Small-time expansions for local jump-diffusion models with infinite jump activity

0.43 MB·English

by  José E. Figueroa-López

Small-time expansions for local jump-diffusion models with infinite
jump activity
Jos´e E. Figueroa-Lo´pez∗ Yankeng Luo† Cheng Ouyang‡
2
July 25, 2012
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0
2

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u
Abstract
J

4 Weconsider a Markov process X which is thesolution of astochastic differential equation driven bya L´evy
2 process Z and an independent Wiener process W. Under some regularity conditions, including non-degeneracy
ofthediffusiveandjump componentsof theprocess aswell assmoothness oftheL´evydensityofZ outsideany

] neighborhood of the origin, we obtain a small-time second-order polynomial expansion for the tail distribution
R
and the transition density of the process X. Our method of proof combines a recent regularizing technique
P
for deriving the analog small-time expansions for a L´evy process with some new tail and density estimates for
.
h jump-diffusion processes with small jumps based on the theory of Malliavin calculus, flow of diffeomorphisms
t forSDEs,andtime-reversibility. Asanapplication,theleadingtermforout-of-the-moneyoptionpricesinshort
a
maturity undera local jump-diffusion model is also derived.
m
[

1 Introduction
3
v
6
The small-time asymptotic behavior of the transition densities of Markov processes {X (x)} with deterministic
8 t t≥0
initial condition X (x) = x has been studied for a long time, with a certain focus to consider either purely-
3 0
3 continuous or purely-discontinuous processes. Starting from the problem of existence, there are several sets of
. sufficient conditions for the existence of the transitiondensity of X (x), denoted hereafterp (·;x). A streamin this
8 t t
0 direction is based on the machinery of Malliavin calculus, originally developed for continuous diffusions (see the
1 monograph [25]) and, then, extended to Markov process with jumps (see the monograph [7]). This approach can
1 also yield estimates of the transition density p (·;x) in small time t. For purely-jump Markov processes, the key
t
:
v assumption is that the L´evy measure of the process admits a smooth L´evy density. The pioneer of this approach
i was L´eandre [19], who obtained the first-order small-time asymptotic behavior of the transition density for fully
X
supported L´evy densities. This result was extended in [17] to the case where the point y cannot be reached with
r
a only one jump fromx but rather with finitely many jumps, while [27] developed a method that can alsobe applied
to L´evy measures with a nonzero singular component (see also [28] and [18] for other related results).
The main result in [19] states that, for y 6=0,
1
lim p (x+y;x)=g(x;y),
t
t→0 t
where g(x;y) is the so-called L´evy density of the process X to be defined below (see (5)). L´eandre’s approach
consisted of first separating the small jumps (say, those with sizes smaller than an ε > 0) and the large jumps of
the underlying L´evy process, and then conditioning on the number of large jumps by time t. Malliavin’s calculus
was then applied to control the resulting density given that there is no large jump. For ε > 0 small enough, the
∗Department of Statistics, Purdue University, W. Lafayette, IN, 47906 USA ([email protected]). Research partiallysupported
bytheNSFGrants#DMS-0906919andDMS-1149692.
†Department ofMathematics,PurdueUniversity,W.Lafayette, IN,47906USA([email protected]).
‡Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
([email protected]).
1
term when there is only one large jump was proved to be equivalent, up to a remainder of order o(t), to the term
resulting from a model in which there is no small-jump component at all. Finally, the terms when there is more
than one large jump were shown to be of order O(t2).
Higher order expansions of the transition density of Markov processes with jumps have been considered quite
recently and only for processes with finite jump activity (see, e.g., [35]) or for L´evy processes with possibly infinite
jump-activity. We focus on the literature of the latter case due to its close connection to the present work. [32]
wasthe firstworkto considerhigher orderexpansionsforthe transitiondensities ofL´evyprocessesusing L´eandre’s
approach. Concretely,the followingexpansionforthe transitiondensities {p (y)} ofaL´evyprocess{Z } was
t t≥0 t t≥0
proposed therein:
d N−1 tn
p (y):= P(Z ≤y)= a (y) +O(tN), (y 6=0, N ∈N). (1)
t t n
dy n!
n=1
X
As in [19], the idea was to justify that each higher order term (say, the term corresponding to k large jumps) can
be replaced, up to a remainder of order O(tN), by the resulting density as if there were no small-jump component.
