## Abstract

The paper presents exact formulae related to the distribution of the first passage time *τ*_{x} of the jump-telegraph process. In particular, the Laplace transform of *τ*_{x} is analysed, when a jump component is in the opposite direction to the crossing level *x* > 0. The case of double exponential jumps is also studied in detail.

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## Acknowledgments

I am very grateful to anonymous referees and the editor for the careful reading of the paper and for the helpful comments that have greatly improved the text.

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## Appendix: Conditions determining appearance of conjugate complex roots

### Appendix: Conditions determining appearance of conjugate complex roots

Consider the compound Poisson process with constant drift, \(X(t)=ct+{\sum }_{n = 1}^{N(t)}Y_{n},\) where *N*(*t*) is a homogeneous Poisson process with parameter *λ*,*λ* > 0; {*Y*_{n}}_{n≥ 1} are independent identically distributed random jumps. For simplicity, we assume that *Y*_{n} has the symmetric Laplace distribution with the density function \(h(y)=\dfrac 12 b\exp (-b|y|), b>0\).

In this particular case, Eq. 3.9 is equivalent to the pair of equations:

Let *c* > 0 (the case *c* < 0 can be analysed similarly). Since for *q* > 0

and *c* > 0, all three roots of the first equation of (A.1) are always real: one negative and two positive, see Fig. 10.

The second equation of (A.1) is equivalent to

This equation always has one negative root. The other two roots are real positive if and only if function *f*(*α*) has a negative local minimum, min*α*> 0*f*(*α*) < 0, which is taken at the stationary point

After the tedious algebra, one can see that the inequality min*α*> 0*f*(*α*) < 0 is equivalent to

where *α*_{∗} is defined by (A.3).

Note that condition (A.4) fails when *b* = (*λ* + *q*)/*c*, see Fig. 11.

If *b*≠(*λ* + *q*)/*c*, then (A.4) is valid only when the trend *c* is far from (*λ* + *q*)/*b*. More precisely, if *c**↓* 0, then *α*_{∗}*↑* +*∞*, in such a way that \(c\alpha _{*}\downarrow \dfrac 23(\lambda +q),\) which ensures (A.4) for a small *c* > 0. In contrast, if *c* → +*∞*, then \(\alpha _{*}\to b/\sqrt {3}\), which again gives (A.4) for a sufficiently large trend *c*.

See Fig. 11: in case (1) (0 < *c* ≪ (*λ* + *q*)/*b*) and in case (3) (*c* ≫ (*λ* + *q*)/*b*) we have two positive real roots; case (2) with moderate *c* corresponds to two conjugate complex roots with a positive real part.

Consider another example. Let *c*_{1} = −*c*_{2} = *c* > 0 and the jump part is the same. In this case, Eq. 3.29 becomes

Note that

Hence, Eq. A.5 always has at least one positive real root, see Fig. 12.

Equation A.5 has two additional positive real roots if and only if there exists *α* > *b* such that

see Fig. 12 (dashed parabola (*λ* + *q*)^{2} − *c*^{2}*α*^{2} with small *c* which corresponds to three real positive roots). Otherwise, Eq. A.5 has one real positive root and a pair of conjugate complex roots with positive real part.

Since, the point of local maximum of *f*(*α*),*α* > *b*, corresponds to

Equation A.5 has three real positive roots if the following relation holds:

which is equivalent to sufficiently small *c*,*c* < (*λ* + *q*)/*b*, such that

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Ratanov, N. First Crossing Times of Telegraph Processes with Jumps.
*Methodol Comput Appl Probab* **22, **349–370 (2020). https://doi.org/10.1007/s11009-019-09709-5

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### Keywords

- Jump-telegraph process
- First passage time
- Laplace transformation
- Double exponential distribution

### Mathematics Subject Classification (2010)

- 60J75
- 60J27
- 60K99