On telegraph processes, their first passage times and running extrema
aa r X i v : . [ m a t h . P R ] F e b ON TELEGRAPH PROCESSES, THEIR FIRST PASSAGE TIMES AND RUNNING EXTREMA
NIKITA RATANOVCHELYABINSK STATE UNIVERSITY, 129, BR.KASHIRIHYKH, CHELYABINSK, 454001, [email protected]
BSTRACT . In this note, we present some ideas for describing the distributions of the running maximum/minimum,first passage times and telegraphic meanders. Explicit formulae for joint distribution of the extrema, the numberof velocity switches and the terminal position are derived using coupled integral equations technique.
Keywords:
Asymmetric piecewise linear process; First passage time; Kac’s scaling; Coupled integralequations; Telegraphic meander; Running maximum1. S
OME PRELIMINARIES
Let ε = ε ( t ) ∈ { , } , t ≥ , be a two-state Markov process defined on the complete probability space ( Ω , F , P ) and governed by two alternating parameters λ , λ , P { ε ( t + d t ) = i | ε ( t ) = i } = exp ( − λ i d t ) + o ( d t ) , d t → , i ∈ { , } . Let P i {·} be the conditional probability P {· | ε ( ) = i } under the given initial state ε ( ) = i . Denote by τ < τ < τ . . . the switching times, τ = . Let N = N ( t ) be a right-continuous process thatcounts the switchings occurring up to time t , N = N ( t ) = inf { n ≥ | τ n < t } , t > , N ( ) = . Consider the piecewise linear process Γ = Γ ( t ) = R t γ ε ( s ) d s , t ≥ , which describes the evolution of theparticle moving with two velocities γ , γ , γ > γ , alternating at random times τ n , n ≥ . Process Γ = Γ ( t ) can be regarded as an inhomogeneous asymmetric (integrated) telegraph process. Throughout the paper, weuse the following notations:(1.1) ξ = ξ ( t , x ) = x − γ t γ − γ , ξ = t − ξ ( t , x ) = γ t − x γ − γ , (1.2) z ( t , x ) = λ λ ξ ( t , x ) ξ ( t , x ) , γ t < x < γ t , and(1.3) θ ( t , x ) = exp ( − λ ξ ( t , x ) − λ ξ ( t , x )) γ − γ . Notice that in the case of homogeneous telegraph process, that is if λ = λ = λ and γ = − γ = γ , thesenotations are simplified to ξ ( t , x ) = γ t + x γ , ξ ( t , x ) = γ t − x γ , z ( t , x ) = λ ( γ t − x ) , θ ( t , x ) = e − λ t γ . For instance, in these terms can be described the well-known joint distribution of Γ ( t ) and N ( t ) , P i ( d x ; t , x ; n ) = P i { Γ ( t ) ∈ d x , N ( t ) = n } , n ≥ , i ∈ { , } . When there is no velocity switching, the particle moves along thestraight line, and the distribution of the particle’s position is singular,(1.4) P ( d x ; t , x ; 0 ) = e − λ t δ γ t ( d x ) , P ( d x ; t , x ; 0 ) = e − λ t δ γ t ( d x ) , The work was supported by Russian Foundation for Basic Research (RFBR) and Chelyabinsk Region, project number 20-41-740020. where δ a ( d x ) is the (atomic) Dirac delta-measure. The probability density functions p i ( t , x ; n ) = P i ( d x ; t , x ; n ) / d x , n ≥ , can be written in terms of (1.1)-(1.3) separately for even and