However,this approachis able to produce the correctexpressionsfor the higher order coefficients a (y),... only in
2
the compound Poissoncase (cf. [12]). The problem was subsequently resolvedin [13] (see Section 6 therein as well
as [12] for a preliminary related result), using a new approach, under the assumption that the L´evy density of the
L´evyprocess{Z } issufficientlysmoothandboundedoutsideanyneighborhoodoftheorigin. Therearetwokey
t t≥0
ideas in [12, 13]. Firstly, instead of working directly with the transition densities, the following analog expansions
for the tail probabilities were first obtained:
N−1 tn
P(Z ≥y)= A (y) +tNR (y), (y >0, N ∈N), (2)
t n t
n!
n=1
X
where sup |R (y)| <∞, for some t >0. Secondly, by considering a smooth thresholding of the large jumps
0<t≤t0 t 0
(so that the density of large jumps is smooth) and conditioning on the size of the first jump, it was possible to
regularizethediscontinuousfunctional 1 and,subsequently,proceedtouseaniteratedDynkin’sformula(see
{Zt≥x}
Section3.2belowformoreinformation)toexpandtheresultingsmoothmomentfunctionsE(f(Z ))asapowerseries
t
in t. (1) was then obtained by differentiation of (2), after justifying that the functions A (y) and the remainder
n
R (y) were differentiable in y.
t
The results and techniques described in the previous paragraph open the door to the study of higher order
expansions for the transition densities of more general Markov models with infinite jump-activity. We take the
analysisonestepfurther andconsiderajump-diffusion modelwithnon-degeneratediffusionandjumpcomponents.
Our analysis can also be applied to purely-discontinuous processes as in [19], but we prefer to consider a “mixture
model”duetoitsrelevanceinfinancialapplicationswherecompellingempiricalevidencesupportsmodelscontaining
bothcontinuousandjumpcomponents(seeSection6belowfordetailedreferencesinthisdirection). Moreconcretely,
we consider the following Stochastic Differential Equations (SDE) driven by a Wiener process {W } and an
t t≥0
independent purely-jump L´evy process {Z } of bounded variation:
t t≥0
t t
Xt(x)=x+ b(Xu(x))du+ σ(Xu(x))dWu+ γ(Xu−(x),∆Zu). (3)
Z0 Z0 u∈(0,tX]:∆Zu6=0
Here, ∆Zu := Zu−Zu− := Zu−limsրtZs denotes the jump Z at time u, and b,σ : R → R,γ : R×R → R are
some suitable deterministic functions. The boundedness of the variation of Z is not essential for our analysis and
is only imposed for the easiness of notation.
As it will be evident from our work, an important difficulty to deal with the model (3) arises from the more
complex interplay of the jump and continuous components. In particular, conditioning on the first “big jump”
of {X (x)} leads us to consider the short-time expansions of the tail probability of a SDE with random initial
s s≤t
value J˜, which creates important, albeit interesting, subtleties. More concretely, in the case of a L´evy process (i.e.
when b, σ, and γ above are state-independent), conditioning on the first big jump naturally leads to analyzing the
small-time expansion of the tail probability P(Xε(x)+J˜ ≥ x+y), where {Xε(x)} stands for the “small jump”
t s
component of {X (x)} (see the end of Section 2 for the terminology). This task is relatively simple to handle
s
2
since the smooth density of J˜ “regularizes” the problem. In contrast, in the general local jump-diffusion model,
conditioning on the first big jump leads to consider P(Xε(x+J˜)≥x+y), a problem that does not allow a direct
t
application of Dynkin’s formula. Instead, to obtain the second-order expansion of the latter tail probability, we
need to rely on smooth approximations of the tail probability building on the theoretical machinery of the flow of
diffeomorphisms for SDEs and time-reversibility.
Under certain regularity conditions on b,σ and γ, as well as the L´evy measure ν of Z, we show the following
second order expansion (as t→0) for the tail distribution of {X (x)} :
t t≥0
t2
P(X (x)≥x+y)=tA (x;y)+ A (x;y)+O(t3), for x∈R, y >0. (4)
t 1 2
2
The assumptionsrequiredfor(4)include boundedness andsufficientsmoothnessofthe SDE’scoefficientsaswellas
non-degeneracyconditions on|∂ γ(x,ζ)| and|1+∂ γ(x,ζ)|. As in [19], the key assumptiononthe L´evymeasureν
ζ x
ofZ isthatthisadmitsadensityh:R\{0}→R thatisboundedandsufficientlysmoothoutsideanyneighborhood
+
ofthe origin. Inthatcase,the leadingtermA (x;y) depends onlyonthe jump componentofthe processasfollows
1
A (x;y)=ν({ζ :γ(x,ζ)≥y})= h(ζ)dζ.
1
Z{ζ:γ(x,ζ)≥y}
ThesecondordertermA (x;y)admitsamorecomplex(butexplicit)representation,whichenablesus,forinstance,
2
to precisely characterize the effects of the drift b and the diffusion σ of the process in the likelihood of a “large”
positive move (say, a move of size more than y) during a short time period t (see Remark 4.3 below for further
details).
Once the asymptotic expansionfor tail distribution is obtained, we proceed to obtain a second order expansion
for the transitiondensity function p (y;x). As expected from taking formaldifferentiation of the tail expansion(4)
t
with respect to y, the leading term of p (x+y;x) is of the form tg(x;y) for y > 0, where g(x;y) is the so-called
t
L´evy density of the process {X (x)} defined by
t t≥0
∂
g(x;y):=− ν({ζ :γ(x,ζ)≥y}) (y >0), (5)
∂y
whilethesecondordertermtakestheform−∂ A (x;y)t2/2. Oneofthemainsubtletiesherearisesfromattempting
y 2
to control the density of X (x) given that there is no “large” jump. To this end, we generalize the result in [19] to
t
the case where there is a non-degenerate diffusion component. Again, Malliavin calculus is proved to be the key
tool for this task.
