HITTING DENSITIES FOR SPECTRALLY POSITIVE STABLE PROCESSES

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1 HITTING DENSITIES FOR SPECTRALLY POSITIVE STABLE PROCESSES THOMAS SIMON Abstract. A multiplicative identity in law connecting the hitting times of completely asymmetric α stable Lévy processes in duality is established. In the spectrally positive case, this identity allows with an elementary argument to compute fractional moments to get series representations for the density. We also prove that the hitting times are unimodal as soon as α 3/2. Analogous results are obtained for the first passage time across a positive level, in a simplified manner.. A multiplicative identity in law Let {X t, t } be a spectrally positive Lévy α stable process ( < α < 2), starting from zero normalized such that (.) E [ e λxt] = e tλα, t, λ. For every x >, consider the first exit times the first hitting times T x = inf{t >, X t > x}, τ x = inf{t >, X t = x}, ˆTx = inf{t >, X t < x} ˆτ x = inf{t >, X t = x}. Notice that since X has no negative jumps, one has ˆτ x = ˆT x a.s. ˆτ x is a positive stable rom variable with index /α - see e.g. Theorem 46.3 in [6]. Also, (.) the optional sampling theorem give the normalization E [ e λˆτx] = e xλ/α, x, λ. On the other h, the process X crosses the level x > by a jump so that X Tx > x τ x > T x a.s. Introducing the overshoot K x = X Tx x, the strong Markov property at time T x entails that {Xt x = X Tx+t X Tx, t } is independent of (T x, K x ). In particular, setting ˆτ y x = inf{t >, Xt x = y} for y >, we have τ x = T x + ˆτ x K x d = T x + K α x ˆ τ d = x a (T + K α ˆ τ ) where ˆ τ is an independent copy of ˆτ the two identities in law follow at once from the self-similarity relationships (T x, τ x, ˆτ x ) d = x α (T, τ, ˆτ ) K x d = xk, 2 Mathematics Subject Classification. 6E5, 6G52. Key words phrases. Exit time - Hitting time - Mellin transform - Running supremum - Series representation - Stable Lévy processes - Unimodality.

2 2 THOMAS SIMON which are themselves simple consequences of the self-similarity of X with index /α. Specialising the above to x = yields the basic identity (.2) τ d = T + K α ˆ τ, which has been known for a long time [6]. Notice that the laws of the three rom variables appearing in the right-h side of (.2) are more or less explicit: the density of ˆ τ is that of a positive /α stable variable, the density of K α comes from a classical computation by Port involving some Beta integral - see e.g. Exercise VIII.3 in [2] - the density of T can be obtained by changing the variable in the main result of [] thanks to the identity T = S α, with the notation S t = sup{x s, s t} for the supremum process. However, lack of explicit information on the law of the bivariate rom variable (T, K ) prevents us from applying (.2) directly to derive an expression of the law of τ. Recently Peskir [4] circumvented this difficulty with an identity of the Chapman-Kolmogorov type linking the laws of X, S τ, providing an explicit series representation at + for the density of τ. In this paper we will follow the method of our previous article [7] in order to derive, firstly, a simple identity in law for τ. Our main result reads: Theorem. One has (.3) τ d = U α ˆτ, where U α is an independent, positive rom variable with density f Uα (t) = Proof: From (.2) we have for any λ E [ e λα τ ] [ ] = E e λα (T + K αˆ τ ) (sin πα)t /α π(t 2 2t cos πα + ) [ [ ]] = E e λα T E e λα K αˆ τ (T, K ) = E [ e (λα T +λk ) ] = E [ e (T λ+k λ ) ], where in the third equality we recalled that ˆ τ is a (/α) stable positive variable with given normalization. On the other h, the double Laplace transform of (T, K ) had been evaluated long ago by Fristedt, yields the simple expression e sλ E [ e ] ( ) ( ) (T λ+k λ ) α dλ = s s α for every s > - see (2.6) in [6]. It is possible to invert the right-h side by noticing that s = e sλ e λ dλ s α = e sλ ( αλ α E α(λ α ) ) dλ for every s >, where E α is the derivative of the Mittag-Leffler function z n E α (z) = Γ( + αn), z C. n=

