Diusions
M.A.H. Dempster
Centre forFinancialResearch
Judge Institute ofManagementStudies
University ofCambridge,Cambridge, England.
mahd2@cam.ac.uk
www-cfr.jims.cam.ac.uk
and
G.Ch. Gotsis
Hellenic CapitalMarket Commission
Syntagma Sq.,Athens,Greece.
May 1998
Abstract
Jumping diusion models for nancial prices and returns are nding increasing
application in the pricing of current contingent claims. Generalizing the theory for
a Markov process with continuous sample paths { characterized in terms of its
in-nitesimal generator { we establish existence and uniqueness of solutions to jump
stochastic dierential equations. We adapt the Stroock and Varadhan approach,
whichdevelops avariantofthe`weaksense'solutionofastochasticdierential
equa-tion by formulating it as a diusion process solution of a martingale problem. Our
approach to deriving theintegro-partial dierential equation forthe value of a
con-tingent claimthroughthecorresponding Kolmogorovforwardequationis illustrated
bya generalizationoftherecent workof Babbsand Webberdescribingxed income
derivative valuationinthe presenceof central bankrate changes.
Keywords: jumpingdiusions,Markovprocesses,martingaleproblem,
The concept of jump-diusion Markov process models of nancial prices and returns is
ndingincreasingapplicationinpricingandhedging evermorecomplexcontingentclaims.
Some of these applicationsinclude equity index jumps [24, 18], central bank rate changes
[3, 19], credit rating changes [7], catastrophe risk [1], etc. On the other hand, usually
either formalstatementsare simply assumed correct orpractically restrictive assumptions
are made in setting out the corresponding integro-partial dierential equation (PDE) for
claim valuation and hedging. In this paper, we marshall appropriate results from the
mathematical literature to establish the existence and uniqueness of a weak solution to a
generaljump stochastic dierential equation (SDE)in terms of the corresponding
martin-galeproblem statedinterms ofthe innitesimal generatorof the Markov solution process.
Although,forexample,non-Markovianversionsof Heath-Jarrow-Mortoncompatibleshort
term interest rate models are possible [5], Markov models are almost universally used in
practice. We illustrate the use of the solution toa martingale problem for a Markov
pro-cess in a typical nancial application under a risk neutral measure to derive rigorously
the integro-PDE satised by the claim value. The machinery treatedin this papercan be
applied ceteris paribus to all the nancial situations mentioned above { and many more
involving point process events.
A numberofauthorshavestudied the existenceand uniquenessof solutionsto
stochas-tic dierential equationswhosesolutions are termed diusion processes. Loosely speaking
the term diusion is attributed to a Markov process which has continuous sample paths
and can be characterized in terms of its innitesimal generator. The latter is specied
in terms of drift and diusion coeÆcients which have natural interpretations in nancial
modelling.
There are several approaches to the study of diusions ranging from the purely
an-alytical to the purely probabilistic. The methodology of stochastic dierential equations
(SDEs) was suggested by P. Levy [17] as an `alternative' probabilistic approach to the
deterministic theory of heat diusion and was carried out by K. Ito. [14]. His strong
so-lution is constructed ona given probability space, with respect to a given ltration and a
given Brownian motionW. The idea of weak solution isanotion inwhich the probability
space, the ltration and the driving Brownian motion are all part of the solution, rather
varia-the search for a diusion process with given drift and dispersion coeÆcients in terms of
the martingale problem. Uniqueness of the solution process is required in the sense of its
nite-dimensional distributions and the continuity of its sample paths. Although nding
this processisequivalenttosolving the relatedstochasticdierential equationinthe weak
sense, the SDE is not explicitly involved. Rather, the innitesimal evolution of suitable
functionals of the solution process { specied by the innitesimal characteristics of the
process through its innitesimal generator { are determined up to anadditive martingale
error term.