odd number of velocity switchings, see e.g. [7,(4.1.10)-(4.1.11)]:(1.5) p i ( t , x ; 2 n + ) = λ i z ( t , x ) n n ! · θ ( t , x ) { γ t < x < γ t } , p i ( t , x ; 2 n + ) = λ λ ξ i ( t , x ) z ( t , x ) n n ! ( n + ) ! · θ ( t , x ) { γ t < x < γ t } , i ∈ { , } , n ≥ . Summing up, one can obtain the distribution of Γ ( t ) , t > , (1.6) P { Γ ( t ) ∈ d x } = e − λ t δ γ t ( d x ) + λ h I ( t , x ) d x + λ ξ ( t , x ) I ( t , x ) i θ ( t , x ) { γ t < x < γ t } , P { Γ ( t ) ∈ d x } = e − λ t δ γ t ( d x ) + λ h I ( t , x ) d x + λ ξ ( t , x ) I ( t , x ) i θ ( t , x ) { γ t < x < γ t } . Here functions I and I are defined by the series,(1.7) I ( t , x ) = ∞ ∑ n = z ( t , x ) n n ! = I (cid:0) √ z (cid:1) | z = z ( t , x ) , I ( t , x ) = ∞ ∑ n = z ( t , x ) n n ! ( n + ) ! = I ( √ z ) √ z | z = z ( t , x ) , where I and I are the modified Bessel functions.The main purpose of this note is to provide explicit formulae for the joint distribution of Γ ( t ) , the runningmaximum/minimum of Γ , and the point at which the extremum is reached. We also introduce and study atelegraphic meander, which resembles the known brownian meanders, see e.g. [12].Some preliminary results on the distribution of the maximum are known beginning with pioneering worksby [10, 11, 5]. These results were later generalised to the inhomogeneous asymmetric case by [1, 2, 7].Recently, a detailed analysis of a homogeneous case with drift was carried out in a series of papers by [3].These results are heavily based on the fact that the arrival times T , . . . , T n of the corresponding homogeneousPoisson process N are uniformly distributed for a given number N ( t ) = n . Unfortunately, the meticulous tech-nique used in these papers does not work if λ = λ . In this truly inhomogeneous case, a simple computationof conditional distributions under given number of velocity turns, { N ( t ) = n } , is unavailable, since arrivalsof the counting process N are distributed not uniformly. An explicit form of the conditional distribution of Γ ( t ) for a given value of N ( t ) can be found at the end of Section 2 of [9].The text is organised as follows. First, we present a generalisation and detailing of known results onthe distribution of first passage times (Section 2). Thereafter, the explicit representation of the meanderdistribution serves as an essential tool for obtaining the main result, Section 3.2. F IRST PASSAGE OF A TELEGRAPH PROCESS THROUGH A THRESHOLD
Let T ( y ) be the time of the first passage by process Γ = Γ ( t ) through the given threshold y , T ( y ) = min { t > | Γ ( t ) = y } , and F i ( d t , t , y ) = P i { T ( y ) ∈ d t } , i ∈ { , } , determine its distribution.Let F i ( d t , t , y ; n ) = P i { T ( y ) ∈ d t , N ( t ) = n } , n ≥ , i ∈ { , } , reflect the joint distribution of the firstpassage time, T ( y ) , and the total number of velocity switchings, N ( t ) , occurring up to