Let us briefly make some remarks about the practical relevance of our results. Short-time asymptotics for the
transition densities and distributions of Markov processes are important tools in many applications such as non-
parametric estimation methods of the model under high-frequency sampling data and numerical approximations
of functionals of the form Φ (x) := E(φ(X (x)). In many of these applications, a certain discretization of the
t T
continuous-time object under study is needed and, in that case, short-time asymptotics are important not only in
developing such discrete-time approximations but also to determine the rate of convergence of the discrete-time
proxies to their continuous-time counterparts.
Asaninstanceoftheapplicationsreferredtointhepreviousparagraph,aproblemthathasreceivedagreatdeal
ofattentioninthelastfewyearsisthestudyofsmall-timeasymptoticsforoptionpricesandimpliedvolatilities(see,
e.g., [16], [10], [14], [6], [11], [31], [34], [15], [24], [13]). As a byproduct of the asymptotics for the tail distributions
(4), we derive here the leading term of the small-time expansion for the arbitrage-free prices of out-of-the-money
European call options. Specifically, let {S } be the stock price process and denote X = logS for each t ≥ 0.
t t≥0 t t
We assume that P is the option pricing measure and that under this measure the process {X } is of the form in
t t≥0
(3). Then, we prove that
1 ∞
lim E(S −K) = S eγ(x,ζ)−K h(ζ)dζ, (6)
t + 0
t→0 t Z−∞(cid:16) (cid:17)+
which extends the analog result for exponential L´evy model (cf. [31] and [34]). A related paper is [21], where (6)
was obtained for a wide class of multi-factor L´evy Markovmodels under certain technical conditions (see Theorem
3
2.1 therein),including the requirementthatlim E(S −K) /t existsinthe “out-of-the-moneyregion”andsome
t→0 t +
stringent integrability conditions on the L´evy density h.
The paper is organized as follows. In Section 2, we introduced the model and the assumptions needed for our
results. The probabilistic tools, such as the iterated Dynkin’s formula as well as tail estimates for semimartingales
with bounded jumps, arepresentedinSection 3. The main resultsofthe paper arethen statedinSections 4 and5,
where the second order expansion for the tail distributions and the transition densities are obtained, respectively.
The application of the expansion for the tail distribution to option pricing in local jump-diffusion financial models
is presented in Section 6. The proofs of our main results as well as some preliminaries of Malliavin calculus on
Wiener-Poisson spaces are given in several appendices.
2 Setup, assumptions, and notation
Throughout, C≥1 (resp., C∞) represents the class of continuous (resp., bounded) functions with bounded and
b b
continuous partial derivatives of arbitrary order n ≥ 1. We let {Z } be a driftless pure-jump L´evy process of
t t≥0
bounded variation with L´evy measure ν and {W } be a Wiener process independent of Z, both of which are
t t≥0
defined onacomplete probabilityspace(Ω,F,P),equipped withthe naturalfiltration(F ) generatedbyW and
t t≥0
Z and augmented by all the null sets in F so that it satisfies the usual conditions (see, e.g., Chapter I in [30]).
As stated in the introduction, in this paper we consider the following local jump-diffusion model
t t
Xt(x)=x+ b(Xu(x))du+ σ(Xu(x))dWu+ γ(Xu−(x),∆Zu), (7)
Z0 Z0 0<u≤t
X
where ∆Zu :=Zu−Zu− :=Zu−limsրtZs denotes the jump Z at time u, and b,σ :R→R and γ :R×R→R are
deterministic functions satisfying suitable conditions under which (7) admits a unique solution. Typical sufficient
conditionsfor(7)tobe well-posedincludelineargrowthandLipschitzcontinuityofthe coefficientsb, σ,andγ (see,
e.g., [3, Theorem 6.2.3], [26, Theorem 1.19]). It is convenient to write Z in terms of its L´evy-Itˆo representation:
t
Z = ∆Z = ζM(du,dζ).
