Dependent jump processes with coupled Lévy measures

Transcription

1 Dependent jump processes wth coupled Lévy measures Naoufel El-Bachr ICMA Centre, Unversty of Readng May 6, 2008 ICMA Centre Dscusson Papers n Fnance DP Copyrght 2008 El-Bachr. All rghts reserved. ICMA Centre The Unversty of Readng Whteknghts PO Box 242 Readng RG6 6BA UK Tel: +44 (0) Fax: +44 (0) Web: Drector: Professor John Board, Char n Fnance The ICMA Centre s supported by the Internatonal Captal Market Assocaton

2 ABSTRACT I present a smple method for the modelng and smulaton of dependent postve jump processes through a seres representaton. Each consttuent process s represented by a seres whose terms are equal to a transformaton of the jump tmes of a standard Posson process. The transformatons are gven by the nverses of the respectve margnal Lévy tal mass ntegral functons. The dependence between the varous consttuent processes s gven by a probablstc copula for the nter-arrval tmes of the varous standard Posson processes MSC code: 60C05, 60G5, 60G55 Keywords: Lévy copulas, Copulas, Lévy processes, Monte-Carlo smulatons Naoufel El-Bachr PhD Student, ICMA Centre, Unversty of Readng, Readng, RG6 6BA, UK. Emal: n.el-bachr@cmacentre.ac.uk

3 INTRODUCTION If Y s a random varable wth dstrbuton functon F then F(Y) s unformly dstrbuted on [0,. Recprocally, f V s unformly dstrbuted on [0, then F (V) has dstrbuton functon F. The counterpart result for Lévy measures states: f (X t ) (t 0) s a jump process wth the tal mass of ts Lévy measure gven by the functon U then {U( X k )} (k ) are dstrbuted as jump tmes of a standard Posson process, where { X k } denotes a sequence of jumps of the process ordered by decreasng magntude. Recprocally, f {Γ k } s a sequence of jump tmes of a standard Posson process then {U (Γ k )} s equal n dstrbuton to a sequence of ordered jumps of the process (X t ). Snce jump tmes of a standard Posson process are unformly dstrbuted across tme, t means that Lévy measure ntegrals are "unformly dstrbuted on [0, )" as probablty-ntegrals are unformly dstrbuted on [0,. A multdmensonal Lévy measure can be constructed by lnkng margnal Lévy measures through a Lévy copula. A Lévy measure ν(b), for B B(R d ) s the expected number per unt tme of jont jumps whose szes belong to B. Any probablstc (ordnary) copula C s a jont dstrbuton functon of standard unform random varables: C(v,..., v n ) = P[V v,..., V n v n. Smlarly, any Lévy copula C L (x,..., x d ) s the expected number of jumps by a vector of standard Posson processes whose tmes of arrval occur jontly before x,..., x d : C L (x,..., x d ) = E[#{k : Γ k x,..., Γ d k x d } (.) Consequently, smulatng paths of a multdmensonal jump process when the dependence s specfed through a Lévy copula s fundamentally related to the smulaton of standard Posson processes. The jump tmes of the Posson processes are dependent n such a way as to satsfy the property n equaton (.) for all (x,..., x d ). There are two equvalent methods to acheve ths requrement: ether through the jont dstrbuton of the jump tmes or through the jont dstrbuton of successve nter-arrval tmes. We can start wth a Lévy copula and derve the mpled dstrbutons as n Tankov (2003a). El-Bachr (2008) dscusses a condtonal samplng technque sutable for ths approach. Alternatvely, ths paper shows that we can construct new Lévy copulas by drectly specfyng these dstrbutons. A postve pure jump Lévy process (X t ) (t>0) has statonary and ndependent postve jumps and s of fnte varaton,.e. lm X tk+ X tk < where 0 = t < t 2 < < t n = t and tk 0 t k t t k = t k+ t k. The dstrbuton of X t for any tme t > 0 s nfntely dvsble and ts characterstc functon satsfes the Lévy-Khntchne formula: E [ e z,x t = e tψ(z), z R d (.2) ( Ψ(z) = ) e z,x ν(dx) (.3) R d where ν s a measure on R d \{0} such that ( x )ν(dx) <. The tal mass of the Lévy measure s defned as U(x,..., x d ) = ν([x, ) [x d, )) for (x,..., x d ) R d \{0} and such that U(x,..., x d ) = 0 f x j = for some j,..., d and U s fnte everywhere except at zero, U(0,..., 0) =. The margnal Lévy measures have as ther tal masses the margns U k (x k ) = U(0,..., 0, x k, 0,..., 0) of the multdmensonal Lévy measure s tal mass. In ths paper, I only consder contnuous tal mass ntegrals. Copyrght 2008 El-Bachr