time T ( y ) . Similarlyto (1.4), if there is no switching, then the distribution of T ( y ) is singular,(2.1) F i { d t ; t , y ; 0 } = e − λ i t δ y / γ i ( d t ) , i ∈ { , } . As before, the probability density functions, f ( t , y ; n ) and f ( t , y ; n ) , f i ( t , y ; n ) = F i ( d t ; t , y ; n ) / d t , n ≥ , i ∈ { , } , can be written separately for different initial states and for an even and odd number of velocity changes. N TELEGRAPH PROCESSES, THEIR FIRST PASSAGE TIMES AND RUNNING EXTREMA 3
Theorem 2.1. • Let y and both velocities γ , γ have the same sign , and ∆ ( y ) be the support of thedistribution of T ( y ) , that is, a segment with ends at the points y / γ and y / γ . Then T ( y ) is a. s.bounded , P { T ( y ) ∈ ∆ ( y ) } = , and (2.2) f i ( t , y ; 2 n + ) = λ i | γ − i | z ( t , y ) n n ! θ ( t , y ) { t ∈ ∆ ( y ) } , f i ( t , y ; 2 n + ) = λ λ | γ i | ξ i ( t , y ) z ( t , y ) n n ! ( n + ) ! θ ( t , y ) { t ∈ ∆ ( y ) } , i ∈ { , } , n ≥ . Further ,(2.3) F ( d t ; t , y ) = e − λ y / γ δ y / γ ( d t )+ λ (cid:16) | γ | I ( t , y ) + λ | γ | ξ ( t , y ) I ( t , y ) (cid:17) θ ( t , y ) { t ∈ ∆ ( y ) } d t , F ( d t ; t , y ) = e − λ y / γ δ y / γ ( d t )+ λ (cid:16) | γ | I ( t , y ) + λ | γ | ξ ( t , y ) I ( t , y ) (cid:17) θ ( t , y ) { t ∈ ∆ ( y ) } d t , In the cases y < < γ ≤ γ and y > > γ ≥ γ , the level y is never reached , T ( y ) = ∞ a . s . • Let the velocities be of opposite signs , γ > > γ , and y > . Then , we havef ( t , y ; 2 n + ) ≡ , f ( t , y ; 2 n + ) ≡ and f ( t , y ; 2 n + ) = λ ξ ( t , y ) · z ( t , y ) n { t > y / γ } n ! θ ( t , y ) · (cid:18) y − γ n + ξ ( t , y ) (cid:19) , (2.4) f ( t , y ; 2 n + ) = λ λ y · z ( t , y ) n n ! ( n + ) ! θ ( t , y ) { t > y / γ } , n ≥ . (2.5) Here ξ ( t , y ) , ξ ( t , y ) , z ( t , y ) and θ ( t , y ) are defined by (1.1) - (1.3). Remark 2.1.
In the case of velocities with opposite signs , γ > > γ , the formulae for f ( t , y ; n ) andf ( t , y ; n ) , n ≥ , with negative threshold y , turn out to be symmetric to (2.4) - (2.5),(2.6) f ( t , y ; n ) | y < = f ↔ ( t , − y ; n ) , f ( t , y ; n ) | y < = f ↔ ( t , − y ; n ) , where f ↔ ( t , · ; n ) and f ↔ ( t , · ; n ) are determined by (2.4) - (2.5) with the following interchange of parameters : γ → − γ , γ → − γ and λ ↔ λ . Note that after these conversions , we also have the interchange ξ ( t , y ) ↔ ξ ( t , − y ) , ξ ( t , y ) ↔ ξ ( t , − y ) . Further , summing up , we obtainF ( d t ; t , y ) = e − λ y / γ δ y / γ ( d t ) + λ λ y I ( t , y ) θ ( t , y ) { t > y / γ } d t , y > , λ ξ ( t , y ) ( − y I ( t , y ) + γ ξ ( t , y ) I ( t , y )) θ ( t , y ) { t > y / γ } d t , y < . (2.7) F ( d t ; t , y ) = λ ξ ( t , y ) (cid:16) y I ( t , y ) − γ ξ ( t , y ) I ( t , y ) (cid:17) θ ( t , y ) { t > y / γ } d t , y > , e − λ y / γ δ y / γ ( d t ) − λ λ y I ( t , y ) θ ( t , y ) { t > y / γ } , y < , (2.8) cf. [9, Theorem 3.1]. Proof.
