Theory of Sochasic Processes Vol. 14 3), no. 2, 28, pp. 139 144 UDC 519.21 JOSEP LLUÍS SOLÉ AND FREDERIC UTZET A FAMILY OF MARTINGALES GENERATED BY A PROCESS WITH INDEPENDENT INCREMENTS An explici procedure o consruc a family of maringales generaed by a process wih independen incremens is presened. The main ools are he polynomials ha give he relaionship beween he momens and cumulans, and a se of maringales relaed o he umps of he process called Teugels maringales. 1. Inroducion In his work, we presen an explici procedure o generae a family of maringales from a process X = {X, } wih independen incremens and coninuous in probabiliy. We exend our resuls exposed in [8], where we deal wih Lévy processes independen and saionary incremens); in ha case, he maringales obained were of he form M = P X,), where P x, ) is a polynomial in x and, and hen hey are ime space harmonic polynomials relaive o X. Here, he maringales consruced are polynomials on X bu, in general, no in. Par of he paper is devoed o define he Teugels maringales of a process wih independen incremens; such maringales, inroduced by Nualar and Schouens [5] for Lévy processes, are a building block of he sochasic calculus wih ha ype of processes. 2. Independen incremen processes and heir Teugels maringales Le X = {X, } be a process wih independen incremens, X =, coninuous in probabiliy and cadlag; such processes are also called addiive processes, and we will indisincly use boh names. Moreover, assume ha X is cenered and has momens of all orders. I is well known ha he law of X is infiniely divisible for all. Le σ 2 be he variance of he Gaussian par of X,andleν be is Lévy measure; for all hese noions, we refer o Sao [6] or Skorohod [7]. Denoe, by ν, he unique) measure on B, ) R ) defined by ν,] B) =ν B), B BR ), where R = R {}. By he sandard approximaion argumen, we have ha, for a measurable funcion f : R R and for every >, fx) νds, dx) < fx) ν dx) <,,] R R and, in his case, fx) νds, dx) = fx) ν dx).,] R R 2 AMS Mahemaics Subec Classificaion. Primary 6G51, 6G44. Key words and phrases. Process wih independen incremens, Cumulans, Teugels maringales. This research was suppored by gran BFM26-6247 of he Miniserio de Educación y Ciencia and FEDER. 139
14 JOSEP LLUÍS SOLÉ AND FREDERIC UTZET Noe ha since, for every, ν is a Lévy measure, ν is σ finie. To prove his, we observe ha ν { x > 1}) <, ν 1/n +1), 1/n]) <, andν [ 1/n, 1/n + 1))) <, n 1. So here is a numerable pariion of R wih ses of finie ν measure, >. Then, we can consruc a numerable pariion of, ) R, each se being wih finie ν-measure. Wrie NC) =#{ :, ΔX ) C}, C B, ) R ), he ump measure of he process, where ΔX = X X. I is a Poisson random measure on, ) R wih inensiy measure ν Sao [6, Theorem 19.2]). Define he compensaed ump measure dñ, x) =dn, x) d ν, x). TheprocessadmisheLévy Iô represenaion X = G + xdñ, x), 2),] R where {G, } is a cenered coninuous Gaussian process wih independen incremens and variance E[G 2 ]=σ2. The relaionship beween he momens of an infiniely divisible law and he momens of is Lévy measure is also well known see Sao [6,Theorem 25.4]). In our case, as he process has momens of all orders, for all, x ν dx) < and x n ν dx) <, n 2. { x >1} R Wrie F 2 ) =σ R 2 + x 2 ν dx) and F n ) = x n ν dx), n 3. 1) R Since { x >1} x ν dx) < and E[X ] =, he characerisic funcion of X can be wrien as φ u) =exp { 1 2 σ2 u2 + e iux 1 iux ) ν dx) }. R I is deduced ha, for n 2, F n ) is he cumulan of order n of X for n = 2, E[X 2 ]=F 2 )). Also σ 2 is coninuous and increasing Sao [6, Theorem 9.8]). Proposiion 1. The funcions F n ), n 2, are coninuous and have finie variaion on finie inervals, and, for n even, hey are increasing. Proof. Consider < u < < v,andwrieu =[u, v]. From he coninuiy in probabiliy of X, lim s, s U Xn s = X n, in probabiliy. Moreover, s U, X s sup r U X r, and since X is a maringale, by Doob s inequaliy, E [ sup r U X r n] C sup E[ X r n ] CE[ X v n ] <. r U So i follows by dominaed convergence ha he funcion E[X n ] is coninuous. Since he cumulans are polynomials of he momens, he coninuiy of all funcions F n ) is deduced.