3 HITTING DENSITIES FOR STABLE PROCESSES 3 Above, the computation involving E α had been made originally by Humbert we refer to the discussion after (5) in [7] for more details. Putting everything together gives an expression of the Laplace transform of τ : (.4) E [ e λα τ ] = F (λ) αf α(λ), λ, where we used the notation F α (x) = E α (x α ) (notice that F (x) = e x ) for every x. We will now prove that the function G α : λ F (λ) αf α(λ) is completely monotone, reasoning as in Theorem in [7]. The well-known curvilinear representation of the analytic function /Γ yields F (λ) = e 2πi Hλ t t λ dt αf α(λ) = αλ 2πi Hλ α e t t α λ dt α for every λ >, where H λ is a Hankel path encircling the disk centered at the origin with radius λ - see e.g. Chapter IX.4 in [4]. We deduce G α (λ) = e t g λ (t)dt, λ > 2πi H λ where ( ) ( ) αλ α g λ (t) = t λ t α λ α is analytic on C/{λ} (, ]. Besides, since g λ (t) (α )/2λ as t λ, it can be continued on the whole complex plane cut on the negative real axis, which entails G α (λ) = e t g λ (t)dt, λ > 2πi H where H is a Hankel path independent of λ. We can then compute, as in Theorem in [7], ( ( ) ( ) ) G α (λ) = lim e s ρ 2πi ρ s + λ + αλα e s e iπα s α λ α s + λ + αλ α ds e iπα s α λ α α sin πα ( ) = e s λ α s α ds π s 2α 2s α λ α cos πα + λ 2α α sin πα ( ) = e λs s α ds, π s 2α 2s α cos πα + so that from (.4) the expression of the density f Uα, E [ e λα τ ] [ ] /α = E λuα e for all λ. Reasoning now exactly as in Theorem 3 in [7] entails finally the desired identity in law τ d = U α ˆτ. Remark. In Theorem of [6], the double Laplace transform of the function (t, x) P[τ x t] was computed for general Lévy processes with no negative jumps. Our result can be viewed as an inversion of this Laplace transform in the particular stable case.

4 4 THOMAS SIMON 2. Fractional moments series representations In this section we will obtain two expressions for the density of τ (a density which is readily known to exist from our main result, see also [2] for the general context) in terms of convergent series, the first one being useful in the neighbourhood of the second in the neighbourhood of +. Finding different series representations or asymptotic expansions for densities of first passage times is a classical issue in probability, see Chapter in [9]. Recall also that this work has been done long ago for ˆτ, respectively by Pollard Linnik - see (4.3) (4.35) in [6]. Our approach is different from the above references relies on Mellin inversion. We will hence first compute the fractional moments of τ, as a simple consequence of our main result: Corollary. For any s ( /α, /α), one has E [τ s ] = Γ( αs) sin(π(α )(s + /α)) Γ( s) sin(π(s + /α)) Proof: From (.3) we have E [τ s ] = E [ˆτ s ] E [Uα] s for every s R. The computation of E [ˆτ s ] is easy classical: E [ˆτ s ] = Γ( s) λ (s+) E [ e λˆτ ] dλ = for every s < /α. Similarly, one can show that E [U s α] = Γ( αs) λ (αs+) G α (λ)dλ = Γ( s) Γ( αs) λ (s+) e λ/α dλ = Γ( αs) Γ( s) λ (αs+) (e λ αλ α E α(λ α ))dλ but I could not carry the computations further with the sole help of Mittag-Leffler functions. Instead, one can rest upon the residue theorem in order to evaluate directly E [U s α] = Setting λ = s + /α + (, 2) sin πα π x s+/α x 2 2x cos πα + dx. ( z) λ f(z) = z 2 2z cos πα + one sees from Formula VIII.(.2.3) in [4] that ( ) sin πα E [Uα] s = (Res xα f + Res xα f) sin πλ where x α = e i(2 α)π x α = e i(α 2)π are the two conjugate roots of x 2 2x cos πα +. After some simple computations, one gets ( ) sin(π(λ )(α )) E [Uα] s sin(π(λ )(α )) sin(π(α )(s + /α)) = = = sin πλ sin π(λ ) sin(π(s + /α)) for every s ( /α, /α), which together with the previous computation for ˆτ completes the proof (notice that /α < /α).