4 Tankov (2003a) defnes a d dmensonal Lévy copula as a d ncreasng grounded functon F : [0, d [0, wth margns F k, k =... d, whch satsfy F k (u) = u, u [0,. He also extends Sklar (959) s theorem showng that any d dmensonal tal mass U can be constructed as a Lévy copula takng the margns of U as arguments. Conversely, f F s a Lévy copula and U,..., U d are one-dmensonal tal masses, then U(x,..., x d ) = F(U (x ),..., U d (x d )) defnes a d dmensonal tal mass. Whle Tankov (2003a) focuses the analyss on Lévy processes wth postve jumps only, Kallsen and Tankov (2006) extends t to general Lévy processes wth both postve and negatve jumps. Lévy copulas for processes wth both postve and negatve jumps can be constructed from postve Lévy copulas but are less tractable. Fortunately, processes wth jumps of the same sgn are already a rch enough class for varous applcatons. For the sake of clarty I focus on the case of processes wth postve jumps only, a.k.a. subordnators. 2 STANDARD POISSON REPRESENTATION FOR LÉVY PROCESSES Ferguson and Klass (972) shows that the ordered jump magntudes X, X 2,... of a Lévy jump process (X t ) have the same dstrbuton as U (Γ ), U (Γ 2 ),..., where {Γ k } denotes the jump tmes of a standard Posson process. Ths justfes the seres representaton of the process (X t ) on the unt tme nterval t [0, : X t = = U (Γ ) {V [0,t} (2.) where {V } s a sequence of..d. random varables unformly dstrbuted on [0,. The proof conssts n showng that the seres converges for each t and the resultng process s a Lévy process wth the same characterstc exponent as X t. The same result follows as the converse of the drect statement n theorem 2., where t s shown that the dstrbuton of U( X ), U( X 2 ),... s the same as the dstrbuton of Γ, Γ 2,.... The alternatve proof 2 presented here has the advantage of makng explct the lnk wth the correspondng theorem for dstrbuton functons of random varables. Theorem 2.. Let (X t ) (t [0,) be a one-dmensonal Lévy process on the unt tme nterval wth Lévy measure densty ν. Denote the tal mass functon of ν by U(x) := ν(y)dy wth the nverse functon x U (x) = nf{u > 0 : U(u) x}. If { X } ( ) s a sequence of ordered jumps (magntudes) of (X t ) along a sample path, then {U( X )} ( ) are dstrbuted as the jump tmes of a standard Posson process. Conversely, f {Γ } ( ) s a sequence of jump tmes of a standard Posson process, then {U (Γ )} ( ) have the same dstrbuton as the ordered jumps { X } of the process (X t ). We restrct the dscusson to processes on the unt tme nterval for smplcty but the case of an arbtrary fnte tme nterval [0, T can be handled by smply replacng the sequence of {Γ } by { Γ T } and the sequence {V } of..d. unforms on [0, by a sequence {TV } of..d. unforms on [0, T. 2 The proof n Fergusson and Klass (972) s more nvolved but more general as t ncludes the case of nfnte varaton and processes wth fxed ponts of dscontnuty. Copyrght 2008 El-Bachr 2

