The Zakai equation of nonlinear filtering for jump-diffusion observation: existence and uniqueness

0.37 MB·English

by  Claudia Ceci

The Zakai equation of nonlinear filtering for jump-diffusion
observation: existence and uniqueness
Claudia Ceci ∗ Katia Colaneri †
October 17, 2012
2
1 Abstract
0
2 This paper is concerned with the nonlinear filtering problem for a general Markovian partially
observed system (X,Y), whose dynamics is modeled by correlated jump-diffusions having common
t
c jump times. At any time t ∈ [0,T], the σ-algebra FtY := σ{Ys : s ≤ t} provides all the available
O
information about the signal Xt. The central goal of stochastic filtering is to characterize the filter,
Y
6 πt, which is the conditional distribution of Xt, given the observed data Ft . It has been proved in [7]
1 thatπ isthe unique probabilitymeasure-valuedprocesssatisfyinga nonlinearstochasticequation,the
so-calledKushner-Stratonovichequation (KS-equation). In this paper the aim is to describe the filter
]
R π in terms of the unnormalized filter ̺, which is solution to a linear stochastic differential equation,
the so-called Zakai equation. We prove equivalence between strong uniqueness for the solution to
P
the Kushner Stratonovich equation and strong uniqueness for the solution to the Zakai one and, as a
.
h
consequence, we deduce pathwise uniqueness for the solutions to the Zakai equation by applying the
t
a Filtered Martingale Problem approach ([25, 7]). To conclude, some particular cases are discussed.
m
[ Keywords: Filtering; Jump-Diffusion Processes

Amsprimary: 93E11; 60J75; 60J60
1
v
9
1 Introduction
7
2
4
The objective of stochastic filtering is to find the best estimate, in some sense, of the state process X of
.
0 a stochastic dynamical system (X,Y), from a partial observation described by a process Y. This subject
1
found several applications throughout the years, which includes a great variety of engineering problems,
2
1 the study of the global climate, the estimation of the economy condition, the identification of tumours
: using digital recording.
v
i
X The literature concerning nonlinear filtering is quite rich; within the first results we mention [31] and
r [27]. We can distinguish three main scenarios based on different dynamics of the observation process.
a
In [24, 29, 25] the process Y gives observation of X in additional Gaussian noise, in [4, 28, 15, 9, 10, 5]
it is studied the case of counting process or marked point process observations and more recently the
case of mixed type observations (marked point processes and diffusions) has been taken into account in
[16,17,6,18,7]. Wewanttofocusonthislastcasewhichisthesameconsidered inthisnote. In[16,17,6]
theinformationflowhasthestructureFm∨Fη,wherem(dt,dz)isamarkedpointprocesswhosedynamics
t t
is influenced by a stochastic factor X and η gives observations of X in additional Gaussian noise. These
particular structures of the observation have a financial motivation; nevertheless in a general framework,
∗Department of Economics, University of Chieti-Pescara, Viale Pindaro 42, I-65127 Pescara, Italy Email:
[email protected].
†Department of Economics, University of Chieti-Pescara, Viale Pindaro 42, I-65127 Pescara, Italy Email:
[email protected] .
1
it may be meaningful to consider the case where the observation flow is generated by a jump-diffusion
process asin themodeldeveloped in [7]and alsointhis paper. For this model, in[7], thefiltering problem
has been studied using the innovation approach and the filter π (defined by π (f) := E[f(t,X )|FY], for
t t t
some suitable functions f) has been characterized as the unique solution to the Kushner-Stratonovich
equation (or KS-equation). The equation found is not only infinite dimensional but also non linear, and
it has a complicated structure that makes it is not suitable for computation. On the other hand, the
problem can be faced from a different point on view, that of the Zakai equation for the unnormalized
filter. This equation still remains infinite dimensional, but it has the advantage to be linear, that is the
reason why it is well suited to be analyzed under numerical approximations, such as, for example the
Galerkin method (see [18], [19] and [20]) or the optimal quantization approach [22]. In particular, in
[18], the Galerkin method is applied to a mixed type observation, given by the pair (η,N) where N is
a compensated Poisson process with unobservable intensity and η gives observations of X in additional
Gaussian noise. e e
In this paper we want to study the filtering problem from this second point of view of the Zakai equa-
tion. To be more precise, we will look for an appropriate Girsanov change of probability measure on
(Ω,FTY,P|FTY), where T is some fixed time horizon, which leads to an equivalent probability measure
dP
P , defined by 0 = Z , such that the associated unnormalized filter, ̺ (dx) := Z−1π (dx), solves
0 dP (cid:12)FTY T t t t
a linear equation of(cid:12) Zakai’s type. Moreover, under additional hypotheses the new probabilty measure
(cid:12)
P defined on (Ω,F(cid:12)Y) coincides with the restriction on FY of a probability measure equivalent to P on
0 T T
F , which allows us to establish the analogy between our approach and the classical one based on the
T
Kallianpur-Striebel formula.
