Generalization of Doob decomposition Theorem

0.32 MB·

by  Nicholas Gonchar

ModernStochastics:Theory andApplications0(0000) 0–0
DOI:
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Generalization of Doob decomposition Theorem.
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R Nicholas Gonchara,∗
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aBogolyubov Institute for Theoretical Phisics of NAS, Kyiv, Ukraine
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t [email protected](N.Gonchar)
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1 Abstract Inthepaper,weintroducethenotionofalocalregularsupermartingale
v relative to a convex set of equivalent measures and prove for it an optional Doob
4 decomposition in the discrete case. This Theorem is a generalization of the famous
7 Doob decomposition onto the case of supermartingales relative to a convex set of
5 equivalent measures.
3
0 Keywords random process, convex set of equivalent measures, optional Doob
. decomposition, regular supermartingale, martingale.
1
0 2010 MSC 60G07, 60G42
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1
:
v 1 Introduction.
i
X
In the paper, we generalize Doob decomposition for supermartingales relative
r
a to one measure onto the case of supermartingales relative to a convex set of
equivalent measures. For supermartingales relative to one measure for contin-
uous time Doob’s result was generalized in papers [12, 13].
Atthebeginning,weprovetheauxiliarystatementsgivingsufficientcondi-
tions ofthe existence ofmaximalelementin amaximalchain,ofthe existence
ofnonzeronon-decreasingprocesssuchthatthe sumofasupermartingaleand
this process is again a supermartingale relative to a convex set of equivalent
measuresneeded for the main Theorems.In Theorem 2 we give sufficient con-
ditionsoftheexistenceoftheoptionalDoobdecompositionforthespecialcase
∗Corresponding author.
www.i-journals.org/vmsta
Preprint submitted to VTeX / Modern Stochastics: Theory and Applications
<January 15, 2016>
2 N. Gonchar
as the set of measures is generated by finite set of equivalent measures with
bounded as below and above the Radon - Nicodym derivatives. After that,
we introduce the notion of a regular supermartingale. Theorem 3 describes
regular supermartingales. In Theorem 4 we give the necessary and sufficient
conditionsofregularityofsupermartingales.Theorem5describesthestructure
ofnon-decreasingprocessforaregularsupermartingale.Thenweintroducethe
notionofa localregularsupermartingalerelativeto a convexsetofequivalent
measures. At last, we prove Theorem 6 asserting that if the optional decom-
positionfor asupermartingaleisvalid,thenit islocalregularone.Essentially,
Theorem6and7givethenecessaryandsufficientconditionsoflocalregularity
of supermartingale.
After that, we prove auxiliary statements nedeed for the description of
local regular supermartingales. Theorem 8 gives the necessary and sufficient
conditions for a special class of nonnegative supermartingales to be local reg-
ular ones. In Theorems 9 and 10 we describe a wide class of local regular
supermartingales.Onthe basisoftheseTheoremsweintroduceacertainclass
of local regular supermartingales and prove Theorem 11 giving the necessary
andsufficientconditionsfornonnegativeuniformlyintegrablesupermartingale
to belong to this class. Using the results obtained we give examples of con-
structionoflocalregularsupermartingales.At last,we provealsoTheorem12
giving possibility to construct local regular supermartingales.
The optional decomposition for supermartingales plays fundamental role
for risk assessment on incomplete markets [6], [7], [8], [9]. Considered in the
paperproblemisgeneralizationofcorrespondingonethatappearedinmathe-
maticalfinanceaboutoptionaldecompositionforsupermartingaleandwhichis
relatedwith constructionof superhedge strategy onincomplete financialmar-
kets.First,the optionaldecompositionforsupermartingaleswasopenedbyEl
Karoui N. and Quenez M. C. [2] for diffusion processes. After that, Kramkov
D. O. [11], [5] proved the optional decomposition for nonnegative bounded
supermartingales.FolmerH.andKabanovYu.M.[3],[4]provedanalogousre-
sult for an arbitrary supermartingale.Recently, BouchardB. and Nutz M. [1]
considered a class of discrete models and proved the necessary and sufficient
conditions for validity of optional decomposition. Our statement of the prob-
lem unlike the above-mentionedone andit is moregeneral:a supermartingale
relative to a convex set of equivalent measures is given and it is necessary to
findconditionsonthe supermartingaleandthe setofmeasuresunderthatop-
tionaldecompositionexists.Generalityofourstatementoftheproblemisthat
wedonotrequirethattheconsideredsetofmeasureswasgeneratedbyrandom
process that is a local martingale as it is done in the papers [1, 2, 11, 4] and
that is important for the proof of the optional decomposition in these papers.