First, let both velocities be positive, γ > γ >
0. In this case, T ( y ) , y > , is a. s. bounded, process Γ ( t ) , t ≥ , is a subordinator, and, hence, f i ( t , y ; n ) d t ∼ p i ( t , y ; n ) d y , d t → , where d y is the incrementof Γ ( t ) corresponding to the time increment d t . Therefore, f i ( t , y ; n ) = γ ε ( t ) p i ( t , y ; n ) , and formulae (2.2), γ > γ > , y > , follow from (1.5). The case of both negative velocities is symmetric. Expressions (2.3)for F ( d t ; t , y ) and F ( d t ; t , y ) follow by summing the formulae (2.2). NIKITA RATANOV CHELYABINSK STATE UNIVERSITY, 129, BR.KASHIRIHYKH, CHELYABINSK, 454001, RUSSIA [email protected]
Let the velocities have opposite signs, γ > > γ , and the threshold is positive, y > . Since the firstcrossing of a positive threshold always occurs at the positive velocity, we have F ( d t ; t , y ; 0 ) | y > ≡ , f ( t , y ; 2 n + ) | y > ≡ , f ( t , y ; 2 n + ) | y > ≡ , n ≥ , y > . Further, note that starting from the state 0 = ε ( ) , the particle first crosses the positive level y only after aneven number of switchings, and the first turn occurs before the time y / γ , that is before the particle reachesthe threshold y . Similarly, if the particle starts from 1 = ε ( ) , it must perform an odd number of switchings,and the first turn must be before the time ξ ( t , y ) . Conditioning on the first velocity switching, we obtain thefollowing sequence of coupled integral equations, for n ≥ , (2.9) f ( t , y ; 2 n ) = Z y / γ λ e − λ τ f ( t − τ , y − γ τ ; 2 n − ) d τ , f ( t , y ; 2 n + ) = Z ξ ( t , y ) λ e − λ τ f ( t − τ , y − γ τ ; 2 n ) d τ , y > . Equations (2.9) can be solved explicitly. For example, for n = , the second equation of this system, dueto (2.1), takes the form(2.10) f ( t , y ; 1 ) = Z t λ e − λ τ F ( d τ ; t − τ , y − γ τ ; 0 )= λ γ γ − γ e − λ ξ ( t , y ) − λ ξ ( t , y ) = λ γ θ ( t , y ) { t > y / γ } . We continue working with system (2.9) using the identities ξ ( t − τ , y − γ τ ) ≡ ξ ( t , y ) − τ , ξ ( t − τ , y − γ τ ) ≡ ξ ( t , y ) , (2.11) ξ ( t − τ , y − γ τ ) ≡ ξ ( t , y ) , ξ ( t − τ , y − γ τ ) ≡ ξ ( t , y ) − τ (2.12)and(2.13) γ ξ ( t , y ) + γ ξ ( t , y ) ≡ y , which are obvious by definition, (1.1).For n = , formula (2.4) is proved, see (2.10) and (2.13). The subsequent formulae in (2.5) and (2.4)follow by induction.Substituting (2.4) (with n − n ) into the first equation of (2.9), due to (2.11) and (2.13) weobtain f ( t , y ; 2 n ) = Z y / γ λ λ ξ − τ (cid:16) y − γ τ − γ n ξ (cid:17) ( λ λ ) n − ( ξ − τ ) n − ξ n − ( n − ) ! d τ · θ ( t , y ) { t > y / γ } = ( λ λ ) n ξ n − ( n − ) ! Z y / γ (cid:18) γ ( ξ − τ ) + n − n γ ξ (cid:19) ( ξ − τ ) n − d τ · θ ( t , y ) { t > y / γ } = ( λ λ ) n ξ n − ( n − ) ! Z ξ − γ ξ / γ (cid:18) γ u n − + n − n γ ξ u n − (cid:19) d u · θ ( t , y ) { t > y / γ } , where ξ = ξ ( t , y ) and ξ = ξ ( t , y ) . After integration, by virtue of (2.13), we obtain f ( t , y ; 2 n ) = ( λ λ ) n ξ n − ( n − ) ! · γ ξ n + γ ξ ξ n − n · θ ( t , y ) = y ( λ λ ) n ( ξ ξ ) n − ( n − ) ! n ! · θ ( t , y ) , t > y / γ , which confirms (2.5). N TELEGRAPH PROCESSES, THEIR FIRST PASSAGE TIMES AND RUNNING EXTREMA 5
Similarly, substituting (2.5) (with n − n ) into the second equation of (2.9), due to (2.12) and(2.13), we get f ( t , n ; 2 n + ) = λ λ Z ξ λ ( y − γ τ ) [ λ λ ξ ( ξ − τ )] n − ( n − ) ! n ! d τ · θ ( t , y ) { t > y / γ } = λ n λ n + ξ n − ( n − ) ! n ! Z ξ ( γ ξ + γ ( ξ − τ )) ( ξ − τ ) n − d τ · θ ( t , y ) { t > y / γ } = λ n λ n + ξ n − ( n − ) ! n ! γ ξ ξ n n + γ ξ n + n + ! · θ ( t , y ) { t > y / γ } = λ n λ n + ξ n − ξ n n ! (cid:18) y − γ n + ξ (cid:19) · θ ( t , y ) { t > y / γ } , which coincides with (2.4). (cid:3) Remark 2.2.