5 HITTING DENSITIES FOR STABLE PROCESSES 5 Remark. The fractional moments of U α τ are finite over the same strip ( /α, /α), whereas those of ˆτ are finite over the larger strip (, /α). We will now derive series representations for the density f τ of τ, inverting the fractional moments which were computed just before. Setting M(z) = E[τ z ] for every z C such that Re(z) ( /α, /α), one has M(iλ) = E[e iλσ ], λ R, where σ = log τ is a real rom variable with finite polynomial moments by Corollary, which is absolutely continuous with continuous density f σ (x) = e x f τ (e x ) over R. The Fourier inversion formula for f σ entails then (2.) f τ (x) = M(it)x it dt, x, 2πx R we will start from this formula to obtain our series representation. The one at zero is particularly simple: Proposition 2. For every x one has f τ (x) = n αx /α+n Γ( αn)γ(/α + n) Proof: Suppose first x <. We compute the integral in (2.) with the help of the usual contour Γ R joining R to R along the real axis R to R along a half-circle in the trigonometric orientation, centered at the origin. Because x <, the integral along the half-circle is easily seen to converge to when R, so that it remains to consider the singularities of L(t) = M(it)x it inside the contour. By Corollary, the latter are located at i(n + /α), n. After stard computations, one finds that ( ) ( ) Γ(2 + αn) sin(nπα)x n+/α αx n+/α Res i(n+/α) L = i = i πγ( + /α + n) Γ( αn)γ(/α + n) for any n. By the residue theorem, this finishes the proof for x (, ) (notice that Res i/α L = ) by analyticity, one can then extend the formula over the whole R +. The series representation at infinity is slightly more complicated than the one at zero, involving actually two series. Up to some painless normalization after some simplifications on the Gamma function, it had already been computed by Peskir [4] with an entirely different, probabilistic argument involving a previous computation made for f S in []. Proposition 3. For every x > one has αx /α n f τ (x) = Γ(αn)Γ(/α n) + x n/α Γ( n/α)n! n n Proof: By analyticity, it suffices to consider the case x >. We compute the integral in (2.) with a contour Γ R which is the reflection of Γ R with respect to the real axis. Because x >, the integral along the half-circle converges to when R, so that one needs again to

6 6 THOMAS SIMON consider the singularities of L inside Γ R, which are located at i(/α n) in/α, n. The term αx /α n Γ(αn)Γ(/α n) n follows then from the computations of Proposition 2 in replacing n by n taking into account the clockwise orientation of Γ R. The last term follows from the simple computation Res i(n+/α) L = ix n/α Γ( n/α)n! Remark. The two above representations are integrable term by term, which yields two series representations for the distribution function of τ : P[τ x] = n αx /α+n Γ( αn)γ( + /α + n) P[τ x] = αx /α n Γ(αn)Γ( + /α n) x n/α Γ( n/α)n! n n One can also differentiate term by term in order to get expansions for the successive derivatives of f τ : for any p one has f τ (p) (x) = αx /α+n p Γ( αn)γ(/α + n p) n f τ (p) (x) = n αx /α n p Γ(αn)Γ(/α n p) + n x n/α p Γ( n/α p)n! The first term of all these expansions give the behaviour of the function at zero respectively at infinity. For instance, one has f τ (x) αx /α Γ( α)γ( + /α) f τ (x) αx /α Γ( α)γ(/α) as x +, whereas f τ (x) x /α 2 Γ(α )Γ(/α) f τ (x) αx /α 3 Γ(α)Γ(/α 2) as x +. Notice finally that since α (, 2), the function f τ is ultimately completely monotone at infinity f τ ultimately completely monotone at zero.