In this paperwe consider the existence and uniqueness of the solution to the
Stroock-Varadhan[23]martingaleproblemforMarkovsemimartingales,i.e.jumpingdiusions. The
next section contains notation and preliminaries, after which the statements and proofs
of the main results are given in Section 3. Loosely speaking, these theorems imply that
the innitesimal characteristics of a jumping diusion { drift, dispersion, jump rate and
post-jump measure {together with the requirement that itssamplepaths beright
contin-uous and leftlimited, uniquely determine the solution process of the martingale problem.
In Section 4, an illustrative application of this practically important result is given to a
contemporaryproblem inmathematicalnance. Howeverthe techniquesdeveloped inthis
paperare much more broadly applicable, as isnoted inthe conclusions of Section 5.
2 Notation and Preliminaries
In this section we review some basic denitions and results from Markov process theory
[8,11, 16, 23].
Throughout the paper (;F;P) denotes a probability space, E is a metric space and
B(E) is the -algebra of Borel subsets of E. A collection fF
t
g := fF
t
; t 2 [0;1)g of
-algebrasof setsof F isaltration iF
t F
t+s
fort;s 2[0;1). Forastochasticprocess
X we dene F X
t
:= fX
s
: 0 s tg to be the algebra which corresponds to the
information known to an observer watching X up to time t. A ltration fF
t
g is said to
satisfy the usual conditions if it is increasing, right-continuous and F
0
PfX(t+s) 2BjF X
t
g=PfX(t+s)2 BjX(t)g (1)
for alls;t0and B 2B(E).
Let L denotea realBanach space with norm kk and let M(E)denote the collection
of all real-valued, Borel measurable functions on E. B(E) M(E) denotes the Banach
space of bounded functions with norm k f k:= sup
x2E
j f(x) j. C(E) denotes the
sub-space of B(E) of bounded continuous functions and C
0
(E) denotes the Banach space of
continuous functions which vanish at innity (in terms of the one point compactication
ofE)equippedwiththisnorm. P(E)denotesthefamilyofBorelprobabilitymeasures onE.
Taking intoconsideration that mostof the stochastic processeswhicharise innancial
applications have the property that they have right and left limits at each time point for
almost every sample path, we denoteby D
E
[0;1)the Skorohod space of rightcontinuous
functions x : [0;1) ! E with left limits. We take D
E
[0;1) to be the path space of all
processes considered here for a suitable space E (usually IR n
) and C
E
[0;1) will denote
the subspace of D
E
[0;1) containingthe continuous functions x:[0;1)!E.
Dening ametric donE wecan induceatopology onthe spaceD
E
[0;1). Itisproven
inEthier and Kurtz [11], Theorem 5.6, p.121, thatwith the topology induced by the
Sko-rohod metric D
E
[0;1) is a separable space if E is separable and is complete if (E;d) is
complete. The processes of interest to us here will take values in a complete, separable
metricspaceE andwill havesamplepathsinD
E
[0;1)equipped withthe Skorohod
topol-ogy.
A one-parameter family fT(t);t 0g of bounded linear operators on a Banach space
L with norm kk is called asemigroup i T(0)=I and T(s+t)=T(s)T(t) for alls;t
0. A semigroup T(t) on L is said to be strongly continuous i lim
t!0
k T(t)f k=k f k
for every f 2 L; it is said to be a contraction semigroup i k T(t) k< 1 for all t 0.
Wedenea semigroupfT(t)gonL tobemeasurable iT()f isameasurable functionon
([0;1);B[0;1))for each f 2L.
