Cramér asymptotics for finite time first passage probabilities of general Lévy processes
aa r X i v : . [ m a t h . P R ] A p r Cram´er asymptotics for finite time first passageprobabilities of general L´evy processes
Zbigniew Palmowski ∗ Martijn Pistorius † October 31, 2018
Abstract
We derive the exact asymptotics of P (sup u ≤ t X ( u ) > x ) if x and t tend to infinity with x/t constant, for a general L´evy process X that admits exponential moments. The proof is based on a renewalargument and a two-dimensional renewal theorem of H¨oglund [9]. The study of boundary crossing probabilities of L´evy processes has appli-cations in many fields, including ruin theory (see e.g. Rolski et al. [13]and Asmussen [2]), queueing theory (see e.g. Borovkov [6] and Prabhu[11]), statistics (see e.g. Siegmund [15]) and mathematical finance (see e.g.Roberts and Shortland [12]).As in many cases closed form expressions for (finite time) first passageprobabilities are either not available or intractable, a good deal of the liter-ature has been devoted to logarithmic or exact asymptotics for first passageprobabilities, using different techniques. Martin-L¨of [10] and Collamore [7]derived large deviation results for first passage probabilities of a generalclass of processes. Employing two-dimensional renewal theory and asymp-totic properties of ladder processes, respectively, H¨oglund [9] and von Bahr[3] obtained exact asymptotics for ruin probabilities of the classical riskprocess (see also Asmussen [2]). Bertoin and Doney [5] generalised the clas-sical Cram´er-Lundberg approximation (of the perpetual ruin probability ofa classical risk process) to general L´evy processes. ∗ University of Wroc law, pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland, E-mail:[email protected] † Imperial College London, Department of Mathematics, South Kensington Campus,London SW7 2AZ, UK, E-mail:[email protected]
1n this paper we obtain the exact asymptotics of the finite time ruinprobability P ( τ ( x ) ≤ t ), where τ ( x ) = inf { t ≥ X ( t ) > x } , for a generalL´evy process X ( t ) ( X (0) = 0), if x and t jointly tend to infinity in fixedproportion, generalising Arfwedson [1] and H¨oglund [9] who treated the caseof a classical risk process. The proof is based on an embedding of the ladderprocess of X and a two-dimensional renewal theorem of H¨oglund [9].The remainder of the paper is organized as follows. In Section 2 themain result is presented, and its proof is given in Section 3. Let X be a L´evy process with non-monotone paths that satisfies E [e α X (1) ] < ∞ for some α > , (2.1)and denote by τ ( x ) = inf { t ≥ X ( t ) > x } the first crossing time of x . Weexclude the case that X is a compound Poisson process with non-positiveinfinitesimal drift, as this corresponds to the random walk case which hasalready been treated in the literature.The law of X is determined by its Laplace exponent ψ ( θ ) = log E [e θX (1) ]that is well defined on the maximal domain Θ = { θ ∈ R : ψ ( θ ) < ∞} .Restricted to the interior Θ o , the map θ ψ ( θ ) is convex and differentiable,with derivative ψ ′ ( θ ). Moreover, ψ ′ (0+) = E [ X (1)] if E [ | X (1) | ] < ∞ . Bythe strict convexity of ψ , it follows that ψ ′ is strictly increasing on (0 , ∞ )and we denote by Γ : ψ ′ (0 , ∞ ) → (0 , ∞ ) its right-inverse function.Associated to the measure P is the exponential family of measures { P ( c ) : c ∈ Θ } defined by their Radon-Nikodym derivativesd P ( c ) d P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t = exp ( cX ( t ) − ψ ( c ) t ) . (2.2)It is well known that under this change of measure X is still a L´evy processesand its new Laplace exponent satisfies ψ ( c ) ( α ) = ψ ( α + c ) − ψ ( c ) . (2.3)Related to X and its running supremum are the local time L of X atits supremum, its right-continuous inverse L − and the upcrossing ladder For θ ∈ Θ \ Θ o , ψ ′ ( θ ) is understood to be lim η → θ,η ∈ Θ o ψ ′ ( η ). H respectively. The Laplace exponent κ of the bivariate (possiblykilled) subordinator ( L − , H ),e − κ ( α,β ) t = E [e − αL − t − βH t ( L − t < ∞ ) ] , (2.4)is related to ψ via the Wiener-Hopf factorisation identity u − ψ ( θ ) = kκ ( u, − θ ) b κ ( u, θ ) , u ≥ , θ ∈ Θ o , (2.5)for some constant k > b κ is the Laplace exponent of the dual lad-der process. Refer to Bertoin [4, Ch. VI] for further background on thefluctuation theory of L´evy processes.Bertoin and Doney [5] showed that, if the Cram´er condition holds, thatis γ >
0, where γ := sup { θ ∈ Θ : ψ ( θ ) = 0 } , (2.6)the Cram´er-Lundberg approximation remains valid for a general L´evy pro-cess: lim x →∞ e γx P [ τ ( x ) < ∞ ] = C γ , (2.7)where C γ ≥ E [e γX (1) | X (1) | ] < ∞ and is thengiven by C γ = β γ / [ γm γ ], where β γ = − log P [ H < ∞ ] , m γ = E [e γH H ( H < ∞ ) ] . Further, Doob’s optional stopping theorem implies the following bound:e γx P ( τ ( x ) < ∞ ) = E ( γ ) [e − γ ( X ( τ ( x )) − x ) ( τ ( x ) < ∞ ) ] ≤ . (2.8)The result below concerns the asymptotics of the finite time ruin proba-bility P ( τ ( x ) ≤ t ) when x, t jointly tend to infinity in fixed proportion. Fora given proportion v the rate of decay is either equal to γvt or to ψ ∗ ( v ) t ,where ψ ∗ is the convex conjugate of ψ : ψ ∗ ( u ) = sup α ∈ R ( αu − ψ ( α )) . We restrict ourselves to L´evy processes satisfying the following condition σ > σ denotes the Gaussian coefficient of X . Recall that a measure iscalled non-lattice if its support is not contained in a set of the form { a + bh, h ∈ Z } , for some a, b >
0. Note that (H) is satisfied by any L´evy processwhose L´evy measure has infinite mass.We write f ∼ g if lim x,t →∞ ,x = vt +o( t / ) f ( x, t ) /g ( x, t ) = 1.3 heorem 1 Assume that ( H ) holds. Suppose that < ψ ′ ( γ ) < ∞ and thatthere exists a Γ( v ) ∈ Θ ◦ such that ψ ′ (Γ( v )) = v . If x and t tend to infinitysuch that x = vt + o( t / ) then P ( τ ( x ) ≤ t ) ∼ ( C γ e − γx , if < v < ψ ′ ( γ ) , D v t − / e − ψ ∗ ( v ) t , if v > ψ ′ ( γ ) ,with C = 1 and D v given by D v = − v log E [e − η v L − ( L − < ∞ ) ] η v E [e Γ( v ) H − η v L − H ( L − < ∞ ) ] × v ) p πψ ′′ (Γ( v )) , where η v = ψ (Γ( v )) . Remark 1 (a) For a spectrally negative L´evy process the joint exponent ofthe ladder process is given by κ ( α, β ) = β + Φ( α ) ( α, β ≥ α ) isthe largest root of ψ ( θ ) = α , and thus D v = D v := vψ (Γ( v )) p πψ ′′ (Γ( v )) , C γ ≡ . (2.9)Indeed, D v = D v × κ ( η v , v ) ∂∂β κ ( η v , β ) | β = − Γ( v ) exp {− κ ( η v , − Γ( v )) } = D v × {− Φ( η v ) + Γ( v ) } = D v since Φ( η v ) = Γ( v ).