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c aiainadtpgahcdetail. typographic 1713–1732 and 3, pagination No. 15, publis Vol. article 2005, original Statistics the Mathematical of of reprint Institute electronic an is This nttt fMteaia Statistics Mathematical of Institute h tutr fteppri sflos nSection In follows. as is paper the of structure The Introduction. 1. e od n phrases. and words Key 1 M 00sbetclassifications. subject 2000 AMS 2004. October revised 2004; March Received upre yteSisNtoa cec onainadCred and Foundation Science National Swiss the by Supported QIAETADASLTL OTNOSMEASURE CONTINUOUS ABSOLUTELY AND EQUIVALENT 10.1214/105051605000000197 yPtikCheridito Patrick By essta a xld n ekle yapotential. a by killed be and explode jump-diffu can two that of cesses distributions the between equivalence and HNE O UPDFUINPROCESSES JUMP-DIFFUSION FOR CHANGES epoieepii ucetcniin o boueconti absolute for conditions sufficient explicit provide We 14 , h ups fti ae st ieepii,easy-to- explicit, give to is paper this of purpose The 15 hneo esr,jm-iuinpoess qiaetme equivalent processes, jump-diffusion measure, of Change , 16 2005 , )aentstse.I Section In satisfied. not are ]) hsrpitdffr rmteoiia in original the from differs reprint This . ui,PrsVI Paris Curie, 1 ai Filipovi Damir , 03,6J5 60J75. 60J25, 60G30, in h naso ple Probability Applied of Annals The 1 19 ,Th´eor`eme], Kazamaki- or IV.3) rator. n acYor Marc and c ´ e ythe by hed 2 3 tSuisse. it rjump-diffusions er 3 iree Marie et Pierre e eitouethe introduce we r eddi the in needed are ia generators. simal tgnrtrinto generator st inpro- sion r,i general, in are, s tosaunique a utions eso how show we nfrainof ansformation rabsolutely or t etconditions ient rcs from process e nsituations in o dffso pro- -diffusion rcs before process nuity tt space state r qecsof equences artingale , X a- 2 P. CHERIDITO, D. FILIPOVIC´ AND M. YOR proof of the paper’s main theorem, which is given in Section 4. In Section 5 we prove a stronger version of the result of Section 2 for a particular set-up, involving the carr´e-du-champ operator. In Section 6 we illustrate the main result by showing how the characteristics of a Cox–Ingersoll–Ross [3] short rate process with additional jumps and a potential can be altered by an absolutely continuous or equivalent change of measure. There exists a vast literature on the absolute continuity of stochastic pro- cesses, and below we quote some related publications. In contrast to many of those works, the primary goal of this paper is to provide results that are based on explicit assumptions which are easy to verify in typical applica- tions. For two applications in finance, see [2] and [1], which contain measure changes for multi-dimensional diffusion models and multi-dimensional jump- diffusion models with explosion and potential, respectively. Itˆoand Watanabe [10], Kunita [18] and Palmowski and Rolski [24] discuss absolute continuity for general classes of Markov processes. Kunita [17] characterizes the class of all absolutely continuous Markov processes with respect to a given Markov process. A special discussion for L´evy processes can be found in Sato [29], Section 33. Dawson [4], Liptser and Shiryaev [21] Kabanov, Liptser and Shiryaev [12], Rydberg [28] and Hobson and Rogers [9] discuss absolute continuity of solu- tions to stochastic differential equations. They are similar in spirit to Kadota and Shepp [13], which contains sufficient conditions for the distribution of a Brownian motion with stochastic drift to be absolutely continuous with respect to the Wiener measure. Pitman and Yor [25] and Yor [33] study mutual absolute continuity of (squared) Bessel processes. Lepingle and M´emin [19] and Kallsen and Shiryaev [14] provide condi- tions for the uniform integrability of exponential local martingales in a gen- eral semimartingale framework (see also Remark 2.7 below), extending the classical results by Novikov [23] and Kazamaki [15]. Discussions of measure changes in a finance context can be found in Sin [30], Lewis [20], Delbaen and Shirakawa [5, 6]. Wong and Heyde [32] give necessary and sufficient conditions for the stochastic exponential of a Brownian motion integral to be a martingale in terms of the explosion time of an associated process. Among various excellent text books that discuss changes of measure in varying degree of generality are, for example, McKean [22], Rogers and Williams [26], Jacod and Shiryaev [11], Revuz and Yor [27] and Strook [31].

