As in Subsection 3.2 letX=(Xt)0≤t≤1be a semimartingale, (Ft) and (Gt) twofinite utility filtrations such thatFt ⊂ Gt, t ∈ [0,1], and letàbe the information drift of (Gt) relative to (Ft). As before we assume that there exist countably generatedfiltrations (Ft0) and (G0t) such that (Ft) and (Gt) are obtained as the the smallest respective extensions satisfying the usual conditions.
Let the decomposition ofXwith respect to (Ft) be given byX=M+ηã M,M, whereM =(Mt)0≤t≤1is an (Ft)−local martingale. To simplify the analyis we will assume throughout this section thatMhas the predictable representation property (PRP) with respect to (Ft). We aim at showing that the utility differenceuG(x)−uF(x) can be interpreted as a conditional entropy of the enlargedfiltration (Gt) with respect to (Ft). If we did not
assume (PRP), then we would obtain that the additional utility is only bounded by this entropy (see [4]).
Recall that the entropy of a measure à relative to a measure ν on a σ-algebraSis defined by
HS(àν)= logdà
dνS dP,ifàνonSand the integral exists,
∞, else.
We first fix a time s ∈ [0,1] and try to measure the entropy of the information contained in G0s relative to thefiltration (Fu0), conditional to theσ−algebraFs. To this end we introduce an auxiliaryfiltration obtained by enlarging (Fu) withG0sat times,
Ku=
F!u if 0≤u<s
r>uFr∨ G0s,ifu∈[s,1]
and we denote by às the information drift of (Ku) relative to M. The conditional entropyof theσ−algebraG0s relative to thefilration (Fu0) on the time interval [s,t],t∈(s,1], will be defined by
H(s,t)=
HG0s(Pt(ω,ã)Ps(ω,ã))dP(ω).
We will now show that 2H(s,t) is equal to the square-integral ofàsonΩì [s,t]. To this end let (Pm)m≥0be an increasing sequence offinite partitions such thatσ(Pm:m≥0)=G0s. Then
H(s,t)=
HG0s(Pt(ω,ã)Ps(ω,ã))dP(ω)
=E
A∈Pm
1AlogPs(ã,A)−1AlogPt(ã,A)
=E
A∈Pm
"
− t
s
ku
Pu(ã,A) 1AdM˜u− t
s
ku
Pu(ã,A) 1AàsudM,Mu
+1 2
t
s
ku
Pu
2
(ã,A) 1AdM,Mu
#,
where the last equation follows from (20). Since ˜Mis a local martingale, we obtain by stopping and taking limits if necessary
H(s,t)=E
A∈Pm
⎡⎢⎢⎢⎢
⎣ t
s
ku
Pu(ã,A) 1AàudM,Mu−1 2
t s
ku
Pu
2
(ã,A) 1AdM,Mu
⎤⎥⎥⎥⎥
⎦.
Lemma 3.1 implies that$
A∈Pm
k
u
Pu
2(ω,A) 1A(ω) is anL2(Pu(ω,ã))-bounded martingale forPM−a.a. (ω,u), and therefore, by Theorem 3.1
limm E t
s
A∈Pm
ku
Pu
2
(ã,A) 1AdM,Mu=E t
s (àsu)2dM,Mu. Similarly we have
limm E t
s
A∈Pm
ku
Pu(ã,A) 1AàsudM,Mu=E t
s (àsu)2dM,Mu. and hence
H(s,t)= 1 2E
t
s (àsu)2dM,Mu. (22)
We are now in a position to introduce a notion of conditional entropy between ourfiltrations (G0t) and (Ft0). For any partition∆: 0 =t0 ≤ t1 ≤ . . .≤tk=1 we will use the abbreviations$
∆=$k
i=1and%
∆=%k
i=1
Definition 4.1. Let (∆n) be a sequence of partitions of [0,1] with mesh|∆n| converging to 0 asn→ ∞. The limit of the sums$
∆nH(ti−1,ti) asn→ ∞ is calledconditional entropyof (G0t) relative to (Ft0) and will be denoted by HG0|F0.