In order to represent the filter in terms of the unnormalized one, we need to prove uniqueness of the
solutions to the Zakai equation. We show the equivalence between strong uniqueness for the KS-equation
and strong uniqueness for the Zakai equation. Then, we deduce pathwise uniqueness for the solutions
to the Zakai equation by pathwise uniqueness results for the solutions to the KS-equation proved in [7]
by applying the Filtered Martingale Problem approach. This method was introduced in [25] and then
generalized in[26]. Inbothofthepapersitisappliedtoprovestronguniquenessforbothoftheequations,
KS and Zakai, in frameworks of signals observed in Gaussian white noise. Here, we extend the uniqueness
result for the Zakai equation in the more general case of jump-diffusion observations. To the authors’
knowledge this is the first time that the dynamics of the unnormalized filter is computed in the case of a
partially observed system (X,Y), where the signal X and the observation Y are described by correlated
jump diffusion processes having common jump times.
The paper is organized as follows. The filtering model is described in Section 2. In Section 3 we derive
the Zakai equation of the nonlinear filtering problem. Technical difficulties introduced by working with
real valued random counting measures instead of counting processes brought us to make the assumption
that there exists a transition function η(t,y,dz) such that the FY-predictable measure η(t,Y ,dz) is
t t−
equivalent to FY-dual predictable projection of the random counting measure m(dt,dz), associated with
t
the jumps of the process Y, π (λφ(dz)), where π denotes the left version of the filter. In Section 4
t− t−
we discuss uniqueness for the solutions of the Zakai equation. In Section 5, we conclude by giving some
examples where pathwise uniqueness for the solutions to the Zakai equation is fulfilled. In particular we
analyze three models. In thefirst the observation process is given by ajump diffusion with jump sizes in a
finite set; inthe second one we consider thecase where theobservation dynamics is driven by independent
point processes with unobservable intensities; in the last one we assume that the state process X is pure
jump process taking values in a countable space. For the first two examples we compute explicitly the
measure η(t,Y ,dz) which ensure us that existence and uniqueness for the Zakai equation hold. Instead
t−
in the third one we derive directly, by a recursive procedure, uniqueness for the solution to the Zakai.
2
2 The partially observed model and preliminary results
Throughout the paper, we consider a partially observed system (X,Y), on a complete filtered probability
space(Ω,{F } ,P), whereT > 0issomefixedtimehorizon. Thedynamics ofthesystemisdescribed
t t∈[0,T]
by the following pair of stochastic differential equations
dX = b (t,X )dt+σ (t,X )dW0+ K (t,X ;ζ)N(dt,dζ); X = x ∈ R
t 0 t 0 t t 0 t− 0 0
 ZZ (2.1)
 dYt = b1(t,Xt,Yt)dt+σ1(t,Yt)dWt1+ K1(t,Xt−,Yt−;ζ)N(dt,dζ); Y0 = y0 ∈ R
ZZ
whereW0 and W1 are two correlated Brownian motions with correlation coefficient ρ ∈ [−1,1] and

N(dt,dζ) is a Poisson random measure on R+ ×Z whose intensity ν(dζ)dt is a σ−finite measure on a
measurable space (Z,Z).
In this model, X represents a signal, also called the state process, which cannot be directly observed and
theprocessY,describedbyacorrelatedprocesshavingcommonjumptimeswithX,givestheobservation.
The coefficients b (t,x), b (t,x,y), σ (t,x) > 0,σ (t,y) > 0, K (t,x;ζ) and K (t,x,y;ζ) are R-valued
0 1 0 1 0 1
measurable functions of their arguments. As in [7] we assume strong existence and uniqueness for the
solutions of the system (2.1). Sufficient conditions are summarized by Assumption C.1 in Appendix C.