2. Statement of the main result. Let E be a closed subset of Rd and E = E ∆ the one-point compactification of E. If not mentioned oth- ∆ ∪ { } erwise, any measurable function f on E is extended to E∆ by setting f(∆) := 0. We let Ω be the space of c`adl`ag functions ω : R E such + → ∆ MEASURE CHANGES FOR JUMP-DIFFUSION PROCESSES 3 that ω(t )=∆ or ω(t) = ∆ implies ω(s) = ∆ for all s t. (Xt)t≥0 is the coordinate− process, given by ≥

Xt(ω) := ω(t), t 0. ≥ It generates the σ-algebra, X := σ(Xs : s 0), F ≥ and the filtration X := σ(Xs : 0 s t), t 0. Ft ≤ ≤ ≥ It follows from Proposition 2.1.5 (a) in [8] that, for all closed subsets Γ of E∆, X inf t Xt Γ or Xt Γ is an ( )-stopping time. { | − ∈ ∈ } Ft Hence,

T := inf t Xt =∆ = inf t Xt =∆ or Xt =∆ ∆ { | } { | − } is an ( X )-stopping time. Note that Ft X· = ∆ on [T , ) ∆ ∞ so that T∆ can be viewed as the lifetime of X. For the handling of explosion, we introduce the ( X )-stopping times Ft ′ T := inf t Xt n or Xt n , n 1, n { | k −k≥ k k≥ } ≥ where denotes the Euclidean norm on Rd and ∆ := . Clearly, T ′ k · k k k ∞ n ≤ T∆, for all n 1. A transition to ∆ occurs either by a jump or by explosion. Accordingly,≥ we define the ( X )-stopping times Ft T , if T ′ = T for some n, T := ∆ n ∆ jump , if T ′ < T for all n,  ∞ n ∆ T , if T ′ < T for all n, T := ∆ n ∆ expl , if T ′ = T for some n,  ∞ n ∆ T ′ , if T ′ < T , T := n n ∆ n , if T ′ = T .  ∞ n ∆ Note that Tjump < Texpl < = ∅, limn→∞ Tn = Texpl, and Tn < T on T{ < .∞}∩{ Hence, T is∞} predictable with announcing sequence expl { expl ∞} expl Tn n (see [11], I.2.15.a). Since,∧ by definition, Ω contains only paths that stay in ∆ after explo- X sion or after a jump to ∆, the filtration ( t ) has the property stated in Proposition 2.1 below, whose proof is givenF in the Appendix. 4 P. CHERIDITO, D. FILIPOVIC´ AND M. YOR

Proposition 2.1. Let T be an arbitrary ( X )-stopping time. Then Ft X X X T = T ∧T = σ T ∧Tn . F F expl F ! n[≥1 Fix a bounded and continuous function χ : Rd Rd such that χ(ξ)= ξ on a neighborhood of 0. Let α, β, γ be measurable mappings→ on E with values d in the set of positive semi-definite symmetric d d-matrices, R and R+, respectively. Furthermore, let µ be a transition× kernel from E to Rd and assume that the functions α( ), β( ), γ( ) and ( ξ 2 1)µ( , dξ) · · · Rd k k ∧ · (2.1) Z are bounded on every compact subset of E. Then, 1 d ∂2f(x) d ∂f(x) f(x) := αij(x) + βi(x) γ(x)f(x) A 2 ∂xi ∂xj ∂xi − i,jX=1 Xi=1 + (f(x + ξ) f(x) f(x),χ(ξ) )µ(x,dξ) Rd − − h∇ i Z defines a linear operator from the space of C2-functions on E with compact 2 support, Cc (E), to the space of bounded measurable functions on E, B(E).

Definition 2.2. We say that a P on (Ω, X ) is a solution of the martingale problem for if, for all f C2(E), F A ∈ c t f M := f(Xt) f(X ) f(Xs) ds, t 0, t − 0 − A ≥ Z0 X is a P-martingale with respect to ( t ). We say that the martingale problem for is well posed if for every probabilityF distribution η on E, there exists a uniqueA solution P of the martingale problem for such that P X−1 = η. A ◦ 0 Remark 2.3. 1. If P is a solution of the martingale problem for , then with respect to P, X is a possibly nonconservative, time-homogenousA jump- diffusion process. The time-homogeneous case can be included in the above set-up by identifying one component of X with time t. 2. If P is a solution of the martingale problem for , then M f is, for all f C2(E), also a P-martingale with respect to ( XA). Indeed, since all ∈ c Ft+ paths of M f are right-continuous, it follows from the backwards martingale convergence theorem that, for all t,s R such that t

3. It is easy to see that if the martingale problem for is well posed, then A for every probability distribution η on E∆, there exists a unique solution P −1 of the martingale problem for such that P X0 = η. A ◦ t t 4. Throughout, we make use of the fact that 0 f(Xu−) dSu = 0 f(Xu) dSu, for a continuous semimartingale S and every measurable function f such R R that the integrals are defined.

Let ˜ be a second linear operator from C2(E) to B(E), given by A c 1 d ∂2f(x) d ∂f(x) ˜f(x) := αij(x) + β˜i(x) γ˜(x)f(x) A 2 i,j=1 ∂xi ∂xj i=1 ∂xi − (2.2) X X + (f(x + ξ) f(x) f(x),χ(ξ) )˜µ(x,dξ), Rd − − h∇ i Z d where β˜ andγ ˜ are measurable mappings from E to R and R+, respectively, andµ ˜ is a transition kernel from E to Rd such that β˜,γ ˜ andµ ˜ satisfy the condition (2.1). Let U be an open subset of E, that is, U = U ′ E for some open subset ∩ U ′ of Rd. Assume that there exist measurable mappings

φ : U Rd, φ : U (0, ) and φ : U Rd (0, ) 1 → 2 → ∞ 3 × → ∞ such that, for all x U, ∈

β˜(x)= β(x)+ α(x)φ1(x)+ (φ3(x,ξ) 1)χ(ξ)µ(x,dξ), Rd − Z (2.3) γ˜(x)= φ2(x)γ(x),

µ˜(x,dξ)= φ3(x,ξ)µ(x,dξ).