A linear operator A on L is said to be dissipative i k f Af k k f k for every
tobethe innitesimalgenerator of a semigroup fT(t)g on L. Wealso dene:
gr
to be the graph of the full generator ^
A of a measurable contraction semigroup fT(t)g on
L with extended domain D( ^
a bounded linear operator on L, then 0
is said to belong to the resolvent set of A, and
R
iscalled the resolvent at 0
An operator A on C(E) is said to satisfy the positive maximum principle i whenever
f 2D(A),x
A process X isamartingale withrespect tothe ltration fF
t
and asub(super) martingale accordingas theequality becomes (). Ifwithrespect to
fF
t
gthereexistsasequenceofstoppingtimes
n
%1suchthatX
()^
n
isamartingalefor
alln=1;2;:::,wesaythat X isalocal martingale. Allmartingalesare local martingales,
but not conversely. An adapted process X is a semimartingale i it has a decomposition
of the form
0 0
tingale and A has sample paths in D
E
[0;1) of nite variation, i.e. of bounded variation
oncompact timesets. Local martingales,nite variationprocesses,sub- and
supermartin-gales and (for E := IR
n
) n-dimensionalstandard Brownian motionW,with uncorrelated
Wiener coordinate processes, are all semimartingales.
Byasolutionofthe martingaleproblem forAwemeanameasurablestochasticprocess
X with values in E and sample paths in := D
E
[0;1) dened in a probability space
(;F;P),where is the Borel-eld relative tothe Skorohod topology, such that for each
(f;g)2A , i.e. g :=Af,
f(X(t)) Z
t
0
g(X(s))ds (4)
is a martingale with respect to the ltration F X
t
for all f 2 B(E). Equivalently, we say
that P solves the martingale problem for A.
When an initial distribution 2 P(E) is specied for the solution process X and (4)
holds, we say that X { equivalently P {is a solution of the martingale problem for (A;)
and without loss of generality write (4) inthe form
f(X(t)) f(X(0)) Z
t
0
g(X(s))ds: (5)
If a solution of the martingale problem for (A;) exists and is unique, we say that the
martingale problem for (A;) iswell-posed.
Denoting by the martingale process X with X(0) given by (5), we have for all
f 2B(E)
f(X(t))=f(X(0))+ Z
t
0
Af(X(s))ds+(t); (6)
which is a stochastic version of the fundamental theorem of calculus due to Dynkin [8]
involving the 0-mean martingale error . Taking expectations conditional on X(0) = x,
x t
@
@t E
x
f(X(t)) AE
x
f(X(t))=0: (7)
Finally,considertheBorelmeasurabledrift vectorfunctionb :[0;1)IR n
!IR n
; (t;x)!
b(t;x)and volatility matrix function:[0;1)IR n
!IR n
2
; (t;x)!(t;x):=(
ij (t;x)).
The diusion matrix function a : [0;1) IR n
! IR n
2
, (t;x) ! a(t;x) := (a
ij
(t;x)) is
dened by the diusion (covariance) coeÆcients
a
ij
(t;x):= n
X
k=1
ik (t;x)
kj (t;x):
We denea weak solution of the stochasticdierential equation (SDE)
dX(t) =b(t;X(t))dt+(t;X(t))dW(t) (8)
tobea triple f(X;W);(;F;P);fF
t
gg
where:-(i) (;F;P)is aprobability space.
(ii) fF
t
g is altration of sub--elds of F satisfying the usual conditions.
(iii) X=X(t);0t<1,isacontinuousadapted IR n
-valuedprocess andXhas sample
paths inC
IR n
[0;1),
(iv) W :=fW(t): 0 t< 1g is ann-dimensional standard Brownian motion and the
following conditions are satised:
(a) Pf R
t
0 [jb
i
(s;X(s))j+ 2
ij
(s;X(s))]ds 1g=1holds forevery 1i;j n and
0t <1, and
(b) the integral version of (8)
X(t) =X(0)+ Z
t
0
b(s;X(s))ds+ Z
t
0
(s;X(s))dW(s) 0t<1 (9)
The following propositions demonstrate the existence ofa unique solution toa martingale
problem for diusions in IR n
and the equivalence of the martingaleproblem solution and
a weak solution to the corresponding stochastic dierential equation. A weak solution to
the SDE (8) induces on C
IR
n[0;1) D
IR
n[0;1) a probability measure P which solves
the martingale problem (A;), for A dened for allf 2B( IR
and conversely.