t u
0<u≤t Z0 ZR0
X
Here, R :=R\{0}andM(du,dζ):=#{u>0:(u,∆Z )∈du×dζ}isthe jumpmeasureofthe processZ,whichis
0 u
aPoissonrandommeasureonR ×R withmeanmeasureEM(du,dζ)=duν(dζ). Wemakethefollowingstanding
+ 0
assumptions about Z:
(C1) The L´evy measure ν of Z has a C∞(R\{0}) strictly positive density h such that, for every ε>0
and n≥0,
(i) (1∧|ζ|)h(ζ)dζ <∞, and (ii) sup |h(n)(ζ)|<∞. (8)
Z |ζ|>ε
Remark 2.1 Condition (8-i) is equivalent to the condition that Z is of bounded variation. This condition is not
essentialfor ourresultsandis imposedonlyfor theeasiness ofnotation. WecouldhavetakenageneralL´evyprocess
Z and replace the last term in (7) with a finite-variation pure-jump process plus a compensated Poisson integral as
follows:
t t
X (x)=x+ b(X (x))du+ σ(X (x))dW
t u u u
Z0 Z0
t t
+ γ(Xu−(x),ζ)M(du,dζ)+ γ(Xu−(x),ζ)M¯(du,dζ), (9)
Z0 Z|ζ|>1 Z0 Z|ζ|≤1
where M¯(du,dζ) is the compensated Poisson measure M(du,dζ)−ν(dζ)du. Condition (8)-(ii) is actually necessary
for the tail probabilities of {X (x)} to admit an expansion in integer powers of time. Indeed, even in the simplest
t t≥0
pure L´evy case (X (x) = Z +x), it is possible to build examples where P(Z ≥ y) converges to 0 at a fractional
t t t
power of t in the absence of (8-ii) (see [22]).
4
Throughout the paper, the generator γ is assumed to satisfy the following conditions:
(C2-a) γ(·,·)∈C≥1(R×R) and γ(x,0)=0 for all x∈R;
b
(C2-b) There exists a constant δ >0 such that |∂ γ(x,ζ)|≥δ, for all x,ζ ∈R.
ζ
Both of the previous conditions were also imposed in [19]. Note that (C2-a) implies that, for any ε > 0, there
exists C <∞ such that
ε
∂iγ(x,ζ)
sup ≤C |ζ|, (10)
∂xi ε
x (cid:12) (cid:12)
(cid:12) (cid:12)
for all |ζ|≤ε and i≥0. Condition (C2-b) is im(cid:12) posed so(cid:12)that, for each x∈R, the mapping ζ →γ(x,ζ) admits an
(cid:12) (cid:12)
inverse function γ−1(x,ζ) with bounded derivatives. Note that (C2-b)together with the continuity of∂γ(x,ζ)/∂ζ
implies that the mapping ζ →γ(x,ζ) is either strictly increasing or decreasing for all x.
We will also require the following boundedness and non-degeneracy conditions:
(C3) The functions b(x) and v(x):=σ2(x)/2 belong to C∞(R).
b
(C4) There exists a constant δ >0 such that, for all x,ζ ∈R,
∂γ(x,ζ)
(i) 1+ ≥δ, (ii)σ(x)≥δ. (11)
∂x
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Remark 2.2 Boundedness conditions of(cid:12)the type (C3(cid:12)) above are not restrictive in practice. Indeed, on one hand,
extremely large values of b and σ won’t typically make sense in a particular financial or physical phenomenon
in mind (e.g. a large volatility value σ could hardly be justified financially). On the other hand, a stochastic
model with arbitrary (but sufficiently regular) functions b and v could be closely approximated by a model with C∞
b
functions b and v. The condition (11-i), which was also imposed in [19], guarantees the a.s. existence of a flow
Φ (x):R→R,x→X (x) of diffeomorphisms for all 0≤s≤t (cf. [19]), where here {X (x)} is defined as
s,t s,t s,t t≥s
in (7) but with initial condition X (x)=x.
s,s
As it is usually the case with L´evy processes, we shall decompose Z into a compound Poisson process and
a process with bounded jumps. More specifically, let φ ∈ C∞(R) be a truncation function such that 1 ≤
ε |ζ|≥ε
φ (ζ) ≤ 1 and let {Z (ε)} and {Z′(ε)} be independent driftless L´evy processes of bounded variation
ε |ζ|≥ε/2 t t≥0 t t≥0
with respective L´evy densities
h (ζ):=φ (ζ)h(ζ) and h¯ (ζ):=(1−φ (ζ))h(ζ). (12)
ε ε ε ε
Without loss of generality, we assume hereafter that
Z =Z (ε)+Z′(ε), for all t≥0. (13)
t t t
The process Z′(ε), that we referredto as the small-jump component of Z, is a pure-jump L´evy process with jumps
bounded by ε. In contrast, the process Z(ε), hereafter referred to as the big-jump component of Z, is a compound
Poisson process with intensity of jumps λ := φ (ζ)h(ζ)dζ and jumps {Jε} with probability density function
ε ε i i≥1
R φ (ζ)h(ζ)
h˘ (ζ):= ε . (14)
ε
λ
ε
Throughout the paper, {Nε} denote the jump counting process of the compound Poisson process {Z (ε)}
t t≥0 t t≥0
and J :=Jε represent a random variable with density h˘ (ζ).
ε
The following result, whose proof is presented in Appendix D, will be needed in what follows.