Let U 1 U 2 be an increasing sequence of open subsets of E such that ⊂ ⊂··· U = U n. We denote U = U ∆ and U n = U n ∆ , n 1. For all n≥1 ∆ ∪ { } ∆ ∪ { } ≥ n 1, we define ≥ S n n Rn := inf t Xt / U or Xt / U . { | − ∈ ∆ ∈ ∆} Note that R′ , if R′ < T , R = n n ∆ n , if R′ = T ,  ∞ n ∆ where

′ n n R := inf t Xt / U or Xt / U , n 1. n { | − ∈ ∈ } ≥ 6 P. CHERIDITO, D. FILIPOVIC´ AND M. YOR

n Since the sets U are open in the topology of E∆, it follows from Proposi- ′ tion 2.1.5(a) of [8] that all Rn, Rn and therefore also,

R∞ := lim Rn = inf t Xt− / U∆ or Xt / U∆ , n→∞ { | ∈ ∈ } Sn := Rn Tn n,n 1, ∧ ∧ ≥ S∞ := lim Sn = R∞ Texpl n→∞ ∧ are ( X )-stopping times. While the sequence T T takes care of Ft 1 ≤ 2 ≤··· a possible explosion of X, the sequence S1 S2 appropriately local- izes the stochastic logarithm of the density process≤ ≤··· for the measure change, see (4.2) below. In view of (2.1) and the convention f(∆) = 0 for measurable functions f,

Sn 1 Λn := α(Xs)φ (Xs), φ (Xs) ds 2 h 1 1 i Z0 Sn + (φ (Xs) log φ (Xs) φ (Xs) + 1)γ(Xs) ds 2 2 − 2 Z0 Sn + (φ3(Xs,ξ) log φ3(Xs,ξ) φ3(Xs,ξ) + 1)µ(Xs, dξ) ds Rd − Z0 Z is well defined for all n 1. With this notation we have the following: ≥ Theorem 2.4. Let P be a solution of the martingale problem for ˜ A and Q a solution of the martingale problem for such that Q F X P F X . A | 0 ≪ | 0 Assume that for ˜, the martingale problem is well posed and that A Λn (2.4) EP[e ] < , ∞ for all n 1. ≥ Then there exists a nonnegative c`adl`ag P-supermartingale (Dt)t≥0 such that, for any ( X )-stopping time T , the following properties hold: Ft (2.5) Q X = DT P X . |FT ∩{T

If Q[T < R∞] = 1 and (DT ∧Tn )n≥1 is P-uniformly integrable, (2.8) then Q X = DT P X . |FT · |FT MEASURE CHANGES FOR JUMP-DIFFUSION PROCESSES 7

Remark 2.5. The following is an easy-to-check sufficient criterion for (2.4): Assume that for every n 1, there exists a finite constant Kn such that, for all x U n, ≥ ∈ (2.9) α(x)φ (x), φ (x) Kn, h 1 1 i≤ (2.10) (φ (x) log φ (x) φ (x) + 1)γ(x) Kn, 2 2 − 2 ≤

(2.11) (φ3(x,ξ) log φ3(x,ξ) φ3(x,ξ) + 1)µ(x,dξ) Kn. Rd − ≤ Z Then (2.4) is satisfied.

Remark 2.6. If P[S = ] = 1, we obtain from (2.5) the loss of mass ∞ ∞ of the P-supermartingale (Dt)t≥0

1 EP[Dt] = 1 Q[t

hφ1(x),ξi Remark 2.7. For φ3(x,ξ)= e , our measure changes are of the same form as the generalized Esscher transforms discussed in [14] (see The- orem 2.19 in [14] or Theorem III.7.23 in [11]).

3. Turning X into a semimartingale. In this section we show some pre- liminary results that we will need in the proof of Theorem 2.4. The notation is the same as in Section 2. For any process Y and stopping time T , we T T denote by Y the stopped process given by Yt := Yt∧T , t 0. Assume that P is a solution of the martingale problem for≥ . Since the co- ordinate process can explode and be killed, it is, in general, notA a semimartin- gale with respect to P. To turn it into a semimartingale, we stop it before it explodes and identify the state ∆ with an arbitrary point ∂ in Rd E. With- out loss of generality, we can assume that such a point exists. If E\ = Rd, we d+1 embed E in R by the map (x1,...,xd) (x1,...,xd, 0) and adjust α, β, µ and χ as follows: For all x E, we extend7→α(x) to a (d + 1) (d + 1)-matrix ∈ × by setting α(x)i,d+1 = α(x)d+1,i := 0 for all i = 1,...,d + 1. β(x) is elon- gated to a (d + 1)-dimensional vector by β(x)d+1 := 0. The measure µ(x, ) is extended to Rd+1 by defining µ(x, Rd+1 Rd) := 0. Finally, the truncation· function χ can be extended to a bounded and\ continuous function from Rd+1 to Rd+1 such that χ(ξ)= ξ on a neighborhood of 0, or simply by setting it equal to zero on Rd+1 Rd. Then, a probability measure P on (Ω, X ) is a solution of the martingale\ problem for in the Rd+1-framework ifF and only if it is in the Rd-framework. A The process