5 Proof. Let (X ε t) (t 0) denote the compound Posson approxmaton of the process (X t ) obtaned by truncatng jumps that are smaller than ε: X ε t = N t ξ = where (N t ) (t 0) s a Posson process wth ntensty θ = U(ε), and (ξ ) ( ) an..d. sequence wth dstrbuton ν(x) θ {x>ε}. The process (Xt) ε s a Lévy process wth characterstc exponent Ψ ε (z) = ( e zx )ν(dx). Furthermore, lm ε 0 Ψ ε (z) = Ψ(z), where Ψ(z) represents the characterstc exponent of the Lévy ε process (X t ). Hence, as ε 0 we have the convergence n law: Xt ε X t by Lévy s contnuty theorem. Snce U(x) U( X ε ) θ θ represents the cumulatve dstrbuton functon of the jump magntudes of X ε t, then s dstrbuted as a unform random varable on [0,. Hence, {U( X ε )} ( ) s a sequence of..d. unforms on [0, θ. An ordered verson of ths sequence has the same dstrbuton as a sequence of jump tmes of a standard Posson process on the nterval [0, θ. When ε 0, we have θ and the convergence n law: X ε t X t. By contnuty of U, we can deduce the convergence n law: U( X ε ) U( X ) and conclude that {U( X )} are dstrbuted as the jump tmes of a standard Posson process. Usng the fact that U (U(x)) = x, the converse statement s deduced from: P[U (Γ ) x = P[U (U( X )) x = P[ X x For any probablstc (ordnary) copula C, C(v,..., v n ) s the probablty that the components of a vector of standard unform random varables are jontly smaller that v,..., v n : C(v,..., v n ) = P[V v,..., V n v n. Theorem 2.2 states a smlar result for Lévy copulas where the probablty s replaced by the expected number of vectors, and the standard unforms are replaced by jump tmes of standard Posson processes,.e. "unforms on [0, ". For any Lévy copula F, F(x,..., x d ) s the expected number of vectors of jump tmes of standard Posson processes whose components are jontly smaller than x,..., x d : F(x,..., x d ) = E[#{k : Γ k x,..., Γ d k x d}. Theorem 2.2. If F s a d-dmensonal Lévy copula, then for all x,..., x d [0, ), F(x,..., x d ) s the expected number k of jump tmes Γ k,..., Γd k of a d-dmensonal vector of standard Posson processes occurrng before x,..., x d respectvely: F(x,..., x d ) = E[#{k : Γ k x,..., Γ d k x d } (2.2) Furthermore, for all k the jont dstrbuton of the vector (Γ k,..., Γd k ) s: P [ Γ k y,..., Γ d k y d = y 0 e y y k (k )! xf(x, y 2,..., y d ) x=y dy (2.3) Copyrght 2008 El-Bachr 3