The idea of using coupled integral equations of the form (2.9) to analyse the properties oftelegraph-like processes (instead of the classic technique which is based on the differential equations) occa-sionally appears in literature, see e.g. [4, Lemma 5.1], [13, formula (5.6)], [14, Chapter 5], [7, (4.1.2)].
Formulae (2.7) - (2.8) were previously derived by a slightly different method, see [2, 9]. At first glance, itseems that formulae (2.5) - (2.4) can be obtained by expanding the Bessel functions in formulae (2.7) - (2.8) .However, to prove this, we need to show that the n-th term of these series must be equal to f · ( t , y ; n ) , whichis not so evident. Remark 2.3.
The distribution of the time to first reach of the threshold y , y > , with a simultaneous velocityreversal at this time follows from (2.14) P i { T ( y ) ∈ d t , N ( t ) = n , N ( t +) = n + } = λ · P i { T ( y ) ∈ d t , N ( t ) = n } , y > , i ∈ { , } , n ≥ . For n = , these equalities hold , because P { T ( y ) ∈ d t , N ( t ) = } ≡ , t > , and P { T ( y ) ∈ d t , N ( t ) = , N ( t +) = } = P { τ ∈ d t and y = γ t } = λ e − λ t δ y / γ ( d t ) , which is λ F ( d t , t , y ; 0 ) . For i = and even n , i = and odd n , equality (2.14) follows in view of the integral equations (2.9) .Equalities (2.14) with i = and odd n ( i = and even n ) are trivial , 0 = . It is interesting to look at formulae (2.7)-(2.8) in light of the Kac’s rescaling. Exactly, let λ , λ → ∞ and(2.15) λ λ → ν , ν > . The Kac condition is assumed for the two states separately: let γ → + ∞ , γ → − ∞ and(2.16) γ √ λ → σ , γ √ λ → − σ , where σ , σ > . Finally, we assume that(2.17) γ λ + γ λ λ + λ → δ . It is known that under conditions (2.15)-(2.17), process Γ ( t ) weakly converges to the scaled Brownianmotion with drift, Σ · W ( t ) + δ t , t > , where(2.18) Σ = σ σ q ( σ + σ ) / . See [9, Section 5] for detailed definitions and comments. In the symmetric case, λ = λ and γ = − γ , wehave δ = σ = σ = Σ , so that the telegraph process Γ ( t ) weakly converges to Σ · W ( t ) , t > . NIKITA RATANOV CHELYABINSK STATE UNIVERSITY, 129, BR.KASHIRIHYKH, CHELYABINSK, 454001, RUSSIA [email protected]
The following result seems natural: under the Kac scaling (2.15)-(2.17), the distribution of T ( y ) convergesto the first passage time distribution of Brownian motion. Corollary 2.1 (Cf [8]) . Under the Kac scaling defined by (2.15) - (2.17),(2.19) F ( d t ; t , y ) , F ( d t ; t , y ) → y √ πΣ t / exp (cid:18) − ( y − δ t ) Σ t (cid:19) , where Σ is given by (2.18) .Proof. We apply the scaling conditions (2.15)-(2.17) to (2.7)-(2.8) and use the asymptotic expansion of theBessel functions I k ( x ) ∼ exp ( x ) √ π x , x → ∞ , see e.g. [6, 8.451]. (cid:3)
3. T
ELEGRAPHIC MEANDERS AND RUNNING EXTREMA