ˆ ½ X := X ½[0,T∆) + ∂ [T∆,∞) X d ˆ is also ( t )-adapted and has right-continuous paths in R . However, XT∆− = ∆ (explosion)F is still possible for this process. 8 P. CHERIDITO, D. FILIPOVIC´ AND M. YOR

Let T bea ( X )-stopping time such that T < T , then Ft expl

(3.1) T < Tn =Ω, { } n[≥1 X and, therefore, (2.1) implies that the following ( t )-predictable processes and random measure are well defined for all ω: F t∧T T B := β(Xs)+ γ(Xs)χ(∂ Xs) ds, t − Z0 t∧T T Ct := α(Xs) ds, Z0 T ν (dt, dξ) := [µ(Xt, dξ)+ γ(Xt)δ∂−Xt (dξ)] ½{t≤T } dt. Condition (2.1) also guarantees that νT satisfies Condition 2.13 on page 77 of [11]. Note that one can choose χ with compact support such that χ(∂ x) = 0 for all x E. In that case, the expression for BT becomes simpler. For− f C2(Rd) (the∈ space of bounded C2-functions on Rd), define the process ∈ b d 2 ˆ T f,T ˆ T ˆ T 1 ∂ f(X ) T ˆ T T := f(X ) f(X0 ) Cij f(X ) B M − − 2 ∂xi ∂xj · −∇ · i,jX=1 (f(Xˆ T + ξ) f(Xˆ T ) f(Xˆ T ),χ(ξ) ) νT − − − h∇ i ∗ (“ ” denotes stochastic integration with respect to a semimartingale and “ ” stochastic· integration with respect to a random measure, for the definition∗ of stochastic integrals with respect to semimartingales and random measures, 2 d 2 see, e.g., [11]). The restriction of a function f Cc (R ) to E is in Cc (E). Recall that by convention, f(∆) = α(∆) = β(∆)∈ = γ(∆) = µ(∆, )=0. Thus, it can easily be checked that ·

(3.2) f,T = M f,T + f(∂)N T , t 0, Mt t t ≥ where t∧T T N := ½ γ(Xs) ds, t 0. t {0

X Lemma 3.1. Let T be an ( t )-stopping time with T < Texpl. Then the process N T is an (( X ), P)-martingale.F Ft+

Tn X Proof. Fix n 1. We first show that N is an (( t+), P)-martingale. ≥ 2 d F Let (fk) be a sequence in C (R ) with 0 fk 1 and fk = 1 on the ball with c ≤ ≤ MEASURE CHANGES FOR JUMP-DIFFUSION PROCESSES 9

fk,Tn X center 0 and radius k, Bk. By Remark 2.3 part 2, M is an (( ), P)- Ft+ martingale for every k. Note that Tn =0 if X0 n. Hence, we have, for all k>n, k k≥ t∧Tn fk,Tn Tn M = fk(X ) fk(X ) fk(Xs) ds t t − 0 − A Z0 Tn = fk(X ) fk(X ) t − 0 t∧Tn + γ(Xs) (fk(Xs + ξ) 1)µ(Xs, dξ) ds. d − R k n − Z0  Z \B −  Clearly, for all ω,

lim fk(Xt∧Tn )= ½{t∧Tn

Notice that T < Texpl implies T∆ t T = Tjump t T . Hence, t∧T { ≤ ∧ } { ≤ ∧ } Lemma 3.1 says that 0 γ(Xs) ds is the predictable compensator for the time of a jump of the stopped process XT to ∆. As a consequence, we obtain R that Tjump = P-almost surely on X0 =∆ if and only if γ(Xt) = 0 P- almost surely∞ for all t. { 6 }

Proposition 3.2. Assume that P is a solution of the martingale prob- X lem for and T is an ( t )-stopping time such that T < Texpl. Then for A2 d f,T F X ˆ T all f Cb (R ), is a local martingale on (Ω, ( t+)t≥0, P) and X is ∈ M X F T T T a semimartingale on (Ω, ( t+)t≥0, P) with characteristics (B ,C ,ν ) with respect to the truncation functionF χ. 10 P. CHERIDITO, D. FILIPOVIC´ AND M. YOR

