The Law of the Hitting Times to Points by a Stable Lévy Process with No Negative Jumps

Let $X=(X_t)_{t \ge 0}$ be a stable Levy process of index $\alpha \in (1,2)$ with the Levy measure $\nu(dx) = (c/x^{1+\alpha}) I_{(0,\infty)}(x) dx$ for $c>0$, let $x>0$ be given and fixed, and let $\tau_x = \inf\{ t>0 : X_t=x \}$ denote the first hitting time of $X$ to $x$. Then the density function $f_{\tau_x}$ of $\tau_x$ admits the following series representation: $$f_{\tau_x}(t) = \frac{x^{\alpha-1}}{\pi ( \Gamma(-\alpha) t)^{2-1/\alpha}} \sum_{n=1}^\infty \bigg[(-1)^{n-1} \sin(\pi/\alpha) \frac{\Gamma(n-1/\alpha)}{\Gamma(\alpha n-1)} \Big(\frac{x^\alpha}{c \Gamma(-\alpha)t} \Big)^{n-1} $$ $$- \sin\Big(\frac{n \pi}{\alpha}\Big) \frac{\Gamma(1+n/\alpha)}{n!} \Big(\frac{x^\alpha}{c \Gamma(-\alpha)t}\Big)^{(n+1)/\alpha-1} \bigg]$$ for $t>0$. In particular, this yields $f_{\tau_x}(0+)=0$ and $$ f_{\tau_x}(t) \sim \frac{x^{\alpha-1}}{\Gamma(\alpha-1), \Gamma(1/\alpha)} (c \Gamma(-\alpha)t)^{-2+1/\alpha} $$ as $t \rightarrow \infty$. The method of proof exploits a simple identity linking the law of $\tau_x$ to the laws of $X_t$ and $\sup_{0 \le s \le t} X_s$ that makes a Laplace inversion amenable. A simpler series representation for $f_{\tau_x}$ is also known to be valid when $x<0$.


Introduction
If a Lévy process X = (X t ) t≥0 jumps upwards, then it is much harder to derive a closed form expression for the distribution function of its first passage time τ (x,∞) over a strictly positive level x, and in the existing literature such expressions seem to be available only when X has no positive jumps (unless the Lévy measure is discrete). A notable exception to this rule is the recent paper [1] where an explicit series representation for the density function of τ (x,∞) was derived when X is a stable Lévy process of index α ∈ (1, 2) having the Lévy measure given by ν(d x) = (c/x 1+α ) I (0,∞) (x) d x with c > 0 given and fixed. This was done by performing a time-space inversion of the Wiener-Hopf factor corresponding to the Laplace transform of (t, y) → P(S t > y) where S t = sup 0≤s≤t X s for t > 0 and y > 0.
Motivated by this development our purpose in this note is to search for a similar series representation associated with the first hitting time τ x of X to a strictly positive level x itself. Clearly, since X jumps upwards and creeps downwards, τ x will happen strictly after τ (x,∞) , and since X reaches x by creeping through it independently from the past prior to τ (x,∞) , one can exploit known expressions for the latter portion of the process and derive the Laplace transform for (t, y) → P(τ y > t). This was done in [6, Theorem 1] and is valid for any Lévy process with no negative jumps (excluding subordinators). A direct Laplace inversion of the resulting expression appears to be difficult, however, and we show that a simple (Chapman-Kolmogorov type) identity which links the law of τ x to the laws of X t and S t proves helpful in this context (due largely to the scaling property of X ). It enables us to connect the old result of [13] with the recent result of [1] through an additive factorisation of the Laplace transform of (t, y) → P(τ y > t). This makes the Laplace inversion possible term by term and yields an explicit series representation for the density function of τ x .
2. The following properties of X are readily deduced from (1) and (2) using standard means (see e.g. [2] and [9]): the law of (X c t ) t≥0 is the same as the law of (c 1/α X t ) t≥0 for each c > 0 given and fixed (scaling property); X is a martingale with EX t = 0 for all t ≥ 0; X jumps upwards (only) and creeps downwards ( in the sense that P( x } is the first passage time of X over x); X has sample paths of unbounded variation; X oscillates from −∞ to +∞ ( in the sense that lim inf t→∞ X t = −∞ and lim sup t→∞ X t = +∞ both a.s.); the starting point 0 of X is regular ( for both (−∞, 0) and (0, +∞)). Note that the constant c = 1/Γ(−α) in the Lévy measure ν(d x) = (c/x 1+α ) d x of X is chosen/fixed for convenience so that X converges in law to 2 B as α ↑ 2 where B is a standard Brownian motion, and all the facts throughout can be extended to a general constant c > 0 using the scaling property of X .
3. Letting f X 1 denote the density function of X 1 , the following series representation is known to be valid (see e.g. (14.30) in [14, p. 88]): for x ∈ IR. Setting S 1 = sup 0≤t≤1 X t and letting f S 1 denote the density function of S 1 , the following series representation was recently derived in [1, Theorem 1]: for x > 0. Clearly, the series representations (3) and (4) extend to t = 1 by the scaling property of X since X t = law t 1/α X 1 and S t := sup 0≤s≤t X s = law t 1/α S 1 for t > 0.
4. Consider the first hitting time of X to x given by for λ > 0 and p > 0. Note that this can be rewritten as follows: for λ > 0 and p > 0. Let IL −1 p denote the inverse Laplace transform with respect to p. Using that 1/(p(p 1/α −λ)) = ∞ n=1 λ n−1 /p 1+n/α and IL −1 for t > 0 where γ(a, x) = x 0 y a−1 e − y d y denotes the incomplete gamma function. Combining (7) with (8) and (9) = α for λ > 0 and t > 0. The first and the third term on the right-hand side of (10) may now be recognised as the Laplace transforms of particular functions considered in [1] and [13] respectively (recall also (2.2) above). The proof of the following theorem provides a simple probabilistic argument (of Chapman-Kolmogorov type) for this additive factorisation (see Remark 1 below).
Theorem 1. Let X = (X t ) t≥0 be a stable Lévy process of index α ∈ (1, 2) with the Lévy measure ν(d x) = (c/x 1+α ) I (0,∞) (x) d x for c > 0, let x > 0 be given and fixed, and let τ x denote the first hitting time of X to x. Then the density function f τ x of τ x admits the following series representation: for t > 0. In particular, this yields: Proof. It is no restriction to assume below that c = 1/Γ(−α) as the general case follows by replacing t in (11) with c Γ(−α) t for t > 0. Since X creeps downwards, we can apply the strong Markov property of X at τ x , use the additive character of X , and exploit the scaling property of X to find where we also use that P(X 1 ≤ 0) = 1/α and F τ x denotes the distribution function of τ x . Note that the second equality in (14) represents a Chapman-Kolmogorov equation of Volterra type (see [11,Section 2] for a formal justification and a brief historical account of the argument). Since τ x = law x α τ 1 by the scaling property of X , we find that (14) reads for x > 0. Hence we see that F τ 1 is absolutely continuous (cf. [10] for a general result on the absolute continuity) and by differentiating in (15) we get for x > 0. Letting t = 1/x α we find that for t > 0. Hence (11) with x = 1 follows by (3) and (4) above. Moreover, since τ x = law x α τ 1 we see that f τ x (t) = x −α f τ 1 (t x −α ) and this yields (11) with x > 0.