Proposition 1. Let a(t;x) and b(t;x) be bounded andBorelmeasurable. Suppose that for
every x2 IR
wherethe probabilitymeasures P
1 andP
2
bothsolvethemartingaleproblemfor(A;)
start-ing from (s;x). Then P
1 =P
2
, i.e. startingfrom (s;x) there isat most one solution to the
martingale problem (A;) with sample paths in C
E
[0;1).
Proof : See Stroock and Varadhan [23], Theorem 6.2.3, p.147.
The next generalpropositionshows existence ofsolution processesXtothe martingale
problem in locally compact spacesE such as IR n
.
Proposition 2. Let E be locally compact and separable and let A be a linear operator on
C
0
(E). Suppose that D(A) is dense in C
0
(E) and that A satises the positive maximum
principle. Denethe linear operatorA
compactication of E, by
(A
solution of the martingale problem for (A ;) with sample paths in D
E
[0;1).
Proof : See Ethier and Kurtz[11], Theorem 5.4, p.199.
Proposition 3. The existence of a solution P on C
IR
n[0;1) to the martingale problem
(A;)forAdenedby(10)isequivalenttotheexistenceofaweaksolutionf(X;W);(;F;P);fF
t gg
to the stochastic integralequation (9) or, formally, the stochastic dierential equation (8)
with initial condition X(0).
Proof : Since amartingale isa local martingale,we may apply Karatzas and Shreve[16],
Proposition 4.6, p.315, to show the existence of the required weak solution. The converse
follows fromIto's lemmaand will be establishedmore generally in Theorem 2below.
3 Main Results
We dene a jumping diusion to be a solution process X in IR n
of the jump stochastic
dierential equation
dX(t)=b(t;X(t ))dt+(t;X(t ))dW(t)+X(t ); (13)
wherethe jump saltus X( ) ofX atthe jump epoch has innitesimal characteristics
at (t;x) 2 [0;1) IR n
given by (t;x) and Q(t;x;). Here 2 B([0;1) IR n
) is the
nonnegativejumpratefunctionandQ: [0;1)IR n
!P(IR n
)isthepost-jump (transition
probability)measure. ThetermX(t )iszeroatnon-jumpepochst2[0;1). The
remain-inginnitesimal characteristics ofXaretheBorelmeasurabledrift b : [0;1)IR n
! IR n
and volatility matrix : [0;1) IR n
! IR n
2
as before.
First we state the generalisation of Ito's rule for semimartingales.
Proposition 4. Suppose X is a semimartingale in IR n
with decompositionof the form
X
t =X
0 +B
t +M
t
forBanadaptedprocessofnitevariationandMalocalmartingale,andletf : IR ! IR
be twice continuously dierentiable with bounded rst and second derivatives. Then
f(X
isthe quadratic covariation, i.e. the
instanta-neous covariance, of X
i
Proof : See Elliott [10], p.132,where the result isobtained for IR . The extension to IR n
is straightforward.
3.1 The martingale problem for time homogeneous jumping
dif-fusions
We consider rst the time homogeneous case of the martingale problem and dene the
bounded post-jump measure Q : E ! P(E) to have the property Q(;B) 2 B(E) for all
B 2B(E).
Proposition 5. Let (E,d) be complete and separable and let grA
1
B(E)B(E), the
jump rate 2B(E), Q be a bounded jump measure and A
2
be given by
A
Suppose thatforevery2P(E) there existsasolution oftheC
E
[0;1)martingaleproblem
for (A
1
;). Then for every 2 P(E) there exists a solution of the D
E
[0;1) martingale
problem for (A
1 +A
Proposition 6. Let L be a family of functions which is dened on a Borel subset of IR n
and is closed under addition, scalar multiplication and weak-convergence. If L contains
all twice continuously dierentiable functions with compact support, then L contains all
bounded Borelfunctions and is separating.