(b) If X is spectrally positive, κ ( α, β ) = [ α − ψ ( − β )] / [ ˆΦ( α ) − β ] (see e.g.[4, Thm VII.4]), where ˆΦ( α ) is the largest root of ψ ( − θ ) = α and we findthat D v = Γ( v ) + ˜Γ( v )Γ( v )˜Γ( v ) 1 p πψ ′′ (Γ( v )) , C γ = ψ ′ (0) ψ ′ ( γ ) , where ˜Γ( v ) = sup { θ : ψ ( − θ ) = ψ (Γ( v )) } , recovering formulas that can befound in Arfwedson [1] and Feller [8] respectively, for the case of a classicalrisk process. 4 emark 2 Heuristically, in the case v > ψ ′ ( γ ), the asymptotics in Thm.1 can be regarded as a consequence of the central limit theorem, that is,under the tilted measure P Γ( v ) , asymptotically τ ( x ) − x/vω √ x follows a standard normal distribution, where by (2.3) and choice of Γ( v ), ω = Var (Γ( v )) [ X ] (cid:0) E (Γ( v )) [ X ] (cid:1) = ψ (Γ( v )) ′′ (0) (cid:0) ψ (Γ( v )) ′ (0) (cid:1) = ψ ′′ (Γ( v )) v . This explains why the asymptotics remain valid if x deviates o( x / ) =o( t / ) from the line vt .In the boundary case v = ψ ′ ( γ ), in which case E (Γ( v )) [ τ ( x )] = t , theexact asymptotics of P ( τ ( x ) ≤ t ) may depend on the way in which x/t tends to v . Note that this case is excluded from Theorem 1. Remark 3
In the case 0 < v < ψ ′ ( γ ), the asymptotics in Theorem 1 are aconsequence of the law of large numbers. To see why this is the case, notethat e γx P ( τ ( x ) ≤ t ) = e γx P ( τ ( x ) < ∞ ) − e γx P ( t < τ ( x ) < ∞ ), where thefirst term tends to C γ in view of (2.7), while for the second term the Markovproperty and (2.8) imply thate γx P ( t < τ ( x ) < ∞ )= Z x −∞ P ( τ ( x ) > t, X ( t ) ∈ dy )e γy e γ ( x − y ) P ( τ ( x − y ) < ∞ ) ≤ Z x −∞ P ( X ( t ) ∈ dy )e γy = P ( γ ) ( X ( t ) ≤ x ) , which tends to 0 as t tends to infinity in view of the law of large numberssince E ( γ ) [ X ( t )] = tψ ′ ( γ ) > x . The proof below deals with the case that v > ψ ′ ( γ ). The idea of the proof is to lift asymptotic results that have been estab-lished for random walks by H¨oglund [9] and Arfwedson [1] to the setting ofL´evy processes by considering suitable random walks embedded in the L´evyprocess (more precisely, in its ladder process). We first briefly recall theseresults following the H¨oglund [9] formulation.5 .1 Review of H¨oglund’s random walk asymptotics
Let (
S, R ) = { ( S i , R i ) , i = 1 , , . . . } be a (possibly killed) random walk start-ing from (0 ,
0) whose components S and R have non-negative increments,and consider the crossing probabilities G a,b ( x, y ) = P ( N ( x ) < ∞ , S N ( x ) > x + a, R N ( x ) ≤ y + b ) ,K a,b ( x, y ) = P ( N ( x ) < ∞ , S N ( x ) > x + a, R N ( x ) ≥ y + b ) , where a ≥ , b ∈ R and N ( x ) = min { n : S n > x } . Let F denote the (possiblydefective) distribution function of the increments of the random walk withjoint Laplace transform φ and set F ( u,v ) (d x, d y ) = e − ux − vy F (d x, d y ) /φ ( u, v ).Let V ( ζ ) = E ζ [( R E ζ [ S ] − S E ζ [ R ]) ] /E ζ [ S ] for ζ = ( ξ, η ) where E ζ denotes the expectation w.r.t. F ζ .For our purposes it will suffice to consider random walks that satisfy thefollowing non-lattice assumption (the analogue of the non-lattice assumptionin one dimension):The additive group spanned by the support of F contains R . (G)Specialised to our setting Prop. 3.2 in H¨oglund (1990) jointly with theremark given on p. 380 therein read as follows: Proposition 1