Lemma 2.3 Under the conditions (C1), (C2) and (C4), the following statements hold:
1. Let γ˜(z,ζ) := γ(z,ζ)+z. Then, for each z ∈ R, the mapping ζ → γ˜(z,ζ) (equiv. ζ → γ(z,ζ)) is invertible
and its inverse γ˜−1(z,ζ) (resp. γ−1(z,ζ)) is C≥1(R×R).
b
5
2. Bothγ˜(z,Jε)andγ(z,Jε)admitdensitiesinC∞(R×R),denotedbyΓ(ζ;z):=Γ (ζ;z)andΓ(ζ;z):=Γ (ζ;z),
b ε ε
respectively. Furthermore, they have the representation:
e e
∂γ −1 ∂γ −1
Γ (ζ;z)=h˘ (γ˜−1(z,ζ)) z,γ˜−1(z,ζ) , Γ (ζ;z)=˘h (γ−1(z,ζ)) z,γ−1(z,ζ) . (15)
ε ε ε ε
∂ζ ∂ζ
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:0) (cid:1)(cid:12) (cid:12) (cid:0) (cid:1)(cid:12)
e (cid:12) (cid:12) (cid:12) (cid:12)
3. The mappings (z,ζ)→P(γ˜(z(cid:12),Jε)≥ζ) and (z,(cid:12)ζ)→P(γ(z,Jε)≥ζ) are C∞((cid:12)R×R). (cid:12)
b
4. The mapping z →u:=z+γ(z,ζ) admits a density, denoted hereafter γ¯(u,ζ), that belongs to C≥1(R×R).
b
Wefinishthissectionwiththedefinitionoftwoimportantprocesses. Throughoutthepaper,welet{X (ε,∅,x)}
s s≥0
be the solution of the SDE:
s s
Xs(ε,∅,x):=x+ b(Xu(ε,∅,x))du+ σ(Xu(ε,∅,x))dWu+ γ(Xu−(ε,∅,x),∆Zu′(ε))
Z0 Z0 0<u≤s
X
s s s
=x+ b(Xu(ε,∅,x))du+ σ(Xu(ε,∅,x))dWu+ γ(Xu−(ε,∅,x),ζ)M′(du,dζ), (16)
Z0 Z0 Z0 Z
where hereafterM′ denotes the jump measureofthe small-jumpcomponentZ′(ε). The lawofthe process(16)can
be interpreted as the law of {X } conditioning on not having any “big” jumps during [0,t]. In other words,
s 0≤s≤t
denoting the law of a process Y (resp. the conditional law of Y given an event B) by L(Y) (resp. L(Y|B)), we
have that, for each fixed t>0,
L {X (x)} Nε =0 =L {X (ε,∅,x)} .
s 0≤s≤t t s 0≤s≤t
(cid:16) (cid:12) (cid:17) (cid:16) (cid:17)
Similarly, for a collection of times 0<s <···<(cid:12) s , let {X (ε,{s ,...,s },x)} be the solution of the SDE:
1 (cid:12) n s 1 n s≥0
s s
X (ε,{s ,...,s },x):=x+ b(X (ε,{s ,...,s },x))du+ σ(X (ε,{s ,...,s },x))dW
s 1 n u 1 n u 1 n u
Z0 Z0
+ γ(Xu−(ε,{s1,...,sn},x),∆Zu′(ε))+ γ(Xs−(ε,{s1,...,sn},x),Jiε),
i
0<Xu≤s i:Xsi≤s
where, as above, {Jε} represents a random sample from the distribution φ (ζ)h(ζ)dζ/λ , independent of Z′.
i i≥1 ε ε
Then, denoting the arrival jumps of Z(ε) by τ <τ <..., it follows that
1 2
L {X (x)} Nε =n,τ =s ,...,τ =s =L {X (ε,{s ,...,s },x)} .
s 0≤s≤t t 1 1 n n s 1 n 0≤s≤t
(cid:16) (cid:12) (cid:17) (cid:16) (cid:17)
The previous two processes will(cid:12)be needed in order to implement L´eandre’sapproachin which the tail distribution
(cid:12)
P(X (x)≥x+y) is expanded in powers of time by conditioning on the number of jumps of Z(ε) by time t.
t
3 Probabilistic tools
Throughout, Cn(I) (resp. Cn) denotes the class of functions having continuous and bounded derivatives of order
b b
0≤k ≤n in an open interval I ⊂R (resp. in R). Also, kgk =sup |g(y)|.
∞ y
3.1 Uniform tail probability estimates
The following general result will be important in the sequel.