6 Equvalently, for all k the condtonal jont dstrbuton of the vector of nter-arrval tmes (τk,..., τd) := k (Γ k Γ k,..., Γd k Γd ) k gven the prevous jump tmes Γ k := (Γ k,..., Γd k ) s gven by: P [ τ k z,..., τ d k z z d /Γ k = e y x F(x + Γ k, z 2 + Γ 2 k,..., z d + Γ d k ) x=y dy (2.4) 0 ( Proof. For some one-dmensonal Lévy tal mass functon U, we have that U U (x) ) = x. Thus, F(x,..., x d ) = F ( ( U U (x ) ) (,..., U d U d (x d ) )). Snce U,..., U d are margnal tal mass functons and F s a Lévy copula, Tankov s extenson to Sklar s theorem mples the exstence of a process (Y t ) wth d-dmensonal Lévy tal mass U such that U(x,..., x d ) = F(U (x ),..., U d (x d )). Hence, F(x,..., x d ) = U ( U (x ),..., U (x d d) ). Usng the defnton of a Lévy measure, we can then deduce: F(x,..., x d ) = E [ #{t [0, : Y t [U (x ), ),..., Y d t [U d (x d ), )} Note the equvalence between the events { Yt [U (x ), )} and {U ( Yt ) (0, x }. We can now use the result of theorem 2. to conclude that: F(x,..., x d ) = E[#{k : Γ k x,..., Γ d k x d } where {Γ k },..., {Γd k } are sequences of standard Posson jump tmes. Denote the condtonal jont dstrbuton of Γ 2 k,..., Γd k condtonal on Γ k = x by P x : P x (x 2,..., x d ) := P [ Γ 2 k x 2,..., Γ d k x d /Γ k = x Snce Γ k,..., Γd k have the same dstrbuton as U ( Yk ),..., U d( Yk d) where { Y k } denotes the sequence of jumps of Y ordered by decreasng magntudes, we can wrte: P x (x 2,..., x d ) = P [ U 2 ( Y 2 k ) x 2,..., U d ( Y d k ) x d /U ( Y k ) = x Ths last condtonal dstrbuton functon has been shown n Tankov (2003b) to satsfy: We can now deduce: P [ U ( Y 2 k ) x 2,..., U d ( Y d k ) x d /U ( Y k ) = x = x F(x, x 2..., x d ) x=x P [ y Γ k y,..., Γ d k y d = P x (y 2,..., y d ) f Γ (x)dx k and replace the probablty densty f Γ of Γ k k by ts expresson to obtan equaton (2.3). To deduce equaton (2.4), note that: P [ [ τ k z,..., τ d k z d /Γ k = P Γ k z + Γ k,..., Γ d k z d + Γ d k and τk = Γ k Γ k s a standard exponental random varable. 0 Copyrght 2008 El-Bachr 4

7 3 EQUIVALENCE BETWEEN THE LÉVY COPULA AND AN ORDINARY COPULA Smulatng paths of a multdmensonal jump process s therefore related to the smulaton of standard Posson processes. The dependence between the components s gven by the jont dstrbuton of ther jump tmes n equaton (2.3) or equvalently by the condtonal jont dstrbuton of ther nter-arrval tmes n equaton (2.4). We can start wth a Lévy copula and derve the mpled dstrbutons. Ths s the procedure followed by Tankov (2003a) generalzng the seres representaton for multdmensonal Lévy processes: X k t = = U k (Γ k ) {V [0,t}, k =,..., d, t [0, (3.) where {Γ },..., {Γ d } are d random sequences ndependent from {V }. The sequence {Γ } represent the jump tmes of a standard Posson process, whle the vector (Γ 2,..., Γd ) condtonally on Γ has dstrbuton functon x F(x,..., x d ) x =Γ. Alternatvely, we can construct new Lévy copulas by drectly specfyng the dstrbutons n equaton (2.3) or equaton (2.4). Snce the dstrbuton functon of exponental random varables and ts nverse are more tractable than those of jump tmes of Posson processes, workng wth the dstrbuton of nter-arrval tmes s preferred and allows the use of probablstc copulas to model the dependence between the nter-arrval tmes: P [ Γ k x,..., Γ d k x d /Γ k = P [ τ k x Γ k,..., τ d k x d Γ d k (3.2) = C ( P[τ k x Γ k,..., P[τ d k x d Γ d k ) where C denotes some sutable probablstc copula whose exstence s guaranteed by Sklar s theorem. The sequences {Γ k },..., {Γd k } are jump tmes of standard Posson processes, thus the nter-arrval tmes of a gven sequence are..d. standard exponental random varables. Intutvely, dependence between the varous sequences necessarly translates nto dependence between the nterarrval exponental random varables of the dfferent Posson processes. Ths s formalzed n theorem 3., where the equvalence between a Lévy copula and a probablstc copula for nterarrval tmes of the standard Posson processes s proved. Theorem 3.. Let (X t ) := (X t,..., X d t ) be a d dmensonal Lévy jump process wth margnal Lévy tal mass functons U,..., U d. The process (X t ) has a unque Lévy copula F : [0, d [0, f and only f there exsts a unque probablstc copula C : [0, d [0, such that for all : P [ U ( X ) x,..., U d ( X d ) x d / X,..., X d = ) C ( e (x U ( X )),..., e (x d U d ( Xd )) (3.3) where X j,..., Xj,... denote the ordered jumps of the process (Xj t). Proof. Let P denote the probablty condtonal on the nformaton X,..., Xd. If F s a Copyrght 2008 El-Bachr 5