2 Discrete case.
We assume that on a measurable space {Ω,F} a filtration F ⊂F ⊂
m m+1
F, m=0,∞,andafamilyofmeasuresM onF aregiven.Further,weassume
that F ={∅,Ω}. A random process ψ ={ψ }∞ is said to be adapted one
0 m m=0
A sample document 3
relative to the filtration {F }∞ if ψ is F measurable random value for
m m=0 m m
all m=0,∞.
Definition 1. An adapted random process f = {f }∞ is said to be a
m m=0
supermartingale relative to the filtration F , m = 0,∞, and the family of
m
measures M if EP|f |<∞, m=1,∞, P ∈M, and the inequalities
m
EP{f |F }≤f , 0≤k ≤m, m=1,∞, P ∈M, (1)
m k k
are valid.
We consider that the filtration F , m = 0,∞, is fixed. Further, for a su-
m
permartingalef weuseasdenotation{f ,F }∞ anddenotation{f }∞ .
m m m=0 m m=0
Bellow in a few theorems, we consider a convex set of equivalent measures
M satisfyingconditions:Radon–NicodymderivativeofanymeasureQ ∈M
1
with respect to any measure Q ∈M satisfies inequalities
2
dQ
0<l≤ 1 ≤L<∞, Q , Q ∈M, (2)
dQ 1 2
2
where real numbers l, L do not depend on Q , Q ∈M.
1 2
Theorem 1. Let {f ,F }∞ be a supermartingale concerning a convex set
m m m=0
of equivalent measures M satisfying conditions (2). If for a certain measure
P ∈ M there exist a natural number 1 ≤ m < ∞, and F measurable
1 0 m0−1
nonnegative random value ϕ , P (ϕ >0)>0, such that the inequality
m0 1 m0
f −EP1{f |F }≥ϕ ,
m0−1 m0 m0−1 m0
is valid, then
l
f −EQ{f |F }≥ ϕ , Q∈M ,
m0−1 m0 m0−1 1+L m0 ε¯0
where
M ={Q∈M, Q=(1−α)P +αP , 0≤α≤ε¯ , P ∈M}, P ∈M,
ε¯0 1 2 0 2 1
L
ε¯ = .
0 1+L
Proof. Let B ∈F and Q=(1−α)P +αP , P ∈M, 0<α<1. Then
m0−1 1 2 2
[f −EQ{f |F }]dQ=
m0−1 m0 m0−1
Z
B
EQ{[f −f ]|F }dQ=
m0−1 m0 m0−1
Z
B
[f −f ]dQ=
m0−1 m0
Z
B
4 N. Gonchar
(1−α) [f −f ]dP +
m0−1 m0 1
Z
B
α [f −f ]dP =
m0−1 m0 2
Z
B
(1−α) [f −EP1{f |F }]dP +
m0−1 m0 m0−1 1
Z
B
α [f −EP2{f |F }]dP ≥
m0−1 m0 m0−1 2
Z
B
(1−α) [f −EP1{f |F }]dP =
m0−1 m0 m0−1 1
Z
B
dP
(1−α) [f −EP1{f |F }] 1dQ≥
m0−1 m0 m0−1 dQ
Z
B
l
(1−α)l ϕ dQ≥(1−ε¯ )l ϕ dQ= ϕ dQ.
m0 0 m0 1+L m0
Z Z Z
B B B
Arbitrariness of B ∈F proves the needed inequality.
m0−1
Lemma1. Anysupermartingale{f ,F }∞ relativetoafamilyofmeasures
m m m=0
M for which there hold equalities EPf = f , m = 1,∞, P ∈M, is a
m 0
martingale with respect to this family of measures and the filtration F , m=
m
1,∞.
Proof. The proof of Lemma 1 see [10].