At any time t the σ- algebra FY := σ{Y : s ≤ t} provides all the available information about the signal
t s
X . Our aim is to characterize the conditional distribution of X given FY, that represents the most
t t t
detailed description of our knowledge of X .
t
In order to describes the jump component of Y we introduce the integer-valued random measure
m(dt,dz)= δ (dt,dz) (2.2)
{s,∆Ys}
s:∆XYs6=0
where δ denotes the Dirac measure at point a. Note that the following equality holds
a
t t
z m(ds,dz)= K (s,ζ)N(ds,dζ) (2.3)
1
Z0 ZR Z0 ZZ
and, in general, for any measurable function g : R → R
t t
g(z) m(ds,dz)= 1I g(K (s,ζ))N(ds,dζ). (2.4)
{K1(s,ζ)6=0} 1
Z0 ZR Z0 ZZ
For all t ∈[0,T], for all A ∈ B(R), we define
d0(t,x) := {ζ ∈Z : K (t,x;ζ) 6= 0}, d1(t,x,y) := {ζ ∈ Z : K (t,x,y;ζ) 6= 0},
0 1
dA(t,x,y) := {ζ ∈ Z : K (t,x,y;ζ) ∈ Ar{0}} ⊆ d1(t,x,y), (2.5)
1
and, finally,
DA = dA(t,X ,Y )⊆ D = d1(t,X ,Y ), D0 = d0(t,X ). (2.6)
t t− t− t t− t− t t−
Normally D0 ∩D 6= ∅ P −a.s. and this models the fact that state process and observation may have
t t
common jump times.
In the sequel we will write b (t),σ (t),K (t,ζ), i = 0,1, for b (t,X ),b (t,X ,Y ), σ (t,X ),σ (t,Y ),
i i i 0 t 1 t t 0 t 1 t
K (t,X ;ζ) and K (t,X ,Y ;ζ) respectively and we will assume the following requirements
0 t− 1 t− t−
3
Assumption 2.1.
T T T
E |K (t,ζ)|ν(dζ)dt <∞, E |b (t)|dt < ∞, E σ2(t)dt < ∞ i= 0,1;
i i i
Z0 ZZ Z0 Z0
T
E ν(D0∪D )dt < ∞. (2.7)
t t
Z0
Note that under these constraints the pair (X,Y) is a Markov process and both of the processes X and
Y have finite first moment.
As proved in Proposition 2.2 of [5], the (P,F )-dual predictable projection, mp(dt,dz), of m(dt,dz) (see
t
[23, 4] for the definition), can be written as
mp(dt,dz) = λ φ (dz)dt, (2.8)
t t
where φ (dz) is a probability measure over (R,B(R)) and ∀A∈ B(R)
t
mp(dt,A) = λ φ (A)dt = ν(DA)dt. (2.9)
t t t
Define the functions λ(t,x,y) := ν(d1(t,x,y)) and φ(t,x,y,dz) := δ (dz)ν(dζ), then the
d1(t,x,y) K1(t,x,y;ζ)
(P,F )-local characteristics of the integer valued counting measure m(dt,dz), given by
t R
(λ ,φ (dz)) = (λ(t,X ,Y ),φ(t,X ,Y ,dz)), (2.10)
t t t− t− t− t−
depend on the state process, and therefore they are not directly observable. In particular, ∀A ∈ B(R),
λ φ (A) =ν(DA)isthe(P,F )−intensity ofthepointprocessN (A) = m((0,t]×A)thatcountsthejumps
t t t t t
of the process Y until time t whose widths belong to A and λ = ν(D ) provides the (P,F )-predictable
t t t
intensity of the point process N = m((0,t]×R) which counts the total number of jumps of Y until t.
t
Similarly, define the function λ0(t,x) := ν(d0(t,x)), the process λ0 := λ0(t,X ) = ν(D0) furnishes the
t t− t
(P,F )-predictable intensity of the point process N0 which counts the total number of jumps of X until
t t
time t. Condition (2.7) imply that the processes N and N0 are both non-explosive and integrable ([4]).