Proposition 3.1 Let M be a Poisson random measure on R ×R with mean measure EM(du,dζ)=ν(dζ)dt and
+ 0
M¯ be its compensated random measure. Let Y :=Y(x) be the solution of the SDE
t t t
Y =x+ ¯b(Y )ds+ σ¯(Y )dW + γ¯(Y ,ζ)M¯(ds,dζ).
t s s s s−
Z0 Z0 Z0 Z
6
Assume that ¯b(x) and σ¯(x) are uniformly bounded and γ¯(x,ζ) is such that, for a constant S <∞, sup |γ¯(y,ζ)|≤
y
S(|ζ|∧1), for ν-a.e. ζ. In particular, the jumps of {Y } are bounded by S. Then there exists a constant C(S,k)
t t≥0
depending on S and k, such that, for any fixed p>0 and all 0≤t≤1,
P sup |Y(x)−x|≥2pS ≤C(S,k)tp.
s
(cid:26)0≤s≤t (cid:27)
Proof. Let V be either tσ¯(Y )dW or t γ¯(Y ,z)M¯(ds,dz). It is clear that, in either case, V is a mar-
t 0 s s 0 s− t
tingale with its jumps bounded by S. Moreover, its quadratic variation hV,Vi is given by either tσ¯2(Y )ds or
R R R 0 s
t γ¯2(Y ,ζ)ν(dζ)dsand,hence,inlightoftheboundednessofσ¯ andtheγ¯,satisfieshV,Vi ≤kt,forsomeconstant
0 s t R
k. By equation (9) in [20], we have
R R
λ2
P sup |V |≥C ≤2exp −λC+ kt(1+exp[λS]) , for all C,λ>0. (17)
s
2
(cid:26)0≤s≤t (cid:27) (cid:20) (cid:21)
Now take C =2pS and λ=|logt|/2S. The rest of the proof is then clear.
Asadirectcorollaryofthe aboveproposition,wehavethefollowingtailprobabilityestimate forthe small-jump
component {X (ε,∅,x)} of X defined in (16).
t t≥
Lemma 3.2 Fix any η > 0 and a positive integer N. Then, under the Conditions (C2-C3) of Section 2, there
exist an ε:=ε(N,η)>0 and C :=C(N,η)<∞ such that
(1) For all t<1,
sup P(|X (ε′,∅,x)−x|≥η)<CtN. (18)
t
0<ε′<ε,x∈R
(2) For all t<1,
∞ ∞
sup P{e|Xt(ε′,∅,x)−x| ≥s}ds= sup P{|X (ε′,∅,x)−x|≥logs}ds<CtN.
t
ε′<ε,x∈RZη ε′<ε,x∈RZη
Proof. The first statement is a special case of Proposition 3.1, which can be applied in light of the boundedness
conditions (C3) as well as the condition (C2-a) that ensured that the components
t t
σ(Xu(ε′,∅,x))dWu, γ(Xu−(ε′,∅,x),ζ)M¯′(du,dζ),
Z0 Z0 Z
aremartingaleswiththeirquadraticvariationboundedbykt,forsomeconstantk >0. Here,M¯′isthecompensated
Poisson measure of the small-jump component Z′(ε′). To prove the second statement, we keep the notation of the
proof of Proposition 3.1. By (17) there exists a constant C >0 such that
∞ ∞ λ2
P{|X (ε,∅,x)−x|≥logs}ds≤C exp −λlogs+ kt(1+exp[λε]) ds
t
2
Zη Zη (cid:20) (cid:21)
Cη λ2
= exp kt(1+exp[λε]) .
(λ−1)ηλ 2
(cid:20) (cid:21)
Now it suffices to take λ=|logt|/2ε and ε=logη/2N.
3.2 Iterated Dynkin’s formula
We now proceed to formally state a second-order iterated Dynkin’s formula for the “small-time component” of X,
{X (ε,∅,x)} , defined in (16). In general, the n-order iterated Dynkin’s formula of a Markov process Y with
t t≥0
initial condition Y =x and generator L takes the generic form:
0 Y
n−1tk tn 1
Ef(Y )= Lkf(x)+ (1−α)n−1E{Lnf(Y )}dα, (19)
t k! Y (n−1)! Y αt
k=0 Z0
X
7
where as usual L0f = f and Lnf = L (Ln−1f), n ≥ 1. For the general jump-diffusion process (9), (19) can be
Y Y Y Y
proved for n = 1 using Itoˆ’s formula (see [26, Theorem 1.23]), while the general case can be proved by induction
undersomewhatstrongconditions. Inwhatfollows,weonlyneedthesecond-orderformulaand,thus,thesubsequent
result will suffice for our purposes. Below, L denotes the infinitesimal generator for {X (ε,∅,x)} which can be
ε t t≥0
written as (c.f. [26])
L f(y):=D f(y)+I f(y), with (20)
ε ε ε
σ2(y)
D f(y):= f′′(y)+b(y)f′(y), I f(y):= (f(y+γ(y,ζ))−f(y))h¯ (ζ)dζ.
ε ε ε
2
Z
Hereafter, h¯ (ζ) := (1−φ (ζ))h(ζ) denotes the density of h truncated outside of a ε-neighborhood of the origin.
ε ε
Let us remark that (20) is well-defined whenever f ∈ C2 in light of condition (C2-a). The proof of the following
b
result is presented in Appendix D.