We define the sets of always negative and always positive trajectories with a given number of switchings,say a negative telegraphic meander ,(3.1) m − ( t , x ; n ) = { Γ ( t ) ∈ d x , M t = , N ( t ) = n } , and a positive telegraphic meander ,(3.2) m + ( t , x ; n ) = { Γ ( t ) ∈ d x , m t = , N ( t ) = n } . where m t : = min u ∈ [ , t ] Γ ( u ) and M t : = max u ∈ [ , t ] Γ ( u ) are running minimum and running maximum.If both velocities are of the same sign, then Γ ( t ) , t > , preserves the sign on an arbitrary time interval. Inthe case of both negative velocities, M t = m t < , ∀ t > . In this case,P i { m − ( t , x ; n ) } = P i { Γ ( t ) ∈ d x , N ( t ) = n } and P i { m + ( t , x ; n ) } ≡ , i ∈ { , } ;if the velocities are positive, we haveP i { m − ( t , x ; n ) } ≡ , and P i { m + ( t , x ; n ) } = P i { Γ ( t ) ∈ d x , N ( t ) = n } , i ∈ { , } , the explicit formulae for P i { Γ ( t ) ∈ d x , N ( t ) = n } are given by (1.4)-(1.5).Let the velocities be of opposite signs, γ > > γ . Therefore, M t | ε ( )= > , m t | ε ( )= < { m − ( t , x ; n ) } ≡ , P { m + ( t , x ; n ) } ≡ . We are interested in explicit expressions for the distributions of negative and positivemeanders,(3.3) G − ( d x , t , x ; n ) = P { m − ( t , x ; n ) } , x < G + ( d x , t , x ; n ) = P { m + ( t , x ; n ) } , x > . For brevity, we focus on the positive meander, i. e. on G + . Note that in the absence of switches, i.e. N ( t ) = , the distribution of Γ ( t ) is singular, see (1.4),(3.4) G + ( d x , t , x ; 0 ) = e − λ t δ γ t ( d x ) , t > . Consider the probability density functions, g + ( t , x ; n ) = P { m + ( t , x ; n ) } / d x , n ≥ . Theorem 3.1.
Let γ > > γ . Functions g + ( t , x ; n ) , x > , are specified explicitly , separately for odd andeven n , by the expressionsg + ( t , x ; 2 n + ) = λ ξ ( t , x ) z ( t , x ) n { < x < γ t } n ! θ ( t , x ) · (cid:18) x − γ n + ξ ( t , x ) (cid:19) / γ , (3.5) g + ( t , x ; 2 n + ) = λ λ z ( t , x ) n { < x < γ t } n ! ( n + ) ! θ ( t , x ) · x / γ , n ≥ . (3.6) Remark 3.1.
By summing up (3.4) - (3.6), we obtain (3.7) G + ( d x , t , x ) = P { Γ ( t ) ∈ d x , m t = } = e − λ t δ γ t ( d x ) + λ γ (cid:20) x ξ ( t , x ) I ( t , x ) + (cid:18) λ x − γ ξ ( t , x ) ξ ( t , x ) (cid:19) I ( t , x ) (cid:21) θ ( t , x ) d x . N TELEGRAPH PROCESSES, THEIR FIRST PASSAGE TIMES AND RUNNING EXTREMA 7
The distribution of the negative meander follows by symmetry : the formulae for g − ( t , x ; n ) , γ t < x < , have the form (3.5) - (3.6) with interchange ↔ , and after summing ,(3.8) G − ( d x , t , x ) = P { Γ ( t ) ∈ d x , M t = } = e − λ t δ γ t ( d x ) + λ γ (cid:20) x ξ ( t , x ) I ( t , x ) + (cid:18) λ x − γ ξ ( t , x ) ξ ( t , x ) (cid:19) I ( t , x ) (cid:21) θ ( t , x ) d x . Proof.