7 HITTING DENSITIES FOR STABLE PROCESSES 7 3. Complements on the first exit time the running supremum In this section, we will derive similar series representations for the density f T of the first exit time T. As mentioned before, this rom variable is connected to the supremum process {S t, t }, so that we will also get series representations for S. Our basic tool is an identity in law analogous to our main result, which was proved in our previous article [7], Theorem 3. This identity reads (3.) T d = T α ˆτ where is an independent rom variable with density given by f Tα (t) = (sin πα)( + t/α ) πα(t 2 2t cos πα + ) over R +. As before, we begin in evaluating the fractional moments of T : Proposition 4. For every s (, /α) one has E [T s ] = Γ( + s) sin(π/α) Γ( + sα) sin(π(s + /α)) Proof. For every s (, /α) we have E [Tα] s = ( [ ] E U s /α α + E [U s α α ] ) = ( ) sin(πs(α )) sin(πs(α ) /α) α sin πs sin(π(s + /α)) sin(πsα) sin(π/α) = α sin(πs) sin(π(s + /α)) where the last equality follows from tedious trigonometric transformations. From (3.), the previous computation made for E [ˆτ s ], the complement formula for the Gamma function, we get as desired. E [T s ] = sin(π/α) sin(π(s + /α) ( ) Γ( sα) sin(πsα)) αγ( s) sin(πs) = Γ( + s) sin(π/α) Γ( + sα) sin(π(s + /α)) Remark. Similarly as above, the fractional moments of T α τ are finite over the same strip (, /α), whereas those of ˆτ are finite over the larger strip (, /α). We notice in passing that this computation allows to show that the law of the independent quotient T / ˆT is Pareto( /α), a fact which had been proved in [5] in the general context of stable processes in duality: Corollary 5. The density of the independent quotient T / ˆT is given by over R +. sin(π/α) πt /α ( + t)

8 8 THOMAS SIMON Proof: By Mellin inversion, it is enough to identify the fractional moments. Proposition 4 the classical computation for the moments of ˆT entail [ ] T s sin(π/α) E = ˆT s sin(π(s + /α)) = sin(π/α)t s πt /α ( + t) dt for every s ( /α, /α), where the second equality follows from Formula VIII.(.2.4) in [4]. We now come back to series representations for f T, which are obtained from Proposition 4 Mellin inversion similarly as in Propositions 2 3. Since we choose exactly the same contours, we will leave all the details to the reader. Notice that contrary to τ, the two-series representation for T is the one at zero that the first series therein defines an entire function. Corollary 6. One has both representations f T (x) = n x /α n αγ(αn )Γ( + /α n), x > f T (x) = x n αγ( αn) + x /α+n Γ( αn)γ(/α + n), x. n n Thanks to the aforementioned identity T d = S α, changing the variable gives corresponding series representations for the density f S = αx (α+) f T (x α ). The first one, at zero, had been obtained originally in [] after solving some fractional integral equation of the Abel type. The second one, at infinity, had been derived in [3] from the results of [] considerations on Wright s hypergeometric function. The same kind of results were subsequently obtained in [] for much more general stable processes with the help of advanced special functions techniques - see Section 7 therein. In the present spectrally positive framework, we stress that up to the Wiener-Hopf factorisation which is a fundamental tool for this type of questions, overall our argument to get series representations for f S only makes use of undergraduate mathematics. Corollary 7. One has both representations f S (x) = n f S (x) = n x αn 2 Γ(αn )Γ( + /α n), x x (αn+) Γ( αn) + αx (αn+2) Γ( αn)γ(/α + n), x >. n Remark. Notice that one really needs the two series to get the behaviour of f T its derivatives at zero. For instance one has f T (x) f x /α T αγ( α) (x) Γ( α)γ(/α) α f τ (x)