Lemma 3.3 Under the Conditions (C1)-(C3) of Section 2, the following assertions hold:
1. For any function f in C2,
b
1
Ef(X (ε,∅,x))=f(x)+t EL f(X (ε,∅,x))dα, (21)
t ε αt
Z0
where the remainder term is such that
1
sup EL f(X (ε′,∅,x))dα ≤K, (22)
ε αt
0<tε<′<1,εx∈R(cid:12)(cid:12)Z0 (cid:12)(cid:12)
(cid:12) (cid:12)
for a constant K <∞ depending only on (cid:12) |ζ|h¯ (ζ)dζ, kf(k)k , kb(k)(cid:12)k , and kv(k)k for k =0,1,2.
ε ∞ ∞ ∞
2. Additionally, if f ∈C4, then R
b
1
Ef(X (ε,∅,x))=f(x)+tL f(x)+t2 (1−α)EL2f(X (ε,∅,x))dα, (23)
t ε ε αt
Z0
where the remainder term is such that
1
sup (1−α)EL2f(X (ε′,∅,x))dα ≤K, (24)
ε αt
0<tε<′<1,εx∈R(cid:12)(cid:12)Z0 (cid:12)(cid:12)
(cid:12) (cid:12)
for a constant K <∞ depending only o(cid:12)n |ζ|h¯ (ζ)dζ, kf(k)k , kb(k)k (cid:12), and kv(k)k for k =0,...,4.
ε ∞ ∞ ∞
R
4 Second order expansion for the tail distributions
We are ready to state our first main result; namely, we characterizethe small-time behavior of the tail distribution
of {X (x)} :
t t≥0
F¯(x,y):=P(X (x)≥x+y), (y >0). (25)
t t
As in [19], the key idea is to use the decomposition (13) by conditioning on the number of “large”jumps occurring
before timet. Concretely,recallingthat{Nε} andλ := φ (ζ)h(ζ)dζ representthejump countingprocessand
t t≥0 ε ε
the jump intensity of the large-jump component process {Z (ε)} of Z, we have
t t≥0
R
∞ (λ t)n
P(X (x)≥x+y)=e−λεt P(X (x)≥x+y|Nε =n) ε . (26)
t t t n!
n=0
X
8
The first term in (26) (when n=0) can be written as
P(X (x)≥x+y|Nε =0)=P(X (ε,∅,x)≥x+y).
t t t
In light of (18), this term can be made O(tN) for an arbitrarily large N ≥1, by taking ε small enough. In order to
deal with the other terms in (26), we use the iterated Dynkin’s formula introduced in Section 3.2. The following is
the main result of this section (see Appendix B for the proof). Below, h and h¯ denote the L´evy densities defined
ε ε
in (12), while g(x;y) denotes the so-called L´evy density of the process {X (x)} defined by
t t≥0
∂
g(x;y):=− h(ζ)dζ, (27)
∂y
Z{ζ:γ(x,ζ)≥y}
for y >0. In light of Lemma 2.3, g admits the representation:
g(x;y)=h(γ−1(x,y)) (∂ γ) x,γ−1(x,y) −1,
ζ
where ∂ζγ is the partial derivative of the function γ(x,ζ(cid:12)) with(cid:0)respect to it(cid:1)s(cid:12)second variable.
(cid:12) (cid:12)
Theorem 4.1 Let x∈R and y >0. Then, under the Conditions (C1-C4) of Section 2, we have
t2
F¯(x,y):=P(X (x)≥x+y)=tA (x;y)+ A (x;y)+O(t3), (28)
t t 1 2
2
as t→0, where A (x;y) and A (x;y) admit the following representations (for ε>0 small enough):
1 2
∞
A (x;y):= g(x;ζ)dζ = h(ζ)dζ,
1
Zy Z{γ(x,ζ)≥y}
A (x;y):=D(x;y)+J (x;y)+J (x;y),
2 1 2
with
∂ ∞
D(x;y)=b(x) g(x;ζ)dζ +g(x;y) +b(x+y)g(x;y)
∂x
(cid:18) Zy (cid:19)
σ2(x) ∂2 ∞ ∂ ∂
+ g(x;ζ)dζ +2 g(x;y)− g(x;y) ,
2 ∂x2 ∂x ∂y
(cid:18) Zy (cid:19)
σ(x+y) ∂
− σ(x+y) g(x;y)+2σ′(x+y)g(x;y) (29)
2 ∂y
(cid:18) (cid:19)
∞ ∞ ∞
J (x;y)= g(x;ζ)dζ + g(x;ζ)dζ −2 g(x;ζ)dζ h¯ (ζ¯)dζ¯,
1 ε
Z Zy−γ(x,ζ¯) Zγ¯(x+y,ζ¯)−x Zy !
∞ ∞
J (x;y)= g(x+γ(x,ζ);ζ′)dζ′h (ζ)dζ −2 g(x;ζ)dζ h (ζ)dζ.
2 ε ε
Z Zy−γ(x,ζ) Zy Z
Remark 4.2 Note that if supp(ν)∩{ζ :γ(x,ζ)≥y}=∅ (so that it is not possible to reach the level y from x with
only one jump), then A (x;y) = 0 and P(X (x) ≥ x+y) = O(t2) as t → 0. If, in addition, it is possible to reach
1 t
the level y from x with two jumps, then J (x;y) 6= 0, implying that P(X (x) ≥x+y) decreases at the order of t2.