Proof : See Dynkin [8], Lemma 5.12, p.160.
Proposition 6 enables us to make use of the following theorem of Ethier and Kurtz
to show the uniqueness of the solution to the martingale problem for jumping diusions
by setting the separating set L := B( IR
), the space of twice continuously
dierentiable functions on IR n
which vanish at innity.
Proposition 7. Let E be separable and let grA B(E)B(E) be linear and dissipative.
Suppose there exists a linear operator A 0
such that grA 0
and L is separating. Let 2 P(E) and suppose X is a solution of the
martin-gale problem for (A;). Then X is the Markov process corresponding to the semigroup on
L generated by the closure of A 0
and uniqueness holds for the solution of the martingale
problem for (A;).
Proof : See Ethier and Kurtz[11], Theorem 4.1, p.182.
We nowstate our rst essentially originaltheorem.
Theorem 1. Consider a diusion process generator A
1
) with locally bounded and Borel measurable drift coeÆcients b
i and
diusion coeÆcients a
ik
, anda jump process generator A
2
with jump rate 2B(IR n
A
). Then there exists a Markov processX which is the unique solution of the
martingale problem for (A=A
1 +A
2 ;).
Proof: In ordertondasolution tothemartingaleproblem usingProposition7wemust
have a dissipative operator, so we need to prove that A:=A
1 +A
2
is dissipative. Indeed
for x
0
:=argmaxf(x) we have that
k
both satisfy the positive maximum principle.
The proof now follows directlyfrom Propositions5, 6and 7.
So the martingale problem (A;) inthe time homogeneous case is well-posed.
Of course { analogous to the C
E
[0;1) case { the solution of the martingale problem
(A:=A
[0;1) meansthe (weak) solution of the jump SDE corresponding
to(;b;) with anadded jump term with innitesimal characteristics and Q.
Theorem 2. A Markov processX in IR n
isthe unique solution of the martingaleproblem
for (A := A
1 +A
2
;) of Theorem 1 if, and only if, X is the unique weak solution of the
time homogeneous form of the jump stochastic dierential equation (13) given in integral
form by
with initial condition X(0).
Remark: The last integral in(20) is ageneralized Itointegral with semimartingale
arbitrary stopping time <1 a.s.
Suppose rst thatX isthe correspondinguniquesolution ofthe local martingale
prob-lem for (A;) stopped at . Then applying Proposition 4 for the stopped version of the
semimartingale
sequence of stopping times %1 a:s: yields the result.
Conversely, if X is aweak solution of (20) withX(0),then using Proposition4 we
may conclude for f 2C
ThusthesemimartingaleincrementsofthelastgeneralizedItointegralhaveexpectation
0and hence the integralrepresents a0-meanmartingale process, as required. Applying
Proposition 6 yields the result for f 2 B(IR n
) and we may conclude from Proposition 7
Let us now consider processes whose parameters vary in time { the common situation in
nancial applications.
In general let grA B([0;1)E)B([0;1)E). Then a measurable E-valued
process X with X(0) is the solution of the martingale problem for (A;) i for each
(f;g)2A
isa0-meanmartingalewithrespecttotheltrationF X
t
. Mostofthebasicresults
concern-ing martingale problems can be extended to the time-dependent case by considering the
space-timeprocess X 0
(t):=(t;X(t)). Itfollows that inthe time-dependent case the linear
operator A
Specializingtothecase E := IR n
ofthe practicalinterest,thedriftanddiusion
coeÆ-cientsarisingfromtheSDE (8)nowalso dependonthe timet anddenethe second-order
dierential operator (10). Thisleads tothe following result.
Theorem 3. Considerthetime-dependentdiusionprocessgeneratorA
1
onB(IR n
)which
is the weak closure of
A
) with locally bounded and Borel measurable drift coeÆcients b
i and
diusion coeÆcients a
ik
and the jump process generator
A
2
f(t;x)=(t;x) Z
with jump rate 2 B([0;1) IR ) and bounded jump transition probability measure
Q:[0;1) IR n
!P( IR n
).