Let us introduce the filter defined as
π (f):= E[f(t,X )|FY] (2.11)
t t t
for any measurable function f(t,x) such that E|f(t,X )| < ∞ ∀t ∈ [0,T]. It is known that π is a
t
probability measure-valued process with càdlàg trajectories (see [25]). We denote by π his left version.
t−
In particular, for all functions F(t,x,y) such that E|F(t,X ,Y )| < ∞ (resp. E|F(t,X ,Y )| < ∞)
t t t− t−
∀t∈ [0,T], we will use the notation
π (F) := π (F(t,·,Y )) resp. π (F) := π (F(t,·,Y )) .
t t t t− t− t−
(cid:16) (cid:17)
Remark 2.2. We recall that for any F -progressively measurable process ψ, satisfying the inequality
t
E T |ψ |dt < ∞, the process E[ T ψ dt|FY] − T π (ψ)dt is a (P,FY)-martingale. In particular, this
0 t 0 t t 0 t t
implies that
R R R
T T
E |π (ψ)|dt = E |ψ |dt < ∞. (2.12)
t t
Z0 Z0
4
Denote by νp(dt,dz) the (P,FY)-predictable projection of the integer-valued measure m(dt,dz); the
t
following proposition, proved in [5], gives a representation of νp(dt,dz) in terms of the filter.
Proposition 2.3. The (P,FY)-predictable projection of the integer-valued measure m(dt,dz) is given by
t
νp(dt,dz) = π (λφ(dz))dt, (2.13)
t−
that is, for any A∈ B(R)
t t
νp((0,t]×A) = π (λφ(A))ds = π ν(dA(.,Y )) ds. (2.14)
s− s− s−
Z0 Z0
(cid:0) (cid:1)
The measure
mπ(dt,dz)= m(dt,dz)−π (λφ(dz))dt (2.15)
t−
iscalledtheFY-compensatedmartingalemeasureandhasthepropertythatforallFY-predictableprocess
t t
indexed by z, H(t,z) satisfying
T T
E H(t,z)π (λφ(dz))dt < ∞ resp. H(t,z)π (λφ(dz))dt < ∞ P −a.s. ,
t− t−
(cid:20)Z0 ZR (cid:21) (cid:18) Z0 ZR (cid:19)
t
the process H(s,z)mπ(ds,dz) is a (P,FY)-martingale (resp. local martingale).
t
Z0 ZR
Finally, assume that
T b (t) T b
E 1 dt < ∞ and E π2 1 dt < ∞, (2.16)
σ (t) t σ
Z0 (cid:12) 1 (cid:12) Z0 (cid:12) 1(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
then we can define the so called innovation process I, which, in our framework, is given by
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
t b (s) b
I := W1+ 1 −π 1 ds. (2.17)
t t σ (s) s σ
Z0 (cid:26) 1 (cid:18) 1(cid:19)(cid:27)
It is not difficult to verify that the process I is a (P,FY)-Brownian motion.
t
Remark 2.4. Let us notice that, by Jensen’s inequality and (2.12), the condition
T b (t) 2
E 1 dt < ∞, (2.18)
σ (t)
Z0 (cid:12) 1 (cid:12)
(cid:12) (cid:12)
which is usually required in the classical approa(cid:12)ch, im(cid:12)plies (2.16).
(cid:12) (cid:12)
The process I and the FY-compensated martingale measure mπ, play a central role in describing the
t
dynamics of the filter. More precisely, in [7], under Assumption 2.1, (2.18) and assuming that the process
t b (s)
L = E − 1 dW1 (2.19)
t σ (s) s
(cid:26) Z0 1 (cid:27)
is a (P,F )-martingale (E denotes the Doléans-Dade exponential), it is proved that the filter is a solution
t
to the Kushner-Stratonovich equation driven by I and mπ. This result can be improved and the theorem
stated below gives the same thesis under weaker hypotheses. In particular, we replace condition (2.18)
with (2.16) and the assumption that L is a (P,F )-martingale with the hypothesis that
t
t b
1
L = E − π dI (2.20)
t s s
σ
(cid:26) Z0 (cid:18) 1(cid:19) (cid:27)
is a (P,FY)-martingale. b
t
5