Proof. Fix n 1. In view of (3.2), Remark 2.3 part 2 and Lemma 3.1, f,Tn X ≥ 2 d is an (( t+), P)-martingale for all f Cc (R ). M F 2 d 2 d ∈ 2 d Now let f Cb (R ). Then ffk Cc (R ), where the fk Cc (R ) are as in the proof of Lemma∈ 3.1, and for all∈ k n, ∈ ≥ f,Tn ffk,Tn |Mt − Mt | Tn Tn f(Xˆ ) ffk(Xˆ ) ≤ | t − t | t∧Tn Tn + f(Xs + ξ) ffk(Xs + ξ) ν (ds, dξ) d R k n | − | Z0 Z \B − t∧Tn ˆ ˆ Tn d f(Xt Tn ) ffk(Xt Tn ) + f ν (ds, R Bk n). ≤ | ∧ − ∧ | k k∞ \ − Z0 Obviously, ˆ ˆ 1 f(Xt Tn ) ffk(Xt Tn ) 0 in L as k , | ∧ − ∧ |→ →∞ and as in the proof of Lemma 3.1, it can be deduced from (2.1) that

t∧Tn Tn d 1 ν (ds, R Bk n) 0 in L as k . \ − → →∞ Z0 f,Tn X Hence, is an (( t+), P)-martingale, for all n 1. This, together with (3.1), M f,T F X ≥ implies that is an (( t+), P)-local martingale. Thus, it follows from [11], Mˆ T FX II.2.42, that X is an (( t+), P)-semimartingale with the claimed character- istics. F 

4. Proof of Theorem 2.4. There exists a nonnegative, X -measurable F0 random variable D0 such that

Q F X = D0 P F X . | 0 · | 0 For each n 1, letµ ˆSn denote the integer-valued random measure associated ≥ ˆ Sn X to the jumps of X (see [11], II.1.16). By Proposition 3.2, its (( t+), P)- compensator is νSn . It can easily be checked that F 1 y log y y + 1 − 1 for y (0, 2] 3 ≤ (y 1)2 ≤ ∈ − and 1 y log y y + 1 − for y 2. 3 ≤ y 1 ≥ − y log y−y+1 (Notice however that limy→∞ y−1 = .) Hence, it follows from (2.4) that ∞ 2 Sn EP[([ψ(X,ξ) 1] ψ(X,ξ) 1 ) ν ] < − ∧ | − | ∗ ∞ MEASURE CHANGES FOR JUMP-DIFFUSION PROCESSES 11 for the nonnegative measurable function ψ : U Rd R defined by

× → + ½ ψ(x,ξ) := φ2(x)½{x+ξ=∂} + φ3(x,ξ) {x+ξ∈E}. Consequently, by [11], II.1.33 c, [(ψ(X ,ξ) 1)] (ˆµSn νSn ) − − ∗ − X is a well defined (( t+), P)-local martingale. Moreover, it follows from (2.4) that F Sn EP α(Xs)φ (Xs), φ (Xs) ds < . h 1 1 i ∞ Z0  Hence, by [11], III.4.5, φ (X) Xˆ Sn,c 1 · is a well-defined continuous (( X ), P)-local martingale, where Xˆ Sn,c denotes Ft+ the continuous martingale part of Xˆ Sn , relative to the measure P. In sum- mary, (4.1) Ln := φ (X) Xˆ Sn,c + [(ψ(X ,ξ) 1)] (ˆµSn νSn ) 1 · − − ∗ − is a well-defined (( X ), P)-local martingale with Ft+ Sn n,c n,c n,c n,c L ,L = L ,L Sn = α(Xs)φ (Xs), φ (Xs) ds h i∞ h i h 1 1 i Z0 and n ˆ ∆Lt = [ψ(Xt−, ∆Xt) 1]½ ˆ Sn > 1. − {∆Xt 6=0} − This latter property assures that the stochastic exponential (Ln) is a X E strictly positive (( t+), P)-local martingale. Moreover, it follows from Th´eor`eme IV.3 F n X of [19], together with (2.4), that (L ) is a uniformly integrable (( t+), P)-martingale, which implies that E F (4.2) Dn := D (Ln) 0E X is a nonnegative, uniformly integrable (( t+), P)-martingale. Obviously, for n m, F ≥ n m D = D for all t Sm. t t ≤ Therefore, for t 0 for all t [0,S ) on D > 0 . t ∈ ∞ { 0 } 12 P. CHERIDITO, D. FILIPOVIC´ AND M. YOR

∞ n Since (DSn )n≥1 = (DSn )n≥1 is a nonnegative martingale, the limit D∞ := lim D∞ 0 S∞ n→∞ Sn ≥ exists P-almost surely, and