Then there exists a Markov process X in IR n
which is the unique solution of the
mar-tingale problem for (A=A
1 +A
2 ;).
Proof : The proof follows directly from Theorem 1 using the space-time process X 0
(t)
dened above.
Finally,westate without proof the straightforward generalization ofTheorem 2.
Theorem 4. A MarkovprocessXin IR n
istheunique solutionof themartingaleproblem
for (A := A
1 +A
2
;) of Theorem 3 if, and only if, X is the unique weak solution of the
jump stochasticdierential equation
dX(t)=b(t;X(t ))dt+(t;X(t ))dW(t)+X(t ) (27)
given in integral form by
X
t =
Z
t
0
b(s;X
s )ds+
Z
t
0
(s;X
s )dW
s +
Z
t
0 d(X
s X
c
s
) (28)
with initial condition X(0).
4 Application to Fixed Income Derivatives
As atypical applicationof the above results tocontingent claimsanalysis we consider the
BabbsandWebber[3]modelforxedincomederivativevaluationinthepresenceofcentral
bank rate changes. They modelled the term structure of interest rates as involving two
correlated state variables : the oÆcial short rate r and the market short rate or state of
the economy X.
We suppose given a probability space (;F;P) and the ltration F
t
generated by a
real-valued Wiener process W. The -algebra F
t
the authorities may possess greater information at time t than is embodied in F
t
. Trade
takes place ina xed nite time interval [0,T].
Ourversionofthemodelinvolvestwoprocesses: theoÆcial shortrate randadiusion
process X in IR n 1
. The vector process X represents the state of economy { including
the market short rate { which, together with the oÆcial short rate r, drives the market's
assessment of the likelihood of the authorities changing the level of r.
We have a savings account whose initial value iscontinuously compounded atinterest
rate r by the factor
P
0
(t)=exp Z
t
0
r(u)du
: (29)
Babbs and Webber characterise the spot (instantaneous) oÆcial interest rate r as a
pure jump process with jumps of xed sizes. At each point in time the probability of a
jump ofsize c
j
occurringinthe next innitesimal timeintervalÆtisgiven by ageneralized
Poisson process N
j
with state-dependent intensity
j :=
j
(X) as
j
Æt+0(Æt).
The process r satises
dr= J
X
j=1 c
j dN
j
; (30)
wherec
1 ;c
2
;:::;c
J
representthejump sizesandN
j
isapointprocesscountingthe number
of jumps of size c
j
since time zero and possessing apredictable intensity
j .
Bounds are needed for both r and the
j
's. Namely, there exists a lower bound r
l ower
and an upper bound r
upper
such that each
j
vanishes when r+c
j 62 [r
l ower ;r
upper ] and
there exists a constant k such that
P[ sup
0tT
max
j=1;2;:::;J
j
tobedrivenbothbyitsown levelrandthestate oftheeconomyX {including themarket
rate { and the joint evolution of r and X is assumed to be Markovian. The dynamics of
X are given by
dX(t)=b(t;r(t );X(t ))dt+(t;r(t );X(t ))dW(t); (31)
whereWisvectorstandard Brownianmotionin IR n 1
and b(;;)and(;;)are locally
bounded and Borel measurable. The evolution of r is given by a pure jump stochastic
dierential equation
dr(t)=r(t ) (32)
withsuitable innitesimal characteristics (t;r;x)and Q(t;r;q;x)allowingarbitraryjump
saltae.
A unique weak solution of the vector jump stochastic dierential equations (31)-(32)
exists.
Theorem 5. Let b and be locally bounded and Borel measurable and let 2 P(IR n
).