Remark 1. If the conditions of Lemma 1 are valid, then there hold equalities
EP{f |F }=f , 0≤k ≤m, m=1,∞, P ∈M. (3)
m k k
Let f = {f ,F }∞ be a supermartingale relative to a convex set of
m m m=0
equivalent measures M and the filtration F , m = 0,∞. And let G be a set
m
of adapted non-decreasing processes g ={g }∞ , g =0, such that f +g =
m m=0 0
{f +g }∞ is asupermartingaleconcerningthe familyofmeasuresM and
m m m=0
the filtration F , m=0,∞.
m
Introduce a partial ordering (cid:22) in the set of adapted non-decreasing pro-
cesses G.
A sample document 5
Definition2. Wesaythatanadaptednon-decreasingprocessg ={g1 }∞ ,
1 m m=0
g1 =0,g ∈G,doesnotexeedanadaptednon-decreasingprocessg ={g2 }∞ ,
0 1 2 m m=0
g2 =0, g ∈ G, if P(g2 −g1 ≥ 0) = 1, m = 1,∞. This partial ordering we
0 2 m m
denote by g (cid:22)g .
1 2
For every nonnegative adapted non-decreasing process g = {g }∞ ∈ G
m m=0
there exists limit lim g which we denote by g .
m ∞
m→∞
Lemma2. LetG˜ beamaximalchaininGandforacertainQ∈M supEQg =
1
g∈G˜
αQ < ∞. Then there exists a sequence gs = {gs }∞ ∈ G˜, s=1,2,..., such
m m=0
that
supEQg =supEQgs,
1 1
g∈G˜ s≥1
where
∞ EQg
EQg = m, g ∈G.
1 2m
m=0
X
Proof. Let 0 < ε < αQ, s = 1,∞, be a sequence of real numbers satisfying
s
conditions ε >ε , ε →0, as s→∞. Then there exists anelement gs ∈G˜
s s+1 s
such that αQ−ε < EQgs ≤ αQ, s = 1,∞. The sequence gs ∈ G˜, s = 1,∞,
s 1
satisfies Lemma 2 conditions.
Lemma3. Ifasupermartingale{f ,F }∞ relativetoaconvexsetofequiv-
m m m=0
alent measures M is such that
|f |≤ϕ, m=0,∞, EQϕ<T <∞, Q∈M, (4)
m
where a real number T does not depend on Q∈M, then every maximal chain
G˜ ⊆G contains a maximal element.
Proof. Let g ={g }∞ belong to G, then
m m=0
EQ(f +ϕ+g )≤f +T, m=1,∞, Q∈M.
m m 0
Then inequalities f +ϕ≥0, m=1,∞, yield
m
EQg ≤f +T, m=1,∞, {g }∞ ∈G.
m 0 m m=0
Introduce for a certain Q∈M an expectation for g ={g }∞ ∈G
m m=0
∞ EQg
EQg = m, g ∈G.
1 2m
m=0
X
Let G˜ ⊆G be a certain maximal chain. Therefore, we have inequality
supEQg =αQ ≤f +T <∞,
1 0 0
g∈G˜
6 N. Gonchar
where Q∈M and is fixed. Due to Lemma 2,
supEQg =supEQgs.
1 1
g∈G˜ s≥1
In consequence of the linear ordering of elements of G˜,
max gs =gs0(k), 1≤s (k)≤k,
0
1≤s≤k
where s (k) is one of elements of the set {1,2,...,k} on which the considered
0
maximum is reached, that is, 1≤s (k)≤k, and, moreover,
0
gs0(k) (cid:22)gs0(k+1).
It is evident that
max EQgs =EQgs0(k).
1 1
1≤s≤k
So, we obtain
supEQgs = lim max EQgs = lim EQgs0(k) =EQ lim gs0(k) =EQg0,
1 1 1 1 1
s≥1 k→∞1≤s≤k k→∞ k→∞
whereg0 = lim gs0(k),andthatthereexists,duetomonotonyofgs0(k).Thus,
k→∞
supEQgs =EQg0 =αQ.
1 1 0
s≥1
Show that g0 = {g0 }∞ is a maximal element in G˜. It is evident that g0
m m=0
belongs to G. For every element g ={g }∞ ∈G˜ two cases are possible:
m m=0
1) ∃k such that g (cid:22)gs0(k).
2) ∀k gs0(k) ≺g.