∞ ∞ ½ Dt := D ½ + D , t [0, ], t {t

(4.4) EP[DT ] 1. ≤ 2 Now, let f Cc (E) and set f(∂)= f(∆) = 0. Then, it follows from (3.2) that f,Sn ∈f,Sn f,Sn X M = . By Remark 2.3 part 2, M is an (( t+), P)-martingale, and obviously,M it has bounded jumps. Therefore, it followsF from Lemma III.3.14 in [11] that M f,Sn ,Ln and M f,Sn , DSn exist and M f,Sn , DSn = DSn h i h i h i − · M f,Sn ,Ln . It can be seen from II.2.36, II.2.43 and the proof of II.2.42 inh [11] thati (4.5) M f,Sn = f,Sn = f(X) Xˆ Sn,c + [f(X + ξ) f(X )] (ˆµSn νSn ) M ∇ · − − − ∗ − is the decomposition of M f,Sn into a continuous and a purely discontinuous (( X ), P)-local martingale part. Hence, Ft+ t f,Sn n M ,L t = f(Xs), α(Xs)φ (Xs) ds h i h∇ 1 i Z0 + ([f(X + ξ) f(X)][ψ(X,ξ) 1]) νSn , − − ∗ t which shows that, for all t 0, ≥ t f,Sn M˜ := f(Xt) f(X ) ˜f(Xs) ds t − 0 − A Z0 t f,Sn f,Sn n f,Sn 1 f,Sn Sn = M M ,L t = M d M , D s. t − h i t − Sn h i Z0 Ds− Thus, it follows from Girsanov’s theorem for local martingales in the form ˜ f,Sn X Sn of [11], III.3.11, that M is an (( t+), D P)-martingale. By the defini- F · ˜ f,Sn X tion of Q and the optional sampling theorem, M is also an (( t ), Q)- martingale. By Remark 2.3, part 3, we can apply Theorem 4.6.1 ofF [8] (ob- serve that for the proof of [8], Theorem 4.6.1, it is only needed that Sn is an ( X )-stopping time, see also [8], Lemma 4.5.16) to conclude that Ft X DSn P = Q on . · FSn MEASURE CHANGES FOR JUMP-DIFFUSION PROCESSES 13

Now, let A X . It can easily be checked that, for all n 1, ∈ FT ≥ X A T

Thus, ½ Q[A T

(4.6)

½ ½ ½ = lim EP[DT ½ T

Q[T

Hence, if Q[T

(4.7) EP[DT ]=1 and DT =0 on T S P-a.s., { ≥ ∞} which proves (2.6). If, in addition, Q F X P F X , then D0 > 0 P-a.s. and it follows from (4.3) | 0 ∼ | 0 that DT > 0 on T

Q F X = DT ∧Tn P F X . | T ∧Tn · | T ∧Tn

Moreover, since limn→∞ Tn = Texpl S∞, we have limn→∞ DT ∧Tn = DT P-a.s. ≥ 1 Hence, if (DT ∧Tn )n≥1 is uniformly integrable, then DT ∧Tn DT in L (P). Therefore, →

Q F X = DT P F X , | T ∧Tn · | T ∧Tn for all n 1, which, by Proposition 2.1, implies (2.8), and the theorem is proved. ≥

5. Carr´e-du-champ operator. Part (2.8) of Theorem 2.4 yields absolute continuity of Q X with respect to P X , also on T Texpl . In this section |FT |FT { ≥ } we consider a special choice of φ1, φ2 and φ3, which even provides equivalence beyond explosion. This is an extension of [27], Section VIII.3, and involves the carr´e-du-champ operator Γ : C2(E) C2(E) B(E) defined by c × c → Γ(f, g) := (fg) f g g f. A − A − A 14 P. CHERIDITO, D. FILIPOVIC´ AND M. YOR

In contrast to above, we now first introduce a probability measure Q such X ˜ that Q P on t+ for all t 0, and then find the appropriate generator for which∼ Q solvesF the martingale≥ problem. A Fix h C2(E). Then H := eh 1 C2(E), and we can define ∈ c − ∈ c t h(Xt)−h(X0) H(Xs) Dt := e exp A ds . − eh(Xs)  Z0  Integration by parts, using d(eh(X))= dM H + H(X) dt, yields A t −h(X0) H(Xs) H −h(Xt−) H (5.1) dDt = e exp A ds dM = Dt−e dM . − eh(Xs) t t  Z0  Since D is uniformly bounded on compact time intervals, we conclude that X D is a strictly positive (( t+), P)-martingale. As in [26], Theorem IV.38.9, it can be deduced from theF Daniell–Kolmogorov extension theorem that there X X exists a probability measure Q on such that Q = Dt P on t+ for all t 0. F · F ≥In view of (3.2) [we set H(∂)=0] and (4.5), we have

M H,Sn = H,Sn = eh(X) Xˆ Sn,c + (eh(X−+ξ) eh(X−)) (ˆµSn νSn ), M ∇ · − ∗ − so that, together with (5.1), we obtain

DSn = ( h(X) Xˆ Sn,c + (eh(X−+ξ)−h(X−) 1) (ˆµSn νSn )), E ∇ · − ∗ − for all n 1. Comparing this to (4.1) suggests that we are in the situation of Theorem≥ 2.4 with (5.2) φ (x)= h(x), φ (x)= e−h(x) and φ (x,ξ)= eh(x+ξ)−h(x), 1 ∇ 2 3 which clearly satisfy (2.9)–(2.11) for all x E and a fixed constant K> 0. ∈ ˜ 2 Theorem 5.1. Q is a solution of the martingale problem for : Cc (E) B(E) given by A → Γ(H,f) (5.3) ˜f := f + , A A eh which equals (2.2) with (2.3) and (5.2).