Remark 2.5. Observe that if L is a (P,F )-martingale then L is a (P,FY)-martingale. In fact, if we
t t
define the probability measure Q equivalent to P over F , such that
0 T
b
e dQ
0
= L , (2.21)
T
dP
(cid:12)
e (cid:12)FT
(cid:12)
then by the Girsanov Theorem, the process (cid:12)
(cid:12)
t b (s)
W1 := W1+ 1 ds (2.22)
t t σ (s)
Z0 1
f
is a (Q ,F )-Wiener process. Taking into account (2.17), we get
0 t
e W1 = I + tπ b1 ds, (2.23)
t t s σ
Z0 (cid:18) 1(cid:19)
f
which implies that W1 is a (Q ,FY)-Wiener process. Again by the Girsanov Theorem we deduce that
0 t
f e dQ
L = 0 = E[L |FY], (2.24)
t dP t t
(cid:12)
e (cid:12)FtY
(cid:12)
b
(cid:12)
then L is a (P,FY)-martingale. (cid:12)
t
Theorem 2.6 (The Kushner-Stratonovich equation). Assume that Assumption 2.1 and (2.16) hold and
b
thatLdefined in(2.20)isa(P,FY)-martingale, then thefilter π solves thefollowingKushner-Stratonovich
t
equation, that is, ∀f ∈ C1.2([0,T]×R)
b
t t t
π (f)= f(0,x )+ π (LXf)ds+ wπ(f,z)mπ(ds,dz)+ hπ(f)dI (2.25)
t 0 s s s s
Z0 Z0 ZR Z0
where
dπ (λφf) dπ (Lf)
wπ(f,z) = t− (z)−π (f)+ t− (z), (2.26)
t dπ (λφ) t− dπ (λφ)
t− t−
∂f
hπ(f) = σ−1(t)[π (b f)−π (b )π (f)]+ρπ σ . (2.27)
t 1 t 1 t 1 t t 0∂x
(cid:18) (cid:19)
Hereby dπt−(λφf)(z)and dπt−(Lf)(z)wemeantheRadon-Nikodymderivativesofthemeasures π (λfφ(dz))
dπt−(λφ) dπt−(λφ) t−
and π (Lf)(dz), with respect to π (λφ(dz)). The operator L¯ f defined by L¯ f(dz):= L¯f(.,Y ,dz) and
t− t− t t t−
∀A∈B(R) L¯f(t,x,y,A) := [f(t,x+K (t,x;ζ))−f(t,x)]ν(dζ) (2.28)
0
ZdA(t,x,y)
takes into account common jump times between the signal X and the observation Y.
Finally, the operator LX given by
∂f ∂f 1 ∂2f
LXf(t,x) = +b (t,x) + σ2(t,x) + {f(t,x+K (t,x;ζ))−f(t,x)}ν(dζ).
∂t 0 ∂x 2 0 ∂x2 0
ZZ
denotes the generator of the Markov process X.
6
Proof. The proof is similar to that of Theorem 3.2 in [7]. We only need to observe that the (P,FY)-
t
martingale representation Theorem in terms of I and mπ, proved in Proposition 2.6 in [7], still holds true
even if we replace the condition (2.18) with (2.16) and the assumption that the process L, defined by
(2.19), is a (P,F )-martingale with the hypothesis that L given in (2.20) is a (P,FY)-martingale. In fact,
t t
it is sufficient introduce the probability measure Q on (Ω,FY), equivalent to the restriction of P over
0 T
FY, defined as b
T
dQ
0
= L . (2.29)
T
dP (cid:12)FTY
(cid:12)
BytheGirsanov Theorem I + tπ b1 dsisa(Q(cid:12)(cid:12) ,FY)b-Wiener processand, taking intoaccount (2.17),
t 0 s σ1 0 t
we obtain that I + tπ b1 Rds=(cid:16)W1(cid:17)+ t b1(s)ds= W1.
t 0 s σ1 t 0 σ1(s) t
We write Fm for the filtra(cid:16)tion(cid:17)generated by the random counting measure m(dt,dz), then, since
t R R
f
dY = z m(dt,dz)+σ (t,Y )dW1,
t 1 t t
R
Z
as in Proposition 2.6 in [7], we can deduce that FY = Fm∨FW1 anfd that every (P,FY)-local martingale
t t tf t
M admits the following decomposition
t
t t
M = M + w(s,z)mπ(ds,dz)+ h(s)dI , (2.30)
t 0 s
Z0 ZR Z0
where w(t,z) is an FY-predictable process and h(t) is an FY-adapted process such that
t t
T T
|w(t,z)|π (λφ(dz))dt < ∞, h(t)2dt < ∞ P −a.s.
t−
Z0 ZR Z0
Finally, as in Theorem 3.2 in [7], by applying the innovation method, we can conclude that the filter π
solves the equation (2.25).