Remark 1.
Note that (14) can be rewritten as follows: for x > 0, and from (2.30) in [1] we know that for λ > 0. In view of (10) this implies that for λ > 0. Recalling (2) we see that (20) is equivalent to for λ > 0. An explicit series representation for f in place of f X 1 in (21) was found in [13] (see also [12]) and this expression coincides with (3) above when x < 0. (Note that (21) holds for all λ ∈ IR and substitute y = −x to connect to [13].) This represents an analytic argument for the additive factorisation addressed following (10) above.

Remark 2.
In contrast to (12) note that for x > 0. This is readily derived from P(τ (x,∞) ≤ t) = P(S t ≥ x) using S t = law t 1/α S 1 and f S 1 (x) ∼ c x −1−α for x → ∞ as recalled in the proof above.
Remark 3. If x < 0 then applying the same arguments as in (14) above with I t = inf 0≤s≤t X s we find that for t > 0. In this case, moreover, we also have P(I t ≤ x) = P(σ x ≤ t) since X creeps through x, so that (23) yields for x < 0 and t > 0. Since X t = law t 1/α X 1 this implies for t > 0 upon using (3) above. Replacing t in (25) by c Γ(−α) t we get a series representation for f τ x in the case when c > 0 is a general constant. The first identity in (25) is known to hold in greater generality (see [4] and [2, p. 190] for different proofs).

Remark 4.
If c = 1/2Γ(−α) and α ↑ 2 then the series representations (11) and (25) with t/2 in place of t reduce to the known expressions for the density function is a standard Brownian motion: for t > 0 and x ∈ IR\{0}.
from where the following identity can be derived (see for x ∈ IR and t > 0 (being valid for any stable Lévy process). By the scaling property of X we have f X s (x) = s −1/α f X 1 (xs −1/α ) for s ∈ (0, t) and x ∈ IR. Recalling the particular form of the series representation for f X 1 given in (3), we see that it is not possible to integrate term by term in (28) in order to obtain an explicit series representation.

Remark 6.
The density function f X 1 from (3) can be expressed in terms of the Fox functions (see [15]), and the density function f S 1 from (4) can be expressed in terms of the Wright functions (see [5,Sect. 12] and the references therein). In view of the identity (17) and the fact that f τ x (t) = x −α f τ 1 (t x −α ), these facts can be used to provide alternative representations for the density function f τ x from (11) above. We are grateful to an anonymous referee for bringing these references to our attention.