Thenthereisauniqueweaksolutionofthevectorjumpstochasticdierentialequation
(31)-(32) corresponding to (;b;;Q;) if, and only if, there exists a unique solution of the
D
IR n
[0;1) martingale problem for (A;), where A isgiven by the weak closure of
A 0
f(t;r;x) =
@f(t;r;x)
@t +
1
2 n
X
i=1 n
X
k=1 a
ik
(t;r;x) @
2
f(t;r;x)
@x
i @x
k +
n
X
i=1 b
i
(t;r;x)
@f(t;r;x)
@x
i
+ (t;r;x) Z
(f(t;q;x) f(t;r;x))Q(t;r;q;x): (33)
Proof : This result is a directapplication of Theorem 4.
In particular the stochastic dierential equation (31) has a weak solution X in the
probability space (C
IR
n[0;1);B(C
IR
system [3] exists.
Proof : Forthe Babbs-Webber model the jump rate is given by
(t;r(t );X(t )):= J
X
j=1
j
(t;r(t );X(t ) (34)
and the discrete jump transition measure isgiven by
j
(t;r(t );X(t )
(t;r(t );X(t ))
for r(t)=c
j
+r(t ); j =1;::: ;n; (35)
and 0 otherwise.
A yield curve can be represented in termsof the prices of zero coupon(pure discount)
bonds of all maturities 2 (0;T]. Babbs and Webber [3] represent the price of a pure
discount bond in terms a partial dierential-dierence equation { a special case of an
integro-partial dierential equation { which the values of the bond must satisfy. We give
here a more generalresult { rst forthe case n:=2 forcomparison with [3].
Let B[t;;r;X] to be the price of a pure discount bond at time t maturing at time
2(0;T].
Theorem 6. For X a scalar process the pure discount bond prices B satisfy the
integro-partial dierential equation
@B[t ;;r;x]
@t
+ b
(t;r;x)
@B[t ;;r;x]
@x
+ 1
2
2
(t;r;x) @
2
B[t ;;r;x]
@x 2
+ (t;r;x) Z
IR
fB[t;;q;x] B[t ;;r;x]gQ
(t ;r;dq;x)
= B[t ;;r;x]r(t ) (36)
where b
:=b
0
and Q
is the risk-adjusted post-jump measure under any appropriate
equivalent martingalemeasure ~
applied to the present value f(t;r;x) := e
is the market price of risk and that under Q
the original oÆcial short
rate process r is replaced by r
t
)ds, a martingale. ~
P
could be taken to be the minimal equivalent martingale measure in the sense of [11] for
the incomplete marketcaused by the unpredictability of oÆcial short rate jumps. In this
case under ~
P the process r 2
hr;ri,centred by itsquadratic variation compensator hr;ri
[10, 20], is a martingale and expected squared losses due to oÆcial short rate jumps are
minimized.
Extending Theorem 6 to the vector process X for arbitrary n, equation (36) must be
replaced by
where wenow consider the pure jump process r to be the rst coordinate of X.
Finally,wenote that this generalization remains valid whenapplied toan appropriate
versionofthe generalvectorjump stochastic dierential equation(27)or(28)inthe
state-mentofTheorem4. Thisallowsanyofthelastn 1coordinatesofXtobediusions,pure
jump processes or jumping diusions with arbitrarytime and state dependent correlation
In this paperwe have given a rigorous pathto the derivation of the integro-PDE satised
by the value of contingent claimsbased onsemimartingale { i.e. jumpingdiusion {price
orrateprocesses. Ourapproachthroughtheuniquesolution oftheStroock-Varadhan
mar-tingaleproblemisequivalenttouniqueweaksolutionof thecorrespondingjumpstochastic
dierentialequationwhichexpresses thenancialanalysts'intuition. Applicationstoclaim
valuation andhedging are numerous forunderlyings whichinclude jumpingequity indices,
oÆcial short rates, credit spreads, catastrophe risks, etc. Note that the results we have
presented { unlikethose inthe previous literature [24, 15,9, 18,21]{allowthe correlated
nancialvariablesunderlyingthemodelstobeanarbitrarycombination ofdiusions,pure
jump processesand jumpingdiusionswithtime andstate varyingcorrelation. Ina
forth-comingpaper we will present some computational results for such processes.
