First Crossing Times of Telegraph Processes with Jumps

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.

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:

$$ c\alpha-\lambda-q=\frac{\lambda b^{2}}{\alpha^{2}-b^{2}}\qquad\text{and}\qquad \lambda+q-c\alpha=\frac{\lambda b^{2}}{\alpha^{2}-b^{2}}. $$

(A.1)

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

$$ c\alpha-\lambda-q|_{\alpha= 0}=-\lambda-q<-\lambda= \frac{\lambda b^{2}}{\alpha^{2}-b^{2}}|_{\alpha= 0} $$

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

Fig. 10

c = c₁ = c₂ > 0 : three real roots of $c\alpha -\lambda -q=\frac {\lambda b^{2}}{\alpha ^{2}-b^{2}},$ two positive and one negative

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

$$ f(\alpha):=c\alpha^{3}-(\lambda+q)\alpha^{2}-cb^{2}\alpha+b^{2}(q + 2\lambda))= 0. $$

(A.2)

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, minα> 0f(α) < 0, which is taken at the stationary point

$$ \alpha_{*}=\frac{\lambda+q+\sqrt{(\lambda+q)^{2}+ 3b^{2}c^{2}}}{3c}. $$

(A.3)

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

$$ \frac{\lambda b^{2}}{c}<(b-\alpha_{*})\left( \alpha_{*}-\frac{\lambda+q}{c}\right), $$

(A.4)

where α_∗ is defined by (A.3).

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

Fig. 11

c = c₁ = c₂ > 0 : the roots of $\lambda +q-c\alpha =\frac {\lambda b^{2}}{\alpha ^{2}-b^{2}}$

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 c₁ = −c₂ = c > 0 and the jump part is the same. In this case, Eq. 3.29 becomes

$$ (\lambda+q)^{2}-c^{2}\alpha^{2}=\frac{\lambda^{2}b^{4}}{(\alpha^{2}-b^{2})^{2}}. $$

(A.5)

Note that

$$ (\lambda+q)^{2}-c^{2}\alpha^{2}|_{\alpha= 0}=(\lambda+q)^{2}> \lambda^{2}=\frac{\lambda^{2}b^{4}}{(\alpha^{2}-b^{2})^{2}}|_{\alpha= 0}. $$

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

Fig. 12

c = c₁ = −c₂ > 0. The roots of $(\lambda +q)^{2}-c^{2}\alpha ^{2}=\frac {\lambda ^{2}b^{4}}{(\alpha ^{2}-b^{2})^{2}}:$ one real positive root and two conjugate complex roots with positive real part (solid parabola); three real positive roots (dashed parabola)

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

$$ f(\alpha):=(\alpha^{2}-b^{2})^{2}\cdot\left[(\lambda+q)^{2}-c^{2}\alpha^{2}\right]>\lambda^{2}b^{4}, $$

see Fig. 12 (dashed parabola (λ + q)² − c²α² 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

$$ \alpha^{2}=\frac{b^{2}}{3}+\frac{2}{3c^{2}}(\lambda+q)^{2}, $$

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

$$ \frac{4}{27}\left( (\lambda+q)^{2}-b^{2}c^{2}\right)^{3}>\lambda^{2}b^{4}c^{4}, $$

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

$$ 3(\lambda/2)^{2/3}b^{4/3}c^{4/3} +b^{2}c^{2}<(\lambda+q)^{2}. $$