Proof. A straightforward calculation yields Γ(f, g)(x)= α(x) f(x), g(x) + γ(x)f(x)g(x) h ∇ ∇ i + (f(x + ξ) f(x))(g(x + ξ) g(x))µ(x,dξ), Rd − − Z which makes it easy to see that (5.3) equals (2.2) with (2.3) and (5.2). MEASURE CHANGES FOR JUMP-DIFFUSION PROCESSES 15

Let f C2(E). Lemma 5.2 below shows that ∈ c t f H M , M t = Γ(f,H)(Xs) ds, t 0. h i ≥ Z0 Therefore, t f M˜ := f(Xt) f(X ) ˜f(Xs) ds t − 0 − A Z0 t t −h(Xs) f H = f(Xt) f(X ) f(Xs) ds e d M , M s − 0 − A − h i Z0 Z0 t f 1 f = M d M , D s, t − D h i Z0 s− and it follows from Girsanov’s theorem for local martingales [11], III.3.11, that M˜ f is an (( X ), Q)-martingale, which proves the theorem.  Ft+ Lemma 5.2. If f, g C2(E), then ∈ c t f g M , M t = Γ(f, g)(Xs) ds. h i Z0 Proof. Literally the same as the proof of Proposition VIII.3.3 in [27]. 

6. Example. We here apply Theorem 2.4 to a one-dimensional diffusion with compound Poisson jumps and a constant killing rate. In [2], it is applied to a multi-dimensional diffusion, and in [1] to a multi-dimensional jump- diffusion model. Let (Ω′, ′, P′) be a probability space that carries the following three independentF random objects: a one-dimensional standard Brownian motion (Wt)t≥0; a (Nt)t≥0 with jump arrival rate λ> 0 and positive jumps that are distributed according to a probability measure m on (0, ); and an exponentially distributed random variable τ with mean 1 > 0. Let∞b 0, b R and σ> 0. It is well known that the SDE γ 0 ≥ 1 ∈ (6.1) dVt = (b0 + b1Vt) dt + σ Vt dWt, V0 = v> 0, has a unique strong solution, V staysp nonnegative, and σ2 (6.2) V never reaches zero if b . 0 ≥ 2 (Cox, Ingersoll and Ross [3] model the short term interest rates by the solution of an SDE of the form (6.1).) It follows from a comparison argument that the same is true for the equation

(6.3) dYt = (b0 + b1Yt) dt + σ Yt dWt + dNt, Y0 = y> 0. p 16 P. CHERIDITO, D. FILIPOVIC´ AND M. YOR

The process ½ Z := Y ½[0,τ) +∆ [τ,∞) takes values in E∆, for E = R+, and its distribution P is a probability mea- sure on the measurable space (Ω, X ) introduced in Section 2. It can be checked that P is a solution of theF martingale problem for f(x)= 1 σ2xf ′′(x) + (b + b x)f ′(x) γf(x) A 2 0 1 − ∞ + [f(x + ξ) f(x)]λm(dξ). − Z0 Let ˜f(x)= 1 σ2xf ′′(x) + (˜b + ˜b x)f ′(x) γ˜(x)f(x) A 2 0 1 − ∞ + [f(x + ξ) f(x)]˜µ(x,dξ), − Z0 ˜ σ2 ˜ 2 where b0 2 , b1 R,γ ˜(x) =γ ˜0 +γ ˜1x, for some (˜γ0, γ˜1) R+ (0, 0) , andµ ˜(x, ≥) is, for∈ all x> 0, a measure on (0, ) of the form∈ µ ˜\( {x,dξ)=} · ∞ [m0(ξ)+m1(ξ)x]λm(dξ), for nonnegative measurable functions m0,m1 : (0, ) R , such that (m (ξ),m (ξ)) R2 (0, 0) for all ξ> 0 and ∞ → + 0 1 ∈ + \ { } ∞ l(m (ξ)+ m (ξ)x)m(dξ) < for all x> 0, 0 1 ∞ Z0 where l(u)= u log u u + 1. It follows from Theorem 2.7 in [7] that the mar- tingale problem for −˜ is well posed. Let Q be the solution of the martingale A problem for ˜ with initial distribution δy. It can be deduced from (6.2) and a comparisonA argument that

(6.4) Q[there exists a t 0 such that Xt =0 or Xt = 0] = 0. ≥ − We set U = (0, ) and Un = (1/n,n), n 1. Since we have no explosion, (6.4) implies that∞Q[S = ] = 1. Furthermore,≥ the measurable mappings ∞ ∞ ˜b b ˜b b φ (x)= 0 − 0 + 1 − 1 , x U, 1 σ2x σ2 ∈ 1 φ (x)= (˜γ +γ ˜ x), x U, 2 γ 0 1 ∈ m (ξ)+ m (ξ)x, if ξ> 0, φ (x,ξ)= 0 1 x U, 3 1, if ξ 0, ∈  ≤ satisfy the conditions (2.9)–(2.11), and for all x U, ∈ 2 ˜b0 + ˜b1x = b0 + b1x + σ xφ1(x),