8 Lévy copula, then theorem 2.2 states that P [ U ( X ) x,..., U d ( X d ) x d = P [ Γ x,..., Γ d x d /Γ = P [ τ x Γ,..., τ d x d Γ d where {Γ k },..., {Γd k } are sequences of jumps tmes of standard Posson processes whose dependence s unquely determned by the Lévy copula such that equaton (2.4) s satsfed. Snce P[τ j x j Γ j = e (x j Γj ), Sklar s theorem mples the exstence of a unque copula C such that equaton (3.3) holds. Conversely, f (Xt,..., Xt d ) s a d-dmensonal jump process such that equaton (3.3) s satsfed, then theorem 2. guarantees that for all j =,..., d, the dstrbuton of U j ( X j ), U j( X2), j... s the same as that of the jump tmes Γ j, Γj 2,... of a standard Posson process. Let the jont transton probablty from Γ k,..., Γd k to Γ k,..., Γd k be gven by: P [ ( ) Γ k x,..., Γ d k x d /Γ k = C e (x Γ k ),..., e (x d Γd k ) We can deduce by convoluton and usng Sklar s theorem that for all k there exsts a unque copula C k such that P [ ( Γ k x,..., Γ d k x d = Ck P[Γ k x,..., P[Γ d k x d ) Let F : [0, d [0, be the functon: F(x,..., x d ) = k= kc k ( P[Γ k x,..., P[Γ d k x d ) We can deduce from the propertes of the copulas C k, k =, 2,... that F s a grounded, d ncreasng functon wth unform margns. Thus, we can conclude that F s a Lévy copula. 4 CONCLUDING REMARKS The result of theorem 3. justfes the use of probablstc (ordnary) copulas to model the dependence between Lévy jump processes by specfyng jont dstrbutons for the ncrements of margnal tal mass ntegrals Γ ks. Ths has some advantages n terms of tractablty over usng Lévy copulas. Indeed, some tractable probablstc copulas have known effcent algorthms for smulaton that avod condtonal samplng. Furthermore, statstcal fttng of Lévy copulas requres ether computatonally expensve smulaton-based estmaton or dervng a complcated lkelhood functon for Γ k,..., Γd k by successve condtonng. On the other hand, estmaton procedures for probablstc copulas are readly avalable. Copyrght 2008 El-Bachr 6

9 REFERENCES [ El-Bachr, N.: Condtonal samplng for jump processes wth Lévy copulas. ICMA centre Dscusson Papers n Fnance DP (2008). [2 Ferguson, T. S., and Klass, M. J.: A representaton of ndependent ncrement processes wthout Gaussan components. The Annals of Mathematcal Statstcs, Vol 43, n. 3, pp (972). [3 Kallsen, J., and Tankov, P.: Characterzaton of dependence of multdmensonal Lévy processes usng Lévy copulas. Journal of Multvarate Analyss, Vol 97, pp (2006). [4 Rosńsk, J.: Seres representatons of Lévy processes from the perspectve of pont processes. In: Lévy Processes-Theory and Applcatons, Barndorff-Nelsen, O., Mkosch, T., and Resnck, S., eds., Brkhäuser (200). [5 Sklar, A.: Fonctons de répartton à n dmensons et leurs marges. Publcatons de l Insttut de Statstque de L Unversté de Pars 8, pp (959). [6 Tankov, P.: Dependence structure of spectrally postve multdmensonal Lévy processes. Workng paper, Ecole Polytechnque (2003a). [7 Tankov, P.: Dependence structure of Lévy processes wth applcatons n rsk management. Rapport Interne 502, CMAP, Ecole Polytechnque (2003b). Copyrght 2008 El-Bachr 7