MARTINGALES AND INDEPENDENT INCREMENT PROCESSES 141 To prove ha F n ) has finie variaion on finie inervals, consider a pariion of [,]: < < < k =. Then k Fn ) F n 1 ) k = x n νds, dx) =1 1, ] R k x n νds, dx) = 1, ] R x n νds, dx) <.,] R =1 =1 Consider he variaions of he process X see Meyer [4]): X 1) = X, X 2) X n) =[X, X] = σ 2 + = <s <s ΔXs ) n, n 3. ΔXs ) 2 By Kyprianou [2, Theorem 2.7], for n 3 he case n = 2 is similar), he characerisic funcion of X n) is exp {,] R e iux n 1 ) νds, dx) } =exp { R e iux 1 ) ν n) dx) }, where ν n) is he measure image of ν by he funcion x x n which is a Lévy measure. So X n) has independen incremens. Also by Kyprianou [2, Theorem 2.7], for n 2, E[X n) ]=F n ) and E [ X n) ) 2 ] = F2n )+ F n ) ) 2. Therefore, combining he independence of he incremens and he coninuiy of F n ), i is deduced ha X n) is coninuous in probabiliy. By Proposiion 1, F n ) has finie variaion on finie inervals. Hence, he process is a semimaringale. X n) = F n )+ X n) F n ) ) The Teugels maringales inroduced by Nualar and Schouens [5] for Lévy processes can be exended o addiive processes. In he same way as in [5], hese maringales are obained cenering he processes X n) : Y 1) = X, Y n) = X n) F n ), n 2, They are square inegrable maringales wih opional quadraic covariaion [Y n),y m) ] = X n+m), and, since F 2n ) is increasing, he predicable quadraic variaion of Y n) is Y n) = F 2n ).
142 JOSEP LLUÍS SOLÉ AND FREDERIC UTZET The formal expression 3. The polynomials of cumulans { exp n=1 u n } κ n = n! n= μ n u n n!. 3) relaes he sequences of numbers {κ n, n 1} and {μ n, n }. When we consider a random variable Z wih momen generaing funcion in some open inerval conaining, hen boh series converge in a neighborhood of, and 3) is he relaionship beween he momen generaing funcion, ψu) =E[e uz ], and he cumulan generaing funcion, log ψu). Moreover, μ n respecively, κ n ) is he momen respecively, he cumulan) of order n of Z, and he well-known relaions beween momens and cumulans can be deduced from 3). The firs hree ones are μ 1 = κ 1, μ 2 = κ 2 1 + κ 2, μ 3 = κ 3 1 +3κ 1 κ 2 + κ 3,... If he random variable Z has only finie momens up o order n, he corresponding relaionship is rue up o his order. There is a general explici expression of he momens in erms of cumulans in Kendall and Suar [1], or formulas involving he pariions of a se, see McCullagh [3]. In general, μ n is a polynomial of κ 1,...,κ n, called Kendall polynomial. Denoe, by Γ n x 1,...,x n ), n 1, his polynomial, ha is, we have μ n =Γ n κ 1,...,κ n ). Also wrie Γ = 1. These polynomials enoy very ineresing properies, as he recurrence formula ha follows from Sanley [9, Proposiion 5.1.7]: We also have Γ n+1 x 1,...,x n+1 )= Γ n x 1,...,x n ) x = = ) n Γ x 1,...,x ) x n+1. 4) ) n Γ n x 1,...,x n ), =1,...,n. 5) Compuing he Taylor expansion of Γ n x 1 + y, x 2,...,x n )ay =,wegehefollowing expression we will need laer: ) n Γ n x 1 + y, x 2,...,x n )= Γ n x 1,...,x n ) y. 6) = = Inerchanging he roles of x 1 and y and evaluaing he funcion a, we obain ) n Γ n x 1,x 2,...,x n )= Γ n,x 2,...,x n ) x 1. 7) 4. A family of maringales relaive o he addiive process The main resul of he paper is he following Theorem: Theorem 1. Le X be a cenered addiive process wih finie momens of all orders. Then he process ) M n) =Γ n X, F 2 ),..., F n )