9 HITTING DENSITIES FOR STABLE PROCESSES 9 as x +. On the other h, only one series is important to get the behaviour of f S its derivatives at infinity: one has f S (x) x (α+) Γ( α) f (p) S (x) x (α+2+p) Γ( (α + + p)) as x + for every p (this is in accordance with the general asymptotics derived in [8], see the final remark therein). 4. Some remarks on unimodality Recall that a real rom variable X is said to be unimodal if there exists a R such that the functions P[X x] P[X > x] are convex respectively in (, a) (a, + ). If X has a density f X, this means that f X increases on (, a] decreases on [a, + ). We refer e.g. to Section 52 in [6] for more on this topic. Differentiating the density of U α : f U t /α α (t) = α(t 2 2t cos πα + ) (( 2 2α)t2 + 2t(α ) cos πα + ), we find that there is a unique positive root, so that the rom variable U α is unimodal. Since the same property is known to hold for ˆτ - see e.g. Theorem 53. in [6] for a much stronger result, in view of our main result it is natural to ask whether the variable τ is unimodal as well. This also seems plausible from the behaviour of f τ f τ at zero. In general, the product of two positive independent unimodal rom variables is not necessarily unimodal. However the notion of multiplicative strong unimodality which had been introduced in [3], will allow us to give a quick positive answer in half of the cases: Proposition 8. For every α 3/2, the rom variable τ is unimodal. Proof: Recalling that ˆτ is unimodal for every α (, 2), from (.3) Theorem 3.6 in [3] is is enough to show that the function t f Uα (e t ) is log-concave over R, in other words that the function g α (t) = log(e t 2 cos πα + e t ) is convex over R. Differentiating twice yields g α(t) = 4( cos πα cosh t) (e t 2 cos πα + e t ) 2 we see that g α is convex over R as soon as α 3/2. When α (3/2, 2) the above proof shows that U α is not multiplicative strongly unimodal (MSU) in the terminology of [3]. In order to get the unimodality of τ when α (3/2, 2), one could of course be tempted to obtain the MSU property for positive stable laws with index β > 2/3. Unfortunately, we showed in [8] that this property does not hold true as soon as β > /2. Nevertheless, in view of the unimodality of ˆτ that of τ for α = 2 - see also Rösler s result for general diffusions on the line [5], we conjecture that τ is unimodal for every value of α. At first sight the problem does not seem easy because of the unstability of the unimodality property under multiplicative convolution. See however our final remark. Differentiating now the density of T α : f T α (t) = t /α α(t 2 2t cos πα + ) 2 (( 2α)t2 2αt 2 /α +2t(α ) cos πα+2αt α cos πα+),

10 THOMAS SIMON we find after some simple analysis that there is also a unique positive root, so that the rom variable T α is unimodal. But contrary to U α, one can check that it is never MSU, so that the above simple argument cannot be applied. Nevertheless we have the Proposition 9. For every α 3/2, the rom variable T is unimodal. Proof: From (3.) the so-called Kanter representation for positive stable laws - see Corollary 4. in [], we have the identity T d = T α L α b α (U) where L is a stard exponential variable, U an independent uniform variable over [, π], b α (u) = (sin(( /α)u)/ sin(u)) α sin(u/α)/ sin(u) is a strictly increasing function from (, π) to R +. Besides, from the beginning of the proof of Theorem 4. in [], we know that the logarithmic derivative of b α is positive strictly increasing, so that the same holds for b α itself, because b α is also positive strictly increasing. By Lemma 4.2 in [] we conclude that b α (U) is unimodal: again, from (3.) Theorem 3.6 in [3], it suffices to show that the density t f Tα L α (e t ) is log-concave over R. We compute f Tα L α (sin πα)x α α(x) = πα(α ) exp (x/u) α u α ( + u /α ) du, x. u 2 2u cos πα + It is easily seen that u +u /α is log-concave over R + (t, u) e t /u log-convex, hence convex, over R R +. In particular, the function (t, u) ( + u /α ) exp ( e t /u ) α is log-concave over R R + by Prékopa s theorem, it is enough to show that the function u u α /(u 2 2u cos πα + ) is log-concave over R +. We compute its second logarithmic derivative, which equals (2α 3)u 4 + 4u 3 (2 α) cos πα (4(2 α) cos 2 πα + 2α)u 2 + 4u cos πα (α )u 2 (u 2 2u cos πα + ) 2 we see that it is negative over R + as soon as α 3/2. Final Remark. The above proof does not work to obtain the unimodality of T when α > 3/2, we do not know as yet how to tackle this situation. However, we can use the same argument to obtain the unimodality of τ when 3/2 < α + / 2 < 2. Details, which are technical, can be obtained upon request. All these questions will be the matter of future research. Acknowledgements. Part of this work was done during a sunny stay at the University of Tokyo I am grateful to N. Yoshida for his hospitality. Ce travail a aussi bénéficié d une aide de l Agence Nationale de la Recherche portant la référence ANR-9-BLAN-84-.

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