Let us observe that the KS-equation is an infinite-dimensional and nonlinear stochastic differential equa-
tion and so, in general, it is difficult to handle. Then it can be useful to characterize the filter in terms of
a simpler equation. For doing so we will determine a probability measure P over (Ω,FY), equivalent to
0 T
the restriction of P onto FY, defined by
T
dP
0
= Z , (2.31)
t
dP (cid:12)FtY
(cid:12)
where Z is a suitable strictly positive (P,FY)-mar(cid:12)tingale, chosen in such a way that the so-called unnor-
t (cid:12)
malized filter ̺, defined by
̺ (dx):= Z−1π (dx) (2.32)
t t t
satisfies a linear stochastic differential equation, the Zakai equation.
Remark 2.7. Note that, if the measure P is the restriction of a probability measure P equivalent to P
0 0
over the whole filtration F then the unnormalized filter can be written as
T
e
̺t(f):= EPe0 f(t,Xt)Zt−1|FtY ,
h i
dP e
0
where Z := . This follows from the well known Kallianpur-Striebel formula
t
dP
(cid:12)
e (cid:12)Ft
(cid:12)
e
(cid:12)
(cid:12) 7
EP0 f(t,X )Z−1|FY
e t t t
π (f) = , (2.33)
t EhP0 Z−1|FY i
e t e t
h i
since EP0 Z−1|FY = Z−1. In order to derive the Zakaei equation under mild conditions we do not require
e t t t
the existenhce of sucih a probability measure P defined on (Ω,F ), as in the classical reference probability
0 T
e
method, but we will work directly with the probability measure P defined on (Ω,FY).
0 T
e
The first step is to mention a complete version of the Girsanov Theorem to be applied on the model
considered in this note.
2.1 Girsanov change of probability
Theorem 2.8. Let ϕ(t) and ψ(t,z) be two processes FY-adapted and FY-predictable respectively such
t t
that
T T
|ϕ(t)|2dt < ∞, |ψ(t,z)|π (λφ(dz))dt < ∞ P −a.s. (2.34)
t
Z0 Z0 ZR
1+ ψ(t,z)m({t},dz) > 0 P −a.s. ∀t ∈ [0,T]. (2.35)
R
Z
Define the process L as
dL = L ϕ(t)dI + ψ(t,z)(m(dt,dz)−π (λφ(dz))dt) . (2.36)
t t− t t−
R
(cid:20) Z (cid:21)
L is a (P,FY)-strictly positive local martingale. If more
t
E[L ]= 1, (2.37)
T
L is a strictly positive (P,FY)-martingale.
t
Then, under (2.37), there exists a probability measure Q defined on (Ω,FY), equivalent to the restriction
T
of P over FY, such that
T
dQ
= L ,
t
dP(cid:12)FtY
(cid:12)
and (cid:12)
(cid:12)
t
(i) the process WQ := I − ϕ(s)ds is a (Q,FY)-Brownian motion
t t t
Z0
(ii) the (Q,FY)-dual predictable projection of the integer-valued measure m(dt,dz) is
t
νQ(dz)dt = (1+ψ(t,z))π (λφ(dz))dt.
t−
Itmay beusefultoinvestigateonwhetherthecondition (2.37)issatisfied, thatis,underwhichhypotheses
L is a strictly positive martingale. For the diffusive case the Novikov criterium provides a sufficient
condition, for the most general case, there exists a similar criterium less known in literature, due to
Protter and Shimbo (Theorem 9 in [30]), that we mention below.
8
Theorem 2.9. Let M be a locally square integrable martingale such that ∆M > −1. If
1
E exp hMc,Mci +hMd,Mdi < ∞, (2.38)
T T
2
(cid:20) (cid:26) (cid:27)(cid:21)
where Mc and Md are the continuous and the purely discontinuous martingale parts of M, then E(M) is
a martingale on [0,T], where T can be ∞.
The following corollary translates the theorem we have just stated in our setting.
Corollary 2.10. Let ϕ(t) and ψ(t,x) be two processes FY-adapted and FY-predictable respectively, sa-
t t
tisfying (2.35). Assume that
E eR0T{12|ϕ(t)|2+RR|ψ(t,z)|2πt(λφ(dz))}dt < ∞, (2.39)
h i
then the process L is a (P,FY)-martingale on [0,T].
t
In the sequel we refer to (2.39) as the Protter-Shimbo condition.
