π + ( x )] d x = C λ − α Γ ( α ) + γ C λ − ( α + ) Γ ( α + )= C λ − α Γ ( α ) [ + λ / λ ] , which gives C = λ + α ( λ + λ ) Γ ( α ) . The proof for the case a = − Remark .1
When γ = and a = , system (23) turn into ( x − ρ ) π ′ ( x ) = − π ( x ) , = − λ π ( x ) + λ π ( x ) , with the boundary conditions (43) . This means π = π = , i.e. in this case there is no invariant proba-bility measure. SCALING
It is known that under the Kac scaling, see [13],that is c , − c → + ∞ , λ , λ → + ∞ such that c / λ , c / λ → σ , (45)the telegraph process T converges in distribution on C ([ , T ] ; R ) (equipped with the sup-norm) to thescaled Brownian motion σ W ( t ) , see the proof in[13, 24]. We apply this idea to a substantially asym-metric telegraph process.First, we speed up the underlying Markov chain.Let ε be driven by alternating switching intensities λ , λ which are high but comparable, i.e. λ , λ → ∞ and λ λ → ν , ν > . (46)With this scaling, the invariant distribution of the un-derlying process ε , π ∗ ( λ , λ ) , (9), becomes π ∗ ( λ , λ ) → π ∗∗ = (cid:0) ( + ν ) − , ν ( + ν ) − (cid:1) . If the velocities c , c remain constants, the process T ( t ) , t > , (3), converges in probability to c ∗ t , T ( t ) = Z t c ε ( s ) d s → c ∗ t , t ∈ [ , T ] , (47)where c ∗ = ~ π ∗∗ · ~ c . ~ a = ( a , a ) , ~ b = ( b , b ) , ~ γ =( γ , γ ) of process M , (6), remain constant. Withthis scaling, process M weakly converges to process M ∗ , which is an ordinary (unmodulated) Ornstein-Uhlenbeck process defined by the Langevin equa-tion, (5), with constant deterministic parametersa ∞ = ~ π ∗∗ ~ a , b ∞ = ~ π ∗∗ ~ b , and γ ∞ = ~ π ∗∗ ~ γ , d M ∗ ( t ) = ( a ∞ − γ ∞ M ∗ ( t )) d t + b ∞ d W ( t ) , t > . See [11, Corollary 5.1].Further, let the scaling condition similar to (45)be satisfied separately for the two states, i. e. let (46)holds, and c → + ∞ , c → − ∞ , so that c √ λ → σ , c √ λ → − σ , (48)where σ , σ >
0. From (46)-(48) it follows that thevelocities are also comparable: c c = c / √ λ c / √ λ p λ / λ → − νσ / σ . We also assume that λ c + λ c λ + λ → δ . (49)The latter limit relation is equivalent to c c / c + λ / λ + λ / λ → δ . Therefore, condition(49) assumes that c / c + λ / λ → . So, condition(49) reads as rate of “similarity” between λ / λ and c / c at infinity. More precisely, c ( c / c + λ / λ ) → δ ( + ν ) . Under the scaling conditions (46)-(49) statedabove, the telegraph process T ( t ) weakly convergesto the Wiener process with drift, see [18], T ( t ) = Z t c ε ( s ) d s → σ W ( t ) + δ t , t > . (50) Here W = W ( t ) is the standard Wiener process and σ = σ σ q ( σ + σ ) / . (51)Assuming (46) to be hold, we apply scaling con-ditions similar to (48)-(49) in several versions. (a) Let a → − ∞ , a → + ∞ , (52)and the consistency condition similar to (48)holds, i. e., a √ λ → − σ , a , a √ λ → σ , a . (53)Let the additional drift be caused by conditionof the form (49), λ a + λ a λ + λ → δ a . (54)Leaving the remaining parameters constant,we see that due to (50), process M ( t ) , (4),weakly converges to the solution M a ( t ) , t > , of the equationd M a ( t ) = ( δ a − γ ∞ M a ( t )) d t + σ a d e W ( t ) + b ∞ d W ( t ) , (55)where e W is the Brownian motion independentof W , and σ a is defined by formula (51) with σ = σ , a and σ = σ , a . (b) Let instead of (52), we assume ~ a and ~ b to beconstant, but γ → − ∞ , γ → + ∞ , (56)with the consistency conditions γ √ λ → − σ , γ , γ √ λ → σ , γ (57)4and additional drift caused by λ γ + λ γ λ + λ → δ γ . (58)With this scaling, similarly to (55), we obtainthe limiting process M b satisfying the equa-tiond M b ( t ) = (cid:0) a ∞ − δ γ M b ( t ) (cid:1) d t − σ γ M b ( t ) d e W ( t ) + b ∞ d W ( t ) , t > , where e W is the Brownian motion independentof W and σ γ is defined by formula (51) with σ = σ , γ and σ = σ , γ . (c) Let both (52)-(54) and (56)-(58) hold simul-taneously, and ~ b be constant.Then M converges to M c determined by theequation, t > , d M c ( t ) = (cid:0) δ a − δ γ M c ( t ) (cid:1) d t − σ γ M c ( t ) d e W ( t ) + σ a d ee W ( t ) + b ∞ d W ( t ) , with three independent Brownian motions, W , e W and ee W . ACKNOWLEDGEMENTS
The work was supported by Russian Foundationfor Basic Research (RFBR) and Chelyabinsk Re-gion, project number 20-41-740020. [1] G. E. Andrews, R. Askey and R. Roy.