is a maringale. MARTINGALES AND INDEPENDENT INCREMENT PROCESSES 143 Proof. Le n 2. We apply he mulidimensional Iô formula o he semimaringales X,F 2 ),...,F n ). By Proposiion 1, he funcions F 2 ),...,F n ) andσ 2 are coninuous and of finie variaion. From 5) and he fac ha [X, X] c = σ 2 and [F,F ] c =, we have ) M n) = n M n 1) n s dx s M s n ) df s) =2 + 1 2 nn 1) M s n 2) dσs 2 ) + Γ n Xs +ΔX s, F 2 s),..., F n s) ) Γ n Xs, F 2 s),..., F n s) ) <s Applying 6), <s =2 nδx s Γ n 1 Xs, F 2 s),..., F n s) )). Γ n Xs +ΔX s, F 2 s),..., F n s) ) = =2 = Then, he umps par given in he expression of M n) is ) n M n ) ) n ) n s ΔXs = Therefore, = =2 ) n ) n M n ) s dx s ) M n ) s d Y s ) + F s) ) M n ) s ΔXs ). ) n M s n 2) dσs 2 2 ) ) n M s n 2) dσs 2 2 ). ) M n) n = =1 ) 2 is a polynomial in X,F 2 ),...,F k ). Taking expecaions and using M n ) s dy s ). 8) Moreover, M k) he relaions beween momens and cumulans, as well as he fac ha he cumulans of X are F n ), n 2, we obain ha E [ M k) ) 2 ] = P F2 ),...,F 2k )), for a suiable polynomial P. Then, for every, we have E [ k)) 2 M s d Y ) ] [ M k)) ] 2 s = E s df 2 s) = P F 2 s),...,f 2k s)) df 2 s) <, So all he sochasic inegrals on he righ-hand side of 8) are maringales. Remark 1. I is worh o noe ha he preceding Theorem implies ha he funcion ) g n x, ) =Γ n x, F 2 ),..., F n ) is a ime space harmonic funcion wih respec o X.By7), g n x, ) = Γ n, F 2 ),...,F n ))x. =
144 JOSEP LLUÍS SOLÉ AND FREDERIC UTZET In general, g n x, ) is a polynomial in x. If F n ), n 2, are polynomials in, hen g n x, ) is a ime space harmonic polynomial; his happens for all Lévy processes wih momens of all orders and for some addiive process; see he example below. Example. Le Λ) :R + R + be a coninuous increasing funcion, and le J be a Poisson random measure on R + wih inensiy measure μa) = A Λd), A BR +). Then he process X = {X, } defined pahwise, X ω) = Jds, ω) Λ), is an addiive process; i is a Cox process wih deerminisic hazard funcion Λ). From he characerisic funcion of X, we deduce ha he Lévy measure is ν dx) =Λ)δ 1 dx), where δ 1 is a Dirac dela measure concenraed in he poin 1. Hence, F n ) =Λ), n 2. Noe ha he condiions we have assumed on Λ are necessary o obain an addiive process, bu i is no necessary hough no very resricive) o assume ha Λ is absoluely coninuous wih respec o he Lebesgue measure. The funcion defined in Remark 1 is ) g n x, ) =Γ n x, Λ),..., Λ). Hence, when Λ) isapolynomial,g n x, ) is a ime space harmonic polynomial. Denoe, by C n x, ), he Charlier polynomial wih leading coefficien equal o 1. Then see [8]) where λ n) 1 =1and g n x, ) = n 1 λ n) k+1 = =k ) n =1 λ n) C x, Λ)), λ ) k, k =1,...,n 1. Bibliography 1. M.Kendall and A.Suar, The Advanced Theory of Saisics, Vol. 1, 4h ediion, MacMillan, New York, 1977. 2. A.E.Kyprianou, Inroducory Lecures on Flucuaions of Lévy Processes wih Applicaions, Springer, Berlin, 26. 3. P.McCullagh, Tensor Mehods in Saisics, Chapman and Hall, London, 1987. 4. P.A.Meyer, Un cours sur les inegrales sochasiques, Séminaire de Probabiliés X, Springer, New York, 1976, pp. 245 4. French) 5. D.Nualar and W.Schouens, Chaoic and predicable represenaion for Lévy processes, Sochasic Process. Appl. 9 2), 19 122. 6. K.Sao, Lévy Processes and Infiniely Divisible Disribuions, Cambridge Universiy Press, Cambridge, 1999. 7. A.V.Skorohod, Random Processes wih Independen Incremens, Kluwer Academic Publ., Dordrech, Boson, London, 1986. 8. J.L.Solé andf.uze, Time-space harmonic polynomials relaive o a Lévy process, Bernoulli 27). 9. R.P.Sanley, Enumeraive Combinaorics, Vol. 2, Cambridge Universiy Press, Cambridge, 1999. Deparamen de Maem aiques, Facula de Ci encies, Universia Auónoma de Barcelona, 8193 Bellaerra Barcelona), Spain. E-mail: llsole@ma.uab.ca, uze@ma.uab.ca