Assume that ( G ) holds, and that there exists a ζ = ( ξ, η ) with φ ( ζ ) = 1 such that v = E ζ [ S ] /E ζ [ R ] , where φ is finite in a neighbour-hood of ζ and (0 , η ) . If x, y tend to infinity such that x = vy + o( y / ) > then it holds that G a,b ( x, y ) ∼ D ( a, b ) x − / e xξ + yη if η > ,K a,b ( x, y ) ∼ D ( a, b ) x − / e xξ + yη if η < , for a ≥ , b ∈ R , where D ( a, b ) = C ( a, b ) · (2 πV ( ζ )) − / , with V ( ζ ) > and C ( a, b ) = 1 | η | E ζ [ S ] e bη Z ∞ a P ζ ( S ≥ x )e ξx d x. Denote by e , e , . . . a sequence of independent exp( q ) distributed randomvariables and by σ n = P ni =1 e i , with σ = 0, the corresponding partial sums,6nd consider the two-dimensional (killed) random walk { ( S i , R i ) , i = 1 , . . . } starting from (0 ,
0) with step-sizes distributed according to F ( q ) (d t, d x ) = P ( H σ ∈ d x, L − σ ∈ d t ) , and write G ( q ) for the corresponding crossing probability G ( q ) ( x, y ) = G , ( x, y ) = P ( N ( x ) < ∞ , R N ( x ) ≤ y ) . Note that F ( q ) is a probability measure that is defective precisely if X driftsto −∞ , with Laplace transform φ given by φ ( u, v ) = Z Z e − ut − vx F ( q ) (d t, d x ) = qq − κ ( u, v ) . The key step in the proof is to derive bounds for P ( τ ( x ) ≤ t ) in termsof crossing probabilities involving the random walk ( S, R ): Lemma 1
Let
M, q > . For x, t > it holds that G ( q ) ( x, t ) ≤ P ( τ ( x ) ≤ t ) ≤ G ( q ) ( x, t + M ) /h (0 − , M ) , (3.1) where h (0 − , M ) = lim x ↑ h ( x, M ) , with h ( x, t ) := P ( H σ > x, L − σ ≤ t ) .Proof: Let T ( x ) = inf { t ≥ H t > x } and note that τ ( x ) = L − T ( x ) . Byapplying the Markov property it follows that P ( τ ( x ) ≤ t ) = P ( T ( x ) < ∞ , L − T ( x ) ≤ t )= ∞ X n =1 P ( σ n − ≤ T ( x ) < σ n , L − T ( x ) ≤ t ) (3.2)= ∞ X n =1 P ( H σ n − ≤ x, H σ n > x, L − T ( x ) ≤ t )= ∞ X n =1 Z Z P ( H σ n − ∈ d y, L − σ n − ∈ d s ) × P ( H σ > x − y, L − T ( x − y ) ≤ t − s ) (3.3)= ∞ X n =0 F ( q ) ⋆n ⋆ f ( x, t ) = ( U ⋆ f )( x, t ) , (3.4)where U = P ∞ n =0 F ( q ) ⋆n , f ( x, t ) = P ( H σ > x, L − T ( x ) ≤ t ) and ⋆ denotesconvolution. Following a similar reasoning it can be checked that G ( q ) ( x, t ) = U ⋆ h ( x, t ) . (3.5)7n view of (3.4) and (3.5), the lower bound in (3.1) follows since f ( x, t ) ≥ h ( x, t ) , taking note of the fact that H σ > x precisely if T ( x ) < σ , while the upperbound in (3.1) follows by observing that for fixed M > h ( x, t + M ) ≥ P ( H σ > x, L − T ( x ) ≤ t, L − σ − L − T ( x ) ≤ M )= P ( H σ > x, L − T ( x ) ≤ t ) P ( L − σ ≤ M )= f ( x, t ) h (0 − , M ) , where we used the strong Markov property of L − and the lack of memoryproperty of σ . (cid:3) Applying H¨oglund’s asymptotics in Proposition 1 yields the following result:
Lemma 2
Let the assumptions of Proposition 1 hold true. If x, t → ∞ suchthat for v > ψ ′ ( γ ) we have x = vt + o( t / ) then G ( q ) ( x, t + M ) ∼ D q,M t − / e − ψ ∗ ( v ) t , M ≥ , where D q,M = v √ πψ ′′ (Γ( v )) C q,M with C q,M = e ψ (Γ( v )) M κ ( ψ (Γ( v )) , c v ψ (Γ( v ))Γ( v ) qq + κ ( ψ (Γ( v )) , , where c v = E [e Γ( v ) H − ψ (Γ( v )) L − H ( L − < ∞ ) ] . Lemma 2 is a consequence of the following auxiliary identities:
Lemma 3
Let u > γ , u ∈ Θ o . φ ( z, − u ) = 1 iff κ ( z, − u ) = 0 iff ψ ( u ) = z (3.6) ψ ′ ( u ) = E ( u ) [ X (1)] = E ( u ) [ H σ ] · ( E ( u ) [ L − σ ]) − (3.7) ψ ′′ ( u ) = E ( u ) [( H σ − ψ ′ ( u ) L − σ ) ] · ( E ( u ) [ L − σ ]) − = ψ ′ ( u ) E ( u ) [( H σ − ψ ′ ( u ) L − σ ) ] · ( E ( u ) [ H σ ]) − (3.8) ψ ∗ ( v ) = v Γ( v ) − ψ (Γ( v )) for v > with Γ( v ) ∈ Θ o . (3.9) Proof : Eq (3.6): Note that for u, z > b κ ( z, u ) >
0. In viewof the identity (2.5) the statement follows.8q (3.7): Note that if u > γ then by the fact that ψ (0) = ψ ( γ ) = 0 andthe strict convexity of ψ it follows that ψ ( u ) >
0. In view of (2.5) it followsthen that κ ( ψ ( u ) , − u ) = 0 for u ∈ Θ o , u > γ . Differentiating with respectto u shows that ψ ′ ( u ) = ∂ κ ( ψ ( u ) , − u )( ∂ κ ( ψ ( u ) , − u )) − . (3.10)Also, note that E ( u ) [ H σ ] = q − E ( u ) [ H ], E ( u ) [ L − σ ] = q − E ( u ) [ L − ] and E ( u ) [ H ] = ∂ κ ( ψ ( u ) , − u ) , E ( u ) [ L − ] = ∂ κ ( ψ ( u ) , − u ) . Eq (3.8) follows as a matter of calculus, by differentiation of (3.10) withrespect to u . Finally, Eq. (3.9) follows from the definition of ψ ∗ . (cid:3) Proof of Lemma 2
The proof follows by an application of Prop. 1 to G ( q ) ( x, t + M ) with( S , R ) = ( H σ , L − σ ) and ζ = ( − Γ( v ) , η v ) . Note that, by (3.6) with u = Γ( v ), φ ( ζ ) = 1, and that η v = ψ (Γ( v )) > v > ψ ′ ( γ ). For this choice of the parameters, E ζ [ S ] = E (Γ( v )) [ H σ ] = c v /q ,and Eqs. (3.9), (3.7),(3.8) imply that ξx + ηt = − ψ ∗ ( v ) t and V ( ζ ) = ψ ′′ (Γ( v )) /ψ ′ (Γ( v )) = ψ ′′ (Γ( v )) /v. To complete the proof we are left to verify the form of the constants. Thecalculation of the C q,M = C (0 , ηM goes as follows: C q,M = q e ψ (Γ( v )) M ψ (Γ( v )) c v (cid:18)Z ∞ e − Γ( v ) x E [e Γ( v ) H σ − ψ (Γ( v )) L − σ ( x ≤ H σ < ∞ ) ]d x (cid:19) = q e ψ (Γ( v )) M ψ (Γ( v ))Γ( v ) c v (cid:16) − E [e − ψ (Γ( v )) L − σ ( L − σ < ∞ ) ] (cid:17) = q e ψ (Γ( v )) M ψ (Γ( v ))Γ( v ) c v (cid:18) − qq + κ ( ψ (Γ( v )) , (cid:19) = q e ψ (Γ( v )) M ψ (Γ( v ))Γ( v ) c v κ ( ψ (Γ( v )) , q + κ ( ψ (Γ( v )) , , in view of the definition (2.4) of κ . Combining all results completes theproof. (cid:3) As final preparation for the proof of Theorem 1 we show that the non-lattice condition holds:
Lemma 4
Suppose that ( H ) holds true. Then F ( q ) satisfies (G). roof : The assertion is a consequence of the following identity betweenmeasures on (0 , ∞ ) (which is itself a consequence of the Wiener-Hopf fac-torisation, see e.g. Bertoin [4, Cor VI.10]) P ( X t ∈ d x )d t = t Z ∞ P ( L − u ∈ d t, H u ∈ d x ) u − d u. (3.11)Fix ( y, v ) ∈ (0 , ∞ ) in the support of µ X (d t, d x ) = P ( X t ∈ d x )d t andlet B be an arbitrary open ball around ( y, v ). Then µ X ( B ) >