Theorem 4.1. The conditional entropyHG0|F0is well defined and it satisfies HG0|F0 = 1
2E 1
0 à2udM,Mu.
Proof. Let (∆n) be a sequence of partitions of [0,1] with mesh|∆|converging to 0 asn→ ∞. For all∆nwe define auxiliaryfiltrations
Dnt =&
s>t
(Fs0∨ G0ti) ift∈[ti,ti+1[.
Since all (Dnt) are subfitrations of (G0t), the respective information driftsàn ofMexist. It follows immediately from Eq. (22) that
∆n
H(ti−1,ti)= 1 2E
t s
(ànu)2dM,Mu.
As it is shown in Theorem 4.4 in [4], the information driftsànconverge in L2(M) to the information driftà. Consequently, the conditional entropy of (G0t) relative to (Ft0) is well defined and equals 12E1
0 à2uM,Mu.
The conditional entropyHG0|F0can be interpreted as a multiplicative inte- gral along thefiltration (G0t). More precisely, if for anys≤t≤1 we define d(s,t, ω, ω)= PPts(ω,ã)(ω,ã)G0
s
(ω), and if∆is a partition of [0,1], then
∆
H(ti−1,ti)=
∆
log Pti(ω,ã) Pti−1(ω,ã)
G0 ti−1
(ω)Pti(ω,dω)
dP(ω)
=
∆
logd(ti−1,ti, ω, ω)dP(ω)
=
log'
∆
d(ti−1,ti, ω, ω)dP(ω)
In the special case where (G0t) is obtained by aninitialenlargment with a random variableG, we have PPst(ω,ã)(ω,ã)G0
s
= PPts(ω,ã)(ω,ã)σ(G)and hence HG0|F0= logP1(ω,dω)
P(dω)
σ(G)(ω)P1(ω,dω)
dP(ω)
=HF1⊗σ(G)(P1(ω,dω)P(dω)P⊗P).
The image of the measure P1(ω,dω)P(dω) under the mapping (ω, ω) → (M(ω),G(ω)) is the joint distribution ofM = (Mt)0≤t≤1 and G. Conse- quently, in the initial enlargement case,HG0|F0 is equal to the entropy of the joint distribution ofMandGrelative to the product of the respective distributions, which is also known as the mutual informationbetweenM and G. To sum up, we obtain a very simple formula for the additional logarithmic utility under initial enlargements.
Theorem 4.2. Let G be a random variable and Gt = !
s>tFs∨σ(G). Then uG(x)−uF(x)coincides with themutual informationbetween M and G.
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A Localization of the L´evy Operators Arising in Mathematical Finances
Mariko Arisawa
GSIS, Tohoku University, Aramaki 09, Aoba-ku, Sendai 980-8579, Japan
1. Introduction
We study the uniform H¨older continuity of the solutions of the following problem.
F(x,∇v(x),∇2v(x))−
RN
[v(x+z)−v(x)
−1|z|<1∇v(x),z]c(z)dz−g(x)=0 x∈RN, (1)
wherec(z)dzis a positive Radon measure, called L´evy density, defined on RNsuch that
RNmin(|z|2,1)c(z)dz<C1, (2)
C2
|z|1+γ<|c(z)|< C3
|z|1+γ ∀z∈RN∩ {|z|<1}, (3)
whereγ∈(0,2),Ci>0 (1<i<3) are constants. We assume that there exists a ”uniform” constantM>1 such that for a constantθ0∈[0,1],
|g(x)−g(y)|<M|x−y|θ0 ∀x,y∈RN, (4)
and
sup
x∈RN
|v|<M.
(5)
The second-order fully nonlinear partial differential operatorFis continu- ous inRN×RN×SN, and assumed to satisfy the following two conditions.