2 t
These observations are consistent with the results in [17] and [28].
Remark 4.3 In the case that the generator γ(x,ζ) does not depend on x, we get the following expansion for
P(X (x)≥x+y):
t
∞ b(x)+b(x+y) σ2(x)+σ2(x+y) t2
t g(ζ)dζ+ g(y)t2− g′(y)+2σ(x+y)σ′(x+y)g(y)
2 2 2
Zy (cid:18) (cid:19)
∞ ∞
+ g(ζ)dζ− g(ζ)dζ h¯ (ζ)dζt2
ε
Z Zy−γ(ζ¯) Zy !
∞ ∞ t2
+ g(ζ′)dζ′h (ζ)dζ −2 g(ζ)dζ h (ζ)dζ +O(t3).
ε ε
2
Z Zy−γ(ζ) Zy Z !
9
The leading term in the above expression is determined by the jump component of the process and it has a natural
interpretation: if within a very short time interval there is a “large” positive move (say, a move by more than
y), this move must be due to a “large” jump. It is until the second term, when the diffusion and drift terms of
the process X(x) appear. If, for instance, b and σ are constants, the effect of a positive drift b > 0 is to increase
the probability of a “large” positive move of more than y by bg(y)t2(1+o(1)). Similarly, since typically g′(y) < 0
when y > 0, the effect of a non-zero spot volatility σ is to increase the probability of a “large” positive move by
σ2|g′(y)|t2(1+o(1)).
2
5 Expansion for the transition densities
Our goal here is to obtain a second-order small-time approximation for the transition densities {p (·;x)} of
t t≥0
{X (x)} . As it was done in the previous section, the idea is to work with the expansion (26) by first showing
t t≥0
that each term there is differentiable with respect to y, and then determining their rates of convergence to 0 as
t → 0. One of the main difficulties of this approach comes from controlling the term corresponding to no “large”
jumps. As in the case of purely diffusion processes, Malliavin calculus is proved to be the key tool for this task.
This analysis is presented in the following subsection before our main result is presented in Section 5.2.
5.1 Density estimates for SDE with bounded jumps
In this part, we analyze the term corresponding to Nε =0:
t
P(X (x)≥x+y|Nε =0)=P(X (ε,∅,x)≥x+y).
t t t
We will prove that, for any fixed positive integer N and η >0, there exist an ε >0 and a constant C <∞ (both
0
only depending on N and η) such that the density p (·;ε,∅,x) of X (ε,∅,x) satisfies
t t
sup p (y;ε,∅,x)<CtN, (30)
t
|y−x|>η,ε<ε0
forall0<t≤1. Theestimate(30)holdstrueforalargerclassofSDEwithboundedjumps. Concretely,throughout
this section, we consider the more general equation
t t t
Xx =x+ b(Xx )ds+ σ(Xx )dW + γ(Xx ,ζ)M¯(ds,dζ), (31)
t s− s− s s−
Z0 Z0 Z0 Z
where,asbefore,M(ds,dζ)isaPoissonrandommeasureonR ×R\{0}withmeanmeasureµ(ds,dζ)=ds×ν(dζ)
+
andM¯ =M−µisitscompensatedmeasure. Asbefore,weassumethatν admitsasmoothdensityhwithrespectto
the LebesguemeasureonR\{0},butmoreover,wenowassumehissupportedina ballB(0,r)forsomer ∈(0,∞).
Note that the solution (X (ε,∅,x)) of (16) perfectly fits within this framework.
t t
The following assumptions on the coefficient functions of (31) are used in the sequel. Note that, when {Xx} is
t
taken to be {X (ε,∅,x)}, these assumptions follow from the assumptions in Section 2.
t
Assumption 5.1 The coefficients b(x),σ(x) and γ(x,ζ) are 4 times differentiable. There exists a constant c > 0
and a function κ∈ Lp(ν) such that:
2≤p<∞
T
1. |∂n b(x)|,|∂n σ(x)|, 1 ∂n γ(x,ζ) ≤c, for 0≤n≤5.
xn xn κ(ζ) xn
(cid:12) (cid:12)
(cid:12) (cid:12)
2. ∂n+kγ(x,ζ) ≤c, fo(cid:12)r 1≤n+k ≤(cid:12)5 and k ≥1.
xnζk
(cid:12) (cid:12)
Ma(cid:12)(cid:12)lliavin calcu(cid:12)(cid:12)lus is the main tool to analyze the existence and smoothness of density for Xtx. For the sake
of completeness, we are presenting some necessary preliminaries of Malliavin calculus on Wiener-Poisson spaces
in Appendix A following the approach in [7]. As described therein, there are different ways to define a Malliavin
operator for such spaces. For our purposes, it suffices to consider the Malliavin operator corresponding to ρ = 0
10