Proof. Let M be given by
dM = ϕ(t)dI + ψ(t,z)[m(dt,dz)−π (λφ(dz))dt]. (2.40)
t t t−
R
Z
T
If (2.35) and (2.39) hold true, then |ϕ(t)|2dt< ∞ and T |ψ(t,z)|2π (λφ(dz))dt < ∞ P −a.s. and M
0 t
Z0
is a (P,FY)-locally square integrable martingale such thatR∆M > −1 P −a.s. ∀t ∈ [0,T], having sharp
t
brackets
T T
hMc,Mci = |ϕ(t)|2dt and hMd,Mdi = |ψ(t,z)|2π (λφ(dz))dt.
T T t
Z0 Z0 ZR
Then the hypothesis (2.39) translates exactly (2.38) in the jump-diffusion case. Therefore, by Theorem
2.9 we get the claimed result.
Remark 2.11. By applying the Cauchy-Schwarz inequality, we are able to split the assumption (2.39)
in two separated sufficient conditions on the continuous part and on the purely discontinuous part of the
martingale M written in (2.40). Indeed, since
1 1
E eR0T(21|ϕ(t)|2+RR|ψ(t,z)|2πt(λφ(dz)))dt ≤ E eR0T|ϕ(t)|2dt 2 E e2R0TRR|ψ(t,z)|2πt(λφ(dz))dt 2 ,
h i h i h i
it can be clearly deduced that, if
E eR0T|ϕ(t)|2dt < ∞, E e2R0TRR|ψ(t,z)|2πt(λφ(dz))dt < ∞
h i h i
then (2.39) is fulfilled.
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3 The Zakai equation
We assume that Assumption 2.1 and (2.16) hold, and, in order to perform a suitable Girsanov change
measure on (Ω,FY), we make the following additional hypothesis.
T
Assumption 3.1. Assume that there exists a transition function η(t,y,dz) such that the FY-predictable
t
measure η(t,Y ,dz) is equivalent to π (λφ(dz)) and
t− t−
T
E η(t,Y ,R)dt < ∞. (3.1)
t−
(cid:20)Z0 (cid:21)
This means that there exists an FY-predicatble process Ψ(t,z) such that
t
π (λφ(dz))dt = (1+Ψ(t,z))η(t,Y ,dz) and 1+Ψ(t,z) > 0 P −a.s. (3.2)
t− t−
Now, we want to introduce a probability measure denoted by P , defined on (Ω,FY), which is equivalent
0 T
to the restriction of P over FY, given by
T
dP
0 = Z = Z0Z1 (3.3)
dP (cid:12)FtY t t t
(cid:12)
(cid:12)
where the processes Z0 and Z1 are described b(cid:12)y the following dynamics
t b
Z0 =E − π 1 dI , (3.4)
t s σ s
(cid:26) Z0 (cid:18) 1(cid:19) (cid:27)
t 1
Z1 =E −1 (m(dt,dz)−π (λφ(dz))dt) . (3.5)
t (cid:18)Z0 ZR(cid:26)1+Ψ(t,z) (cid:27) t− (cid:19)
As usual E denotes the Doléans-Dade exponential. Let us observe that Z is a strictly positive (P,FY)-
t
local martingale, nevertheless, if we want to define the probability measure P via the equation (3.3) we
0
make the following requirement.
Assumption 3.2. Assume that Z is a (P,FY)-martingale.
t
1
Remark 3.3. Setting U(t,z) = − 1, by Theorem 2.9, a sufficient condition, which implies
1+Ψ(t,z)
that Assumption 3.2 is fulfilled, is given by
1 T b T
E exp π2 1 ds+ U2(s,z)π (λφ(dz))ds < ∞. (3.6)
(cid:20) (cid:26)2Z0 s (cid:18)σ1(cid:19) Z0 ZR s (cid:27)(cid:21)
Moreover, let us observe that if the following conditions hold
b
π 1 ≤ C , |U(t,z)| ≤ C P −a.s. ∀t∈ [0,T] ∀z ∈ R,
t 1 2
σ
(cid:12) (cid:18) 1(cid:19)(cid:12)
(cid:12) (cid:12) T
(cid:12) (cid:12)
(cid:12) (cid:12) πt(λ) dt ≤ C3 P −a.s.,
Z0
b (t)
1
for C ,i = 1,2,3 positive constants, then (3.6) is verified. Clearly, it is sufficient that the ratio as
i
σ (t)
1
well as the (P,F )-intensity, λ , of the point process N , and U(t,z) are P −a.s.-bounded processes to
t t t
make these conditions true.
10