(Degenerate ellipticity) :
F(x,p,X)≥F(x,p,Y) if X<Y,
23
∀x∈RN, ∀p∈RN, ∀X,Y∈SN. (6)
(Continuity I) : There are modulus of continuity functionswandηfrom R+∪ {0} →R+∪ {0}such that limσ↓0w(σ)=0, limσ↓0η(σ)=0, and
|F(x,p,X)−F(y,p,X)|<w(|x−y|)|p|q+η(|x−y|)||X||
(7)
∀x,y∈RN, ∀p∈RN, ∀X∈SN, whereq≥1.
We study this problem in the framework of the viscosity solutions for the integro-differential equations, the definition of which is introduced in Arisawa [5] (see also [6] and [7]). The definition is the following. In order to get rid of the singularity of the L´evy measure, we shall use the following superjet (resp. subjet) and its residue. Let ˆx∈ RN, and let (p,X)∈ J2R,+Nu( ˆx) (resp. (p,X)∈ J2R,−Nu( ˆx)) be a second-order superjet (resp. subjet) ofu at ˆx.
Then, for anyδ >0 there existsε >0 such that u( ˆx+z)<u( ˆx)+p,z+1
2Xz,z+δ|z|2 if |z|<ε (8)
(resp.
v( ˆx+z)≥v( ˆx)+p,z+1
2Xz,z −δ|z|2 if |z|<ε (9)
) holds. We use this pair of numbers (ε, δ) satisfying (8) (resp. (9)) for any (p,X) ∈ J2,+RNu( ˆx) (resp. (p,X) ∈ JR2,−Nv( ˆx)) in the following definition of viscosity solutions.
Definition 1.1. Letu∈USC(RN) (resp.v∈LSC(RN)). We say thatu(resp.v) is a viscosity subsolution (resp. supersolution) of (1), if for any ˆx∈RN, any (p,X)∈ J2,+RNu( ˆx) (resp.∈ J2,−RNv( ˆx)), and any pair of numbers (ε, δ) satisfying (8) (resp. (9)), the following holds for any 0< ε<ε
F( ˆx,p,X)−
|z|<ε
1
2(X+2δI)z,zc(z)dz
−
|z|≥ε
[u( ˆx+z)−u( ˆx)−1|z|<1z,p]c(z)dz<0.
(resp.
F( ˆx,p,X)−
|z|<ε
1
2(X−2δI)z,zc(z)dz
−
|z|≥ε[v( ˆx+z)−v( ˆx)−1|z|<1z,p]c(z)dz≥0.
Ifuis both a viscosity subsolution and a viscosity supersolution , it is called a viscosity solution.
In the framework of the viscosity solutions in Definition 1.1, we have the existence and the comparison results in [5], [6] and [7]. For the convenience of the readres, we shall give typical comparison results and the proof in§2 in below.
Then, we claim the uniform H¨older continuity ofuin the following two cases.
(I)N=1.
(II)N≥2, andFsatisfies the following uniform ellipticity.
(Uniform ellipticity) : There existsλ0>0 such that
F(x,p,X)−F(x,p,Y)≥λ0(Y−X) if X<Y,
∀x∈RN, ∀p∈RN, ∀X,Y∈SN. (10)
In the case of (I), we claim that for anyθ∈ (0,min{1, θ0+γ}) there exists Cθ >0 such that
|v(x)−v(y)|<Cθ|x−y|θ ∀x,y∈RN, (11)
whereCθ >0 depends only onMandC1. (See Theorem 3.1 in below.) In the case of (II), we claim that for any θ ∈ (0,1), there existsCθ >0 such that (11) holds. (See Theorem 3.2 in below.) (These results hold for more general problem
F(x,∇v(x),∇2v(x))+sup
α∈A{−
RN
[v(x+z)−v(x)
−1|z|<1∇v(x),z]c(x,z, α)dz−g(x, α)}=0 x∈RN, which we do not treat here.)