Special Func-tions. Encyclopedia of Mathematics and its Applica-tions (Vol 71), Cambridge University Press, 1999. [2] O. E. Barndorff-Nielsen and N. Shephard. Non-Gaussian Ornstein-Uhlenbeck-based models andsome of their uses in financial economics.
J. R. Stat.Soc. Ser. B Stat. Methodol. , 63(2), 167-241, 2001.[3] F. Biagini, Yaozhong Hu, B. Øksendal and TushengZhang.
Stochastic Calculus for Fractional Brown-ian Motion and Applications.
Springer-Verlag, Lon-don, 2008.[4] Ph. Broadbridge, A. D. Kolesnik, N. Leonenko andA. Olenko. Random spherical hyperbolic diffusion.
J. Stat. Phys.
Ricerche DiMatematica
The Langevin Equation: with applications tostochastic problems in physics, chemistry and elec-trical engineering , 2nd ed, World Scientific, 2004.[7] H. Fink and C. Kl¨uppelberg. Fractional L´evy-drivenOrnstein-Uhlenbeck processes and stochastic differ-ential equations.
Bernoulli
Phys. Rev. A
46, 707(R),1992.[9] S. K. Foong and S. Kanno. Properties of the telegra-pher’s random process with or without a trap,
Stoch.Proc. Appl.
Table of In-tegrals, Series and Products . Academic Press,Boston, 1994.[11] G. Huang, H.M. Jansen, M. Mandjes, P. Spreijand K. De Turk. Markov-modulated Ornstein-Uhlenbeck processes.
Adv. Appl. Probab.
48, 235–254, 2016.[12] M. Jacobsen. Laplace and the origin of the Ornstein-Uhlenbeck process.
Bernoulli , 2(3), 271–286, 1996. [13] M. Kac. A stochastic model related to the telegra-pher’s equation. Rocky Mountain J Math
4: 497–509, 1974. Reprinted from: M. Kac,
Some stochas-tic problems in physics and mathematics.
Collo-quium lectures in the pure and applied sciences, No.2, hectographed, Field Research Laboratory, So-cony Mobil Oil Company, Dallas, TX, pp. 102–122,1956.[14] M. L. Kleptsyna and A. Le Breton. Statistical anal-ysis of the fractional Ornstein- Uhlenbeck type pro-cess.
Stat. Inference Stoch. Process , 5, 229–248,2002.[15] A. D. Kolesnik and N. Ratanov.
Telegraph Pro-cesses and Option Pricing.
Springer-Verlag Heidel-berg, New York, Dordrecht, London, 2013.[16] Yu. A. Kutoyants.
Statistical Inference for ErgodicDiffusion Processes.
Springer-Verlag London 2004.[17] H. Larralde. First-passage probabilities and meannumber of sites visited by a persistent randomwalker in one- and two-dimensional lattices.
Phys.Rev. E
Journal of Applied Probability ,51(2), (2014), 569–589.[19] R. A. Maller, G. M¨uller and A. Szimayer.
Ornstein-Uhlenbeck processes and extensions.
In: MikoschT., Kreiß JP., Davis R., Andersen T. (eds)
Handbookof Financial Time Series . Springer-Verlag Berlin,Heidelberg, pp 421–437, 2009.[20] J. Masoliver and G. H. Weiss. First passage timesfor a generalized telegrapher’s equation.
Physica A
Phys. Rev. E
Stochastic Processes and Applica-tions. Diffusion Processes, the Fokker-Planck andLangevin Equations.
Springer-Verlag New YorkHeidelberg Dordrecht London, 2014.[23] L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity.I. Mean and variance of the ring time,
Biol. Cybern.
35, 1–9, 1979.[24] N. Ratanov. Telegraph evolutions in inhomogeneousmedia.
Markov Proc. Related Fields , , No 1, 53-68,1999.[25] N. Ratanov. A jump telegraph model for option pric-ing. Quantitative Finance , 7(5), 575-583, 2007.[26] N.Ratanov Ornstein-Uhlenbeck processes ofbounded variation.
Methodol. Comput. Appl.Probab.
BioSystems , 2020.[28] N. Ratanov. First passage time for mean-revertingprocesses with bounded variation. The 5th Interna-tional Conference on Stochastic Methods (ICSM-5), Proceedings, 168-172, 2020.[29] N. Ratanov, A. Di Crescenzo and B. Martinucci.Piecewise deterministic processes following two al-ternating patterns.
J. Appl. Prob.
56: 1006–1019,2019.[30] S. M. Ross.
Stochastic Processes. (2d Ed.) Wiley,1995.[31] G. E. Uhlenbeck and L. S. Ornstein, On the theoryof Brownian motion.
Phys. Rev. , 823–841,1930.[32] O. Vasicek. An equilibrium characterization of theterm structure. J. Financ. Econ. .5(2), 177–188,1977.[33] G. H. Weiss. First passage time problems for one-dimensional random walks,
J. Stat. Phys.
24, 587–594, 1981. [34] G. H. Weiss. Aspects and Applications of the Ran-dom Walk.
North-Holland, Amsterdam, 1994.[35] G. H. Weiss Some applications of persistent randomwalks and the telegrapher’s equation
Physica A