The proof of the theorem is based on the following observation: each meander path γ ( s , x ) , s ∈ [ , t ] , considered in reverse time, that is, γ ( t − s , x ) , s ∈ [ , t ] , matches the trajectory starting at x with first passagethrough the origin at time t . Therefore, the probability density functions g + ( t , x ; n ) can be obtained similarlyto (2.4)-(2.5).In contrast to the proof of Theorem 2.1, the coupled integral equations for g + ( t , x ; · ) are written out byconditioning on the last velocity change, cf Remark 2.3. We have(3.9) g + ( t , x ; 2 n + ) = Z x / γ λ e − λ s g + ( t − s , x − γ s ; 2 n + ) d s , g + ( t , x , n + ) = Z ξ ( t , x ) λ e − λ s g + ( t − s , x − γ s ; 2 n ) d s . Formulae (3.5)-(3.6) for always positive paths can be proved by induction similarly to the proof of Theorem2.1. Formula (3.7) follows by summing up (3.5)-(3.6). (cid:3)
The above preparation allows to receive the joint distribution of the running extrema m t ( M t ) , the time ζ mt ( ζ Mt ) to reach the running extremum, and the terminal position Γ ( t ) , reached after N ( t ) = n velocityswitchings.If both velocities are positive, γ > γ > , then ζ mt = , m t = ζ Mt = t , M t = Γ ( t ) a. s. Therefore, inthis case, for i ∈ { , } , P i { ζ mt ∈ d s , m t ∈ d y , Γ ( t ) ∈ d x , N ( t ) = n } = δ ( d s ) δ ( d y ) p i ( t , x ; n ) d x , P i (cid:8) ζ Mt ∈ d s , M t ∈ d y , Γ ( t ) ∈ d x , N ( t ) = n (cid:9) = δ t ( d s ) δ x ( d y ) p i ( t , x ; n ) d x , γ t < x < γ t ;similarly, for 0 > γ > γ , P i { ζ mt ∈ d s , m t ∈ d y , Γ ( t ) ∈ d x , N ( t ) = n } = δ t ( d s ) δ x ( d y ) p i ( t , x ; n ) d x , P i (cid:8) ζ Mt ∈ d s , M t ∈ d y , Γ ( t ) ∈ d x , N ( t ) = n (cid:9) = δ ( d s ) δ ( d y ) p i ( t , x ; n ) d x , γ t < x < γ t , the formulae for p i ( t , x ; n ) are given by (1.5). In these two cases, the distributions of h ζ mt , m t , Γ ( t ) i and h ζ Mt , M t , Γ ( t ) i can be obtained by summing up, see (1.6).Let the velocities be of opposite signs, γ > > γ . The singular components of these distributions corre-sponding to { ζ mt = } and { ζ Mt = } are represented as follows. Each time the running minimum or runningmaximum is zero, m t = M t =
0, the trajectory follows the meander, and(3.10) P { ζ mt = , m t = , Γ ( t ) ∈ d x , N ( t ) = n } = G + ( d x ; t , x ; n ) , x > , P (cid:8) ζ Mt = , M t = , Γ ( t ) ∈ d x , N ( t ) = n (cid:9) = G − ( d x ; t , x ; n ) , x < , see (3.3)-(3.6). Therefore,(3.11) P { ζ mt = , m t = , Γ ( t ) ∈ d x } = G + ( d x ; t , x ) , P (cid:8) ζ Mt = , M t = , Γ ( t ) ∈ d x (cid:9) = G − ( d x ; t , x ) , see (3.7)-(3.8). NIKITA RATANOV CHELYABINSK STATE UNIVERSITY, 129, BR.KASHIRIHYKH, CHELYABINSK, 454001, RUSSIA [email protected]