0; in viewof the identity (3.11) it follows that there exists a set A with positiveLebesgue measure such that P (( L − u , H u ) ∈ B ) > u ∈ A andthus P (( L − σ , H σ ) ∈ B ) >
0. Since B was arbitrary we conclude that ( y, v )lies in the support of F ( q ) . To complete the proof we next verify that if aL´evy process X satisfies (H) then µ X satisfies (G). To this end, let X satisfy(H). Suppose first that its L´evy measure ν has infinite mass or σ >
0. Then P ( X t = x ) = 0 for any t > x ∈ R , according to Sato [14, Thm. 27.4]. Thus, the support of P ( X t ∈ d x ) is uncountable for any t >
0, so that µ X satisfies (G). If ν has finite mass then it is straightforward to verify that P ( X t ∈ d x ) is non-lattice for any t > ν is, and that then µ X satisfies(G). (cid:3) Proof of Theorem 1:
Suppose that v > ψ ′ ( γ ) (the case v < ψ ′ ( γ ) wasshown in Remark 3). Writing l ( t, x ) = t / e ψ ∗ ( v ) t P ( τ ( x ) ≤ t ), Lemmas 1, 2and 3 imply that s = lim sup x,t →∞ ,x = tv +o( t / ) l ( t, x ) ≤ D q,M /h (0 − , M ) ,i = lim inf x,t →∞ ,x = tv +o( t / ) l ( t, x ) ≥ D q, . By definition of h and D q,M it directly follows that, as q → ∞ , D q, → D v , D q,M → D v e ψ (Γ( v )) M and h (0 − , M ) = P ( L − σ ≤ M ) → . Letting M ↓ s = i = D v , and the proof is complete. (cid:3) Acknowledgments
We thank Florin Avram for helpful advice and useful comments. We aregrateful to the referee for supplying the short direct proof (Remark 3) forthe case v < ψ ′ ( γ ). This work is partially supported by EPSRC grantEP/D039053/1 and the Ministry of Science and Higher Education of Polandunder the grant N N2014079 33 (2007-2009).10 eferences [1] Arfwedson, G. (1955) Research in collective risk theory. II. Skand. Ak-tuarietidskr. , 53–100.[2] Asmussen, S. (2000) Ruin Probabilities.
World Scientific, Singapore.[3] von Bahr, B. (1974) Ruin probabilities expressed in terms of ladderheight distributions,
Scand. Actuar. J. L´evy processes.
Cambridge University Press.[5] Bertoin, J. and Doney, R.A. (1994) Cram´er’s estimate for L´evy pro-cesses.
Stat. Prob. Lett. , 363–365.[6] Borovkov, A.A. (1976) Stochastic processes in queueing theory .Springer, Berlin.[7] Collamore, J.F. (1996) Large deviations for first passage times.
Ann.Prob. , 2065–2078.[8] Feller, W. (1971) An introduction to probability theory and its applica-tions. Vol II. 2nd ed.
Wiley.[9] H¨oglund, T. (1990) An asymptotic expression for the probability of ruinwithin finite time.
Ann. Prob. , 378–389.[10] Martin-L¨of, A. (1986) Entropy estimates for the first passage time of arandom ralk to a time dependent barrier. Scand J. Statist. , 221–229.[11] Prabhu, N.U. (1997) Insurance, queues, dams . Springer.[12] G. O. Roberts and C. F. Shortland (1997)
Math. Finance , 83–93.Pricing Barrier Options with TimeDependent Coefficients[13] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999) StochasticProcesses for Insurance and Finance.
Wiley, Chichester.[14] Sato, K. (1999)
L´evy processes and infinitely divisible distributions.