aa r X i v : . [ m a t h . P R ] A ug Subordination of Predictable Compensators
Henry Chiu ∗ August 9, 2015
Abstract
We consider general subordination and obtain the formula of the sub-ordinated predictable compensator. An example of application is given.
Introduction
The idea of subordination (i.e. obtaining a new stochastic process by a ran-dom time change) was first introduced by Salomon Bochner in 1949 and is widelyapplied in the modelling of random phenomena such as stock price movements(e.g. the old Wall Street adage that "it takes volume Z to move prices X Z ").In many applications, the subordinated process X Z is discontinuous.A central tool in the study of discontinuous process is the predictable com-pensator that arose from the general theory of stochastic processes [1]. Thepredictable compensator, which can be seen as a generalisation of the Lévy mea-sure, gives a tractable description of the jump structure of a general stochasticprocess. It is an indispensable tool in many applications, for example, when per-forming an equivalent change of measure, an important operation in financialmathematics.For a general time changed Markov process, the formula of the associatedpredicable compensator is not known. The purpose of this paper is to obtainsuch a formula. Results
It is widely known that when the time of a Lévy process X is changed byan increasing Lévy process Z independent of X , the subordinated process X Z is a Lévy process and the subordinated predictable compensator ( µ X Z ) P of therandom jump measure of X Z can be obtained by [2,Thm 30.1]: ( µ X Z ) P ( dt, dy ) = γ ( µ X ) P ( dt, dy ) + Z R + P Xz ( dy )( µ Z ) P ( dt, dz ) , (1) ∗ [email protected]; Institut für Mathematik, Humboldt-Universität zu Berlin P Xt ( dy ) is the distribution of X t and γt = Z t − P s ≤ t ∆ Z s . Extensionof (1) to the case where Z is an additive subordinator has been considered in[5,Prop.1].When X is replaced by, for example, more general diffusion process, (1)will no longer be applicable (the subordinated predictable compensator shall nolonger be deterministic). We extend (1) to the case where X is a quasi left-continuous [1,Def.I.2.25] Markov process and Z is an increasing process andgive an example of application. Definitions and Framework
Let X and Z be two independent real-valued càdlàg processes defined on acomplete probability space (Ω , F , P ) and Z be increasing (i.e. non-decreasing).Denote X Z for the process obtained by a time-change of X by Z . Let F bea right-continuous filtration in F such that X Z is F -adapted, a non-negativerandom measure ( µ X Z ) P on B ( R + × R ) is called the F -predictable compensatorof the random jump measure of X Z [1,Thm.II.1.8.(i)] & [1,Thm.I.2.2.(i)] if forall F -stopping times T and B ∈ B ( R \{ } ) E X t ≤ T I B (∆( X Z ) t ) = E Z T Z B ( µ X Z ) P ( ω, dt, dy ) and that the integral process R t ∧ T R B ( µ X Z ) P ( ω, ds, dy ) is F -predictable.Denote (and respectively for Z and X Z ) F X := ( F Xt ) t ≥ for the right-continuous and completed canonical filtration of X , F Xt − := W s Let A ( ω ) ≥ be H t (resp. H t − )-measurable, then there exists a D ⊗ -measurable H ( u, v ) ≥ such that A ( ω ) = H ( X · ( ω ) , Z · ( ω )) P -a.s. and ω H ( X · ( e ω ) , Z · ( ω )) (6)is F Zt (resp. F Zt − )-measurable for P -a.s. e ω ∈ Ω held fixed. If in addition, A is F t (resp. F t − )-measurable, then ω H ( X · ( ω ) , Z · ( e ω )) (7)is F XZ t ( e ω ) (resp. F XZ t − ( e ω ) − )-measurable for P -a.s. e ω ∈ Ω held fixed. Proof. Π t ( ω ) := ( X · ( ω ) , Z ·∧ t ( ω )) , then Π t is a random variable defined on (Ω , H t , P ) taking values in ( D × , D ⊗ ) and one sees H t = ((Π t ) − D ⊗ ) P bythe construction of H t . If we denote Z ·∧ t − for the map ( s Z s ∧ t − ) ∈ D ,then H t − = ((Π t − ) − D ⊗ ) P and hence if A ( ω ) = P a i I A i ( ω ) for A i ∈ H t (resp. H t − ) then A ( ω ) = P a i I B i (Π t ( ω )) (resp. I B i (Π t − ( ω )) ) P -a.s. for some B i ∈ D . The first claim holds on simple A .If in addition, A i ∈ F t (resp. F t − ), we define a D ⊗ -measurable map Π t ( u, v ) := ( u ◦ v, v )( · ∧ t ) and Π( ω ) t := (Π t ◦ Π t )( ω ) = ( X Z ·∧ t ( ω ) ( ω ) , Z ·∧ t ( ω )) .Observe also ( s ( u ◦ v )( s ∧ t − )) ∈ D , one sees F t = (Π − t D ) P and F t − =(Π − t − D ) P hence, A ( ω ) = P a i I B i (Π t ( ω )) (resp. I B i (Π t − ( ω )) ) P -a.s. for some B i ∈ D . Since I B i ( X Z ·∧ t ( ω ) ( ω ) , Z ·∧ t ( ω )) = 1 I B i (Π t ( X · ( ω ) , Z ·∧ t ( ω ))) and thatthe path s X Z s ∧ t − ( ω ) ( ω ) ≡ X Z t − ( ω ) − ( ω ) for s ≥ t , we see that the secondclaim also holds on simple A . 3f ( H n ) n ≥ and H are D ⊗ -measurable, then H n ◦ Π t → H ◦ Π t P -a.s. on Ω ⇔ H n → H P ◦ (Π t ) − -a.s. on D × . By a monotone class argument, theclaims follow. Proposition 2 Let A ( ω, t ) ≥ be Q -measurable, then there exists a D ⊗ ⊗ R + -measurable H (( u, v ) , t ) ≥ such that A ( ω, t ) = H ( X · ( ω ) , Z · ( ω ) , t ) up to a P -evanescent setand ( ω, t ) H ( X · ( e ω ) , Z · ( ω ) , t ) (8)is P Z -measurable for P -a.s. e ω ∈ Ω held fixed. If in addition, A is P -measurable,then ( ω, t ) H ( X · ( ω ) , Z · ( e ω ) , t ) (9)is P XZ ( e ω ) -measurable for P -a.s. e ω ∈ Ω held fixed. Proof. The claims clearly holds for all Q -measurable (resp. P -measurable) A ofthe form A t = A I { } ( t )+ P i ∈ N A t i I ( t i ,t t +1 ] ( t ) for H t i − (resp. F t i − )-measurable A t i as a direct consequence of Proposition 1. Observe also if ( H n ) n ≥ and H are D ⊗ ⊗ R + -measurable then H n (Π t ( ω ) , t )) → H (Π t ( ω ) , t )) on Ω × R + up to a P -evanescent set ⇔ H n (( u, v ) , t ) → H (( u, v ) , t ) on D × × R + up to a P ◦ (Π t ) − -evanescent set. By a monotone class argument, the claims follow. Theorem Let X be a quasi left-continuous Markov process with transition kernel P Xt ( x, s, dy ) and Z be an increasing process independent of X . Denote X Z the process ob-tained by a time-change of X by Z and Z ct := Z t − P s ≤ t ∆ Z s then ( µ X Z ) P ( ω, dt, dy ) is changed as follows: ( µ X ) P ( ω, dZ ct , dy ) + Z R + P Xz ( X Z t − , Z t − , { X Z t − } + dy )( µ Z ) P ( ω, dt, dz ) . (10) Proof. Let A × B ∈ P ⊗ B ( R \{ } ) , W := 1 I A × B , I := { t ≥ | ∆ Z t = 0 } . Observe ∆( X Z ) t = ∆( X ) Z t on I and ∆( X Z ) t = X Z t − +∆ Z t − X Z t − − on I c and by (2),we can write F ( X · , Z · ) = ( W ∗ µ X Z ) ∞ = X t ∈ I I A I B (∆( X ) Z t ) + X t ∈ I c I A I B ( X Z t − +∆ Z t − X Z t − − ) . Let Z − denote the left-continuous generalized inverse of Z , by (4) & (9) put I A = H ( X · , Z · , t ) then ( ω, t ) H ( X · ( ω ) , Z · ( e ω ) , Z − t ( e ω )) is P X -measurable for4 -a.s. e ω ∈ Ω held fixed [1,Prop.I.2.4]. Together with the quasi left-continuityof X , [1,Thm.II.1.8] & [Cor.II.1.19], it follows E P t ∈ I I A I B (∆( X ) Z t ) (see also(2) for notation) = E Z E X X t ∈ Z ( I ) H ( X · ( ω ) , Z · ( e ω ) , Z − t ( e ω ))1 I B (∆ X t )= E Z E X Z Z ( I ) Z R H ( X · ( ω ) , Z · ( e ω ) , Z − t ( e ω ))1 I B ( y )( µ X ) P ( ω, dt, dy )= E Z E X Z I Z R H ( X · ( ω ) , Z · ( e ω ) , t )1 I B ( y )( µ X ) P ( ω, dZ t ( e ω ) , dy )= E Z E X Z R + Z R H ( X · ( ω ) , Z · ( e ω ) , t )1 I B ( y )( µ X ) P ( ω, dZ ct ( e ω ) , dy )= E Z R + × R W ( ω, t, y )( µ X ) P ( ω, dZ ct ( ω ) , dy ) . Since X has no fixed times of discontinuity and that I c is countable and by (8),it follows E P t ∈ I c I A I B ( X Z t − +∆ Z t − X Z t − − )= E X E Z X t ∈ I c H I B ( X Z t − +∆ Z t − X Z t − )= E X E Z Z R + Z R + H I B ( X Z t − + z − X Z t − )( µ Z ) P ( ω, dt, dz )= E Z Z R + Z R + E X [ H I B ( X Z t − + z − X Z t − )]( µ Z ) P ( e ω, dt, dz ) . By (4), (9) and [1,Prop.I.2.4], we see that for P -a.s. e ω ∈ Ω held fixed, the map ω H ( X · ( ω ) , Z · ( e ω ) , t ) is F XZ t − ( e ω ) -measurable for all t ≥ . Together with theMarkov property of X we have E X [ H I B ( X Z t − + z − X Z t − )]= E X h H E X [1 I B ( X Z t − ( e ω )+ z − X Z t − ( e ω ) ) |F XZ t − ( e ω ) ] i = E X [ HP Xz ( X Z t − ( e ω ) , Z t − ( e ω ) , { X Z t − ( e ω ) } + B )] hence E P t ∈ I c I A I B ( X Z t − +∆ Z t − X Z t − − )= E Z E X Z R + × R + H Z B P Xz ( X Z t − , Z t − , { X Z t − } + dy )( µ Z ) P ( ω, dt, dz )= E Z R + × R W Z R + P Xz ( X Z t − , Z t − , { X Z t − } + dy )( µ Z ) P ( ω, dt, dz ) . Define v ( ω, dt, dy ) :=( µ X ) P ( ω, dZ ct , dy ) + Z R + P Xz ( X Z t − , Z t − , { X Z t − } + dy )( µ Z ) P ( ω, dt, dz ) E ( W ∗ µ X Z ) ∞ = E ( W ∗ v ) ∞ . It is clear that v ( ω, dt, dy ) defines a non-negative random measure on R + × R and that ( W ∗ v ) t is F -predictable (3). If T is a F -stopping time, put A := { ( ω, t ) : 0 ≤ t ≤ T ( ω ) } ∈ P , (10) follows. Example We calculate the compensator ( µ X Z ) P of the random jump mea-sure of X Z with X and Z taken to be, respectively, a skew Brownian motion(diffusion process) and a tempered stable subordinator independent of X . Thecompensator of the random jump measure of Z is ( µ Z ) P ( dt, dz ) = dt cz α e − λz I { z> } ( dz ) (11)for c, λ > and α ∈ [0 , . The case α = 0 corresponds to a Gamma subordina-tor. By [3,(17)], the transition function of a skew Brownian motion X can bewritten as P Xt ( x, dy ) = 1 √ πt (cid:18) e − ( | y − x | )22 t + β sgn ( y ) e − ( | y | + | x | )22 t (cid:19) dy (12)for β ∈ [ − , . The case β = 0 corresponds to the standard Brownian motion.Using the modified Bessel function K v ( x ) for the integral representation Z ∞ z v e − a z − b z dz = 2( ab ) v K v ( ab ) , (13)the compensator formula (10) and φ ( X Z t − ( ω ) , y ) := | X Z t − ( ω ) | + | X Z t − ( ω ) + y | ,we obtain ( µ X Z ) P ( ω, dt, dy ) = 2 c √ π √ λ | y | ! / α K / α (cid:16) √ λ | y | (cid:17) dtdy + β c √ π sgn ( X Z t − + y ) √ λφ ( X Z t − , y ) ! / α × K / α (cid:16) √ λφ ( X Z t − , y ) (cid:17) dtdy and for the Gamma case α = 0 , ( µ X Z ) P ( ω, dt, dy ) = ce −√ λ | y | | y | + β sgn ( X Z t − + y ) ce −√ λφ ( X Zt − ,y ) φ ( X Z t − , y ) ! dtdy. (14)We see that ( µ X Z ) P is deterministic and time-independent if and only if β = 0 ,in this case X Z is a time-changed Brownian motion. If in addition, Z is aGamma process (i.e. α = β = 0 ) then X Z is a Variance Gamma process [4]with Lévy measure v ( dy ) = ce −√ λ | y | | y | dy and (14) reduces to ( µ X Z ) P ( ω, dt, dy ) = dtv ( dy ) . (15)6 eferences [1] Jacod, J. and Shiryaev, A.N. (2003) Limit Theorems for Stochastic Pro-cesses . 2ed., Springer.[2] Sato, K. (1999) Lévy processes and infinitely divisible distributions . Cam-bridge studies in advanced mathematics, Cambridge.[3] Lejay, Antoine. (2006) On the Constructions of the Skew Brownian Motion .Probability Surveys, Vol. 3, 413-466.[4] Küchler, U. and Tappe, S. (2008) Bilateral Gamma distributions and pro-cesses in financial mathematics . Stochastic Processes and their Applications,Vol. 118, Issue 2, 261-283.[5] Mijatovic, A. and Pistorius, M. (2010)