Further, since Γ ( t ) arrives at the minimum, m t (maximum, M t ) with the negative (positive) velocity, thenegative running minimum x is reached at the last time, that is ζ mt = t , with probabilities(3.12) P { ζ mt = t , m t = Γ ( t ) ∈ d x , N ( t ) = n } = ( , if n is even , f ( t , x ; n ) d x , if n is odd , P { ζ mt = t , m t = Γ ( t ) ∈ d x , N ( t ) = n } = ( , if n is odd , f ( t , x ; n ) d x , if n is even , x < ζ Mt = t , (3.13) P (cid:8) ζ Mt = t , M t = Γ ( t ) ∈ d x , N ( t ) = n (cid:9) = ( , if n is odd , f ( t , x ; n ) d x if n is even , P (cid:8) ζ Mt = t , M t = Γ ( t ) ∈ d x , N ( t ) = n (cid:9) = ( , if n is even , f ( t , x ; n ) d x if n is odd , x > , where f ( t , x ; n ) and f ( t , x ; n ) are given by (2.1), (2.4)-(2.6). Summing up, we obtain(3.14) P { ζ mt = t , m t = Γ ( t ) ∈ d x } = λ ξ ( t , x ) ( − x I ( t , x ) + γ ξ ( t , x ) I ( t , x )) θ ( t , x ) { γ t < x < } d x − γ , P { ζ mt = t , m t = Γ ( t ) ∈ d x } = (cid:16) e − λ x / γ δ γ t ( d x ) − λ λ x I ( t , x ) (cid:17) θ ( t , x ) { γ t < x < } d x − γ , P (cid:8) ζ Mt = t , M t = Γ ( t ) ∈ d x (cid:9) = (cid:16) e − λ x / γ δ γ t ( d x ) + λ λ x I ( t , x ) (cid:17) θ ( t , x ) { < x < γ t } d x γ , P (cid:8) ζ Mt = t , M t = Γ ( t ) ∈ d x (cid:9) = λ ξ ( t , x ) ( x I ( t , x ) − γ ξ ( t , x ) I ( t , x )) θ ( t , x ) { < x < γ t } d x γ . Compare formulae (3.12)-(3.14) with similar formulae obtained in the symmetric case, see [3].The “regular” component of the distribution is obtained by weighing the compound paths consisting ofa passage to a minimum (maximum) value, switching the velocity at this moment, then moving along themeander above (below) the reached level in the remaining time. The corresponding probabilities for 0 < s < t turn out to beP { ζ mt ∈ d s , m t ∈ d y , Γ ( t ) ∈ d x , N ( t ) = n } = λ ∑ ≤ k ≤ [ n / ] F ( d s , s , y ; 2 k − ) G + ( d x , t − s , x − y ; n − k ) d y , n ≥ , P { ζ mt ∈ d s , m t ∈ d y , Γ ( t ) ∈ d x , N ( t ) = n } = λ ∑ ≤ k ≤ [ n / ] F ( d s , s , y ; 2 k ) G + ( d x , t − s , x − y ; n − k − ) d y , n ≥ , < s < t , y < ∧ x , and P (cid:8) ζ Mt ∈ d s , M t ∈ d y , Γ ( t ) ∈ d x , N ( t ) = n (cid:9) = λ ∑ ≤ k ≤ [( n − ) / ] F ( d s , s , y ; 2 k ) G − ( d x , t − s , x − y ; n − k − ) d y , n ≥ , P (cid:8) ζ Mt ∈ d s , M t ∈ d y , Γ ( t ) ∈ d x , N ( t ) = n (cid:9) = λ ∑ ≤ k ≤ [ n / ] F ( d s , s , y ; 2 k − ) G − ( d x , t − s , x − y ; n − k ) d y , n ≥ , < s < t , y > ∨ x . Here F ( d s ; · , · ; · ) and F ( d s ; · , · ; · ) are written out in (2.1), (2.4)-(2.6), and G ∓ ( d x ; · , · ; · ) are determined byformulae (3.3)-(3.6). N TELEGRAPH PROCESSES, THEIR FIRST PASSAGE TIMES AND RUNNING EXTREMA 9