Inthefirstcaseg (cid:22)g0.Inthesecondonefrom2)wehaveg0 (cid:22)g.Atthesame
time
EQgs0(k) ≤EQg. (5)
1 1
By passing to the limit in (5), we obtain
EQg0 ≤EQg. (6)
1 1
The strict inequality in (6) is impossible, since EQg0 = supEQg. Therefore,
1 1
g∈G˜
EQg0 =EQg. (7)
1 1
The inequality g0 (cid:22)g and the equality (7) imply that g =g0.
A sample document 7
Let M be a convex set of equivalent probability measures on {Ω,F}. In-
troduce into M a metric |Q −Q |= sup|Q (A)−Q (A)|, Q , Q ∈M.
1 2 1 2 1 2
A∈F
Lemma 4. Let {f ,F }∞ be a supermartingale relative to a compact con-
m m m=0
vex set of equivalent measures M satisfying conditions (2). If for every set
of measures {P ,P ,...,P }, s < ∞, P ∈ M, i = 1,s, there exist a natural
1 2 s i
number1≤m <∞,anddependingonthissetofmeasuresF measurable
0 m0−1
nonnegative random variable ∆s , P (∆s >0)>0, satisfying conditions
m0 1 m0
f −EPi{f |F }≥∆s , i=1,s, (8)
m0−1 m0 m0−1 m0
then the set G of adapted non-decreasing processes g ={g }∞ , g =0, for
m m=0 0
which {f +g }∞ is a supermartingale relative to the set of measures M
m m m=0
contains nonzero element.
Proof. For any point P ∈M let us define a set of measures
0
MP0,ε¯0 ={Q∈M, Q=(1−α)P +αP, P ∈M, 0≤α≤ε¯ }, (9)
0 0
L
ε¯ = .
0 1+L
Prove that the set of measures MP0,ε¯0 contains some ball of a positive radius,
thatis,thereexistsarealnumberρ >0suchthatMP0,ε¯0 ⊇C(P ,ρ ), where
0 0 0
C(P ,ρ )={P ∈M, |P −P|<ρ }.
0 0 0 0
Let C(P ,ρ˜) = {P ∈ M, |P −P| < ρ˜} be an open ball in M with the
0 0
centeratthe pointP ofaradius0<ρ˜<1.Consideramapofthe setM into
0
itself given by the law: f(P)=(1−ε¯ )P +ε¯ P, P ∈M.
0 0 0
′ ′
The mapping f(P) maps an open ball C(P ,δ) = {P ∈ M,|P −P| < δ}
2 2
′
with the center at the point P of a radius δ > 0 into an open ball with the
2
′
center at the point (1−ε¯ )P +ε¯ P of the radius ε¯ δ, since |(1−ε¯ )P +
0 0 0 2 0 0 0
′ ′
ε¯ P −(1−ε¯ )P −ε¯ P|=ε¯ |P −P|<ε¯ δ. Therefore, an image of an open
0 2 0 0 0 0 2 0
set M ⊆M is an open set f(M )⊆M, thus f(P) is an open mapping. Since
0 0
f(P ) = P , then the image of the ball C(P ,ρ˜) = {P ∈ M, |P −P| < ρ˜}
0 0 0 0
is a ball C(P ,ε¯ ρ˜)= {P ∈M, |P −P|< ε¯ ρ˜} and it is contained in f(M).
0 0 0 0
Thus, inclusions MP0,ε¯0 ⊇ f(M)⊇C(P ,ε¯ ρ˜) are valid. Let us put ε¯ ρ˜=ρ .
0 0 0 0
Then we have MP0,ε¯0 ⊇ C(P ,ρ ), where C(P ,ρ ) = {P ∈ M, |P −P| <
0 0 0 0 0
ρ }. Consider an open covering C(P ,ρ ) of the compact set M. Due to
0 0 0
P0∈M
compactness of M, there exists a Sfinite subcovering
v
M = C(Pi,ρ ) (10)
0 0
i=1
[
withthecenteratthepointsPi ∈M, i=1,v,andacoveringbysetsMP0i,ε¯0 ⊇
0
C(Pi,ρ ), i=1,v,
0 0
v
M = MP0i,ε¯0. (11)
i=1
[
8 N. Gonchar
Consider the set of measures Pi ∈M, i=1,v. From Lemma 4 conditions,
0
there exist a natural number 1 ≤ m < ∞, and depending on the set of
0
measures Pi ∈ M, i = 1,