As for the case other than (I) and (II), that isN≥2 andFis not necessarily uniformly elliptic (i.e. (10) is not satisfied), we study the following two problems in the torusTNinstead of (1). Thefirst one is, forλ >0,
λv(x)+H(∇v(x))−
RN
[v(x+z)−v(x)
−1|z|<1∇v(x),z]c(z)dz−g(x)=0 x∈TN. (12)
And the second one is
λv(x)+F(x,∇v(x),∇2v(x))−
RN
[v(x+z)−v(x)
−1|z|<1∇v(x),z]c(z)dz−g(x)=0 x∈TN, (13)
whereλ > 0. HereHis afirst-order nonlinear operator, andFis a fully nonlinear degenerate elliptic operator, satisfying the following conditions.
(Periodicity) :
H(ã,p), F(ã,p,X), and g(ã) are periodic in x∈TN, for ∀p∈RN, ∀X∈SN. (14)
(Partial uniform ellipticity) : There exists a constantλ1>0 such that F(x,p,X)≥F(x,p,Y)+λ1Tr(Y−X) ∀x∈TN, ∀p∈RN,
∀X,Y∈SN, X=
X X12
X21X22
, Y=
Y Y12
Y21Y22
, (15)
where X<Y(X,Y∈SM), 0<M<N.
(Continuity II) : There are modulus of continuity functionsw andη fromR+∪ {0} →R+∪ {0}such that limσ↓0w(σ)=0, limσ↓0η(σ)=0, and
|F(x,p,X)−F(y,p,X)|<w(|x−y|)|p|q+η(|x−y|)||X||
∀x,y∈TN, ∀p=(p,p)∈RM×Rm, ∀X=
X X12
X21 X22
∈SN, where X∈SM, M+m=N, q≥1.
(16)
Roughly speaking, we claim that for anyθ∈ (0, θ0) (θ0 > 0), there exists Cθ>0 such that
|v(x)−v(y)|<Cθ
λ |x−y|θ ∀x,y∈TN, (17)
whereCθ>0 is independent onλ >0. (See Theorems 4.1 and 4.2 in below.) The method to derive the above uniform H¨older continuity (11) and the H¨older continuity (17) is based on the argument used in the proof of the comparison result. (See Ishii and Lions [21], for the similar argument in the PDE case.)
Next, we shall state the strong maximum principle for the L´evy op- erator. In [18], for the second-order uniformly elliptic integro-differential operator
− N i,j=1
aij ∂2v
∂xi∂xj − N
i=1
bi∂v
∂xi−
RN
[v(x+z)−v(x)− ∇v(x),z]c(x,z)dz x∈RN, (18)
the strong maximum principle was given, where λ0I<(aij)1<i,j<N<Λ0I (0 < λ0<Λ0). See also, Cancelier [13] for another type of the maximum principle. Here, we shall give the strong maximum principle inRNwith- out assuming the uniform ellipticity of the partial differential operatorFin (1) (see Theorem 5.1 in below, and M. Arisawa and P.-L. Lions [9]).
Finally, we shall apply these regularity results (11), (17) and the strong maximum principle, to study the so-called ergodic problem. In the case of the Hamilton-Jacobi-Bellman (HJB) operator
sup
α∈A{−
N i,j=1
aij(x, α) ∂2u
∂xi∂xj − N
i=1
bi(x, α)∂u
∂xi− f(x, α)},
the ergodicity of the corresponding controlled diffusion process, for exam- ple in the torusTN=RN\ZN, can be studied by the existence of a unique real numberdfsuch that the following problem admits a periodic viscosity solutionu:
df +sup
α∈A{−
N i,j=1
aij(x, α) ∂2u
∂xi∂xj − N
i=1
bi(x, α)∂u
∂xi −f(x, α)}=0 x∈TN. We refer the readers to M. Arisawa and P.-L. Lions [8], M. Arisawa [2], [3], for more details. From the analogy of the diffusion case, here we shall formulate the ergodic problem for the integro-differential equations as follows.
(Ergodic problem) Is there a unique number df depending only on f(x) such that the following problem has a periodic viscosity solutionu(x) defined onTN?
df+F(x,∇u,∇2u)−
RN
[u(x+z)
−u(x)−1|z|<1∇u(x),z]c(z)dz− f(x)=0 x∈TN.
The results on the existence of the above numberdf is stated in Theorem 6.1 in below.