γ˜(x)= φ2(x)γ,

µ˜(x,dξ)= φ3(x,ξ)λm(dξ). MEASURE CHANGES FOR JUMP-DIFFUSION PROCESSES 17

Therefore, Theorem 2.4 applies, and we obtain that

Q X P X |FT ≪ |FT 2 for all ( X )-stopping times T < . Moreover, if b σ , then Ft ∞ 0 ≥ 2 P[S = ] = 1 P[there exists a t 0 such that Xt =0 or Xt = 0] = 1, ∞ ∞ − ≥ − and Theorem 2.4 yields that

Q X P X |FT ∼ |FT X for all ( t )-stopping times T < . If we identify ∆ with 1, the process Z becomesF the semimartingale ∞ −

ˆ ½ Z := Y ½ . [0,τ) − [τ,∞) ˆc It can be seen from (6.3) that dZt = ½{0≤t<τ}σ√Yt dWt. The random mea- sureµ ˆ associated to the jumps of Zˆ is an integer-valued random measure 2 on R+ with compensator

ν(dt, dξ)= ½ dt (λm(dξ)+ γδ Zt (dξ)). {0≤t<τ} × −1−

Since the distribution of ½ ψ(Zt−,ξ)= φ2(Zt−)½{Zt−+ξ=−1} + φ3(Zt−,ξ) {Zt−+ξ≥0}, and the stochastic exponential D′ = (φ (Z) Zˆc + (ψ(Z ,ξ) 1) (ˆµ ν)) E 1 · − − ∗ − only depend on the distribution of Z, it follows from (4.7) that ′ EP′ [Dt]= EP[Dt] = 1. ′ ′ ′ ′ Hence, D is a P -martingale, and for all t 0, Dt P is a probability measure on (Ω′, ′) under which the distribution≥ of the stopped· process Zt is equal F σ2 ′ ′ ′ ′ to Q X . If b , then D > 0 P -almost surely for all t R , and D P |Ft 0 ≥ 2 t ∈ + t · is equivalent to P′.

APPENDIX

Proof of Proposition 2.1. It is clear that

X X X (A.1) T T ∧T σ T ∧Tn . F ⊃ F expl ⊃ F ! n[≥1 To show the reverse inclusions, we first prove that

X X (A.2) σ Tn . F ⊂ F ! n[≥1 18 P. CHERIDITO, D. FILIPOVIC´ AND M. YOR

Note that for all t 0, and all Borel subsets B of E, ≥ Xt B = Xt B T >t = ( Xt B Tn >t ), { ∈ } { ∈ } ∩ { expl } { ∈ } ∩ { } n[≥1 and for all n 1, ≥ X Xt B Tn >t . { ∈ } ∩ { } ∈ FTn Hence,

X (A.3) Xt B σ Tn . { ∈ }∈ F ! n[≥1 Moreover, for all t 0, ≥ Xt =∆ = T t T t { } { expl ≤ } ∪ { jump ≤ }

= Tn t ( Tjump t Texpl >t ) { ≤ }! ∪ { ≤ } ∩ { } n\≥1

= Tn t ( Tjump t Tn >t ). { ≤ }! ∪ { ≤ } ∩ { } n\≥1 n[≥1 It can easily be checked that, for all n 1, ≥ X Tn t and T t Tn >t belong to . { ≤ } { jump ≤ } ∩ { } FTn Hence,

X Xt =∆ σ Tn , { }∈ F ! n[≥1 which, together with (A.3), implies (A.2). For every set A X , we write ∈ FT A = [A T < Texpl ] [A T Texpl ] (A.4) ∩ { } ∪ ∩ { ≥ }

= A T < Tn [A T Texpl ]. " ∩ { }# ∪ ∩ { ≥ } n[≥1 Observe that, for all n 1, ≥ X (A.5) A T < Tn . ∩ { } ∈ FT ∧Tn For every class of subset of Ω, we define G T T := G T T G . G ∩ { ≥ expl} { ∩ { ≥ expl} | ∈ G} MEASURE CHANGES FOR JUMP-DIFFUSION PROCESSES 19

It follows from (A.2) that

X (A.6) A T Texpl σ Tn T Texpl , ∩ { ≥ }∈ F ! ∩ { ≥ } n[≥1 and it can easily be checked that

X X X σ Tn T Texpl σ Tn T Texpl σ T ∧Tn . F ! ∩ { ≥ }⊂ F ∩ { ≥ }! ⊂ F ! n[≥1 n[≥1 n[≥1 Hence, (A.4), (A.5) and (A.6) imply that

X X T σ T ∧Tn , F ⊂ F ! n[≥1 which, together with (A.1), proves the proposition. 

Acknowledgments. We are grateful to the referees and the Associate Editor for many valuable comments.

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P. Cheridito D. Filipovic´ Department of Operations Research Department of Mathematics and Financial Engineering University of Munich Princeton University Theresienstrasse 39 Princeton, New Jersey 08544 80333 Munich USA Germany e-mail: [email protected] e-mail: fi[email protected] MEASURE CHANGES FOR JUMP-DIFFUSION PROCESSES 21

M. Yor Laboratoire de Probabilites´ et Modeles` aleatoires´ Universite´ Pierre et Marie Curie, Paris VI 4 Place Jussieu 75252 Paris Cedex 05 France