Acknowledgements
This work is based in part on the second author's doctoral dissertation [13 ] written under the
supervision of therst author while both were membersof theInstitute for Studies inFinance
and theDepartment of Mathematics intheUniversity ofEssex, Colchester, England.
References
[1] Aase, K.K.(1988). Contingent claims valuation when securityprice is a combination of an
Ito process and a randompoint process.Stoch. Proc.Appl. 28185-220.
[2] Ahn, C.M.and H.E. Thompson(1988). Jumpdiusionprocesses andthe termstructureof
interest rates. J.of Finance 43155{174.
[3] Babbs, S.H. and N.J. Webber (1994). A theory of theterm structure with an oÆcial short
rate. Unpublished Working Paper, Financial Options Research Center, University of W
ar-wick.
[4] Babbs, S.H. andN.J. Webber(1995). Whenwesay jump:::.Risk 8 49-53.
[5] Baxter, M. and A. Rennie (1997). Financial Calculus. Cambridge University Press,
New York.
[7] DuÆe, D.and K. Singleton (1997).Modelingtermstructures of defaultable bonds.
Unpub-lished Working Paper, Graduate Schoolof Business,StanfordUniversity.
[8] Dynkin, E.B.(1965). Markov Processes I.Springer-Verlag,Berlin.
[9] El-Jahel, L.,H. Lindbergand W. Perraudin.Interest ratedistributions,yield curve
modeli-ing and monetarypolicy. Mathematicsof Derivative Securities. M.A.H. Dempsterand S.R.
Pliska,eds. CambridgeUniversity Press,423{453.
[10] Elliott, R.J.(1982). Stochastic Calculus and Applications. Applications ofMathematics 18,
Springer-Verlag,New York.
[11] Ethier,S.andT.Kurtz(1986).Markov Processes{CharacterizationandConvergence.Wiley,
New York.
[12] Follmer, H.and M. Schweizer (1991). Hedgingof contingent claimsunderincomplete
infor-mation.AppliedStochasticAnalysis.M.H.A.DavisandR.J.Elliott,eds.GordonandBreach,
New York,389{414.
[13] Gotsis, G. (1996). Pricing risk for interest rate derivatives. PhD. Thesis, Department of
Mathematics, Universityof Essex, April1996.
[14] Ito, K. (1950). Stochastic dierential equations in a dierential manifold. Nagoya Math. J.
1 35{47.
[15] Jarrow, R.A.andD.Madan(1995).Optionpricingusingthetermstructureofinterestrates
to hedge systematicdiscontinuities inassetreturns.Mathematical Finance 5 311{336.
[16] Karatzas, I. and S. Shreve (1979). Brownian Motion and Stochastic Calculus.
Springer-Verlag, Berlin.
[17] Levy,P.(1948). Processus Stochastiques etMovement Brownien. Gauthier-Villars, Paris.
[18] Merton,R.C. (1976).Optionpricing whenunderlying stockreturnsarediscontinuous. J.of
Financial Economics 3 125{144.
[19] Picque, M. and M. Pontier (1970). Optimal portfolio fora small investor in a market with
discontinuousprices.Appl. Math. Optimization 22287-310.
[20] Rogers,L.C.G.andD.Williams(1987).Diusions,MarkovProcessesandMartingales.Vol.2:
processes.Mathematical Finance 177{94.
[22] Stroock,D.W.andS.R.S.Varadhan.(1969).DiusionProcesseswithcontinuouscoeÆcients
I &II. Comm. Pure&Appl. Math.2 345-400 &479-530.
[23] Stroock,D.W.andS.R.S.Varadhan.(1979).MultidimensionalDiusionProcesses.
Springer-Verlag, Berlin
[24] Zhang, X.L. (1997). Valuation of American options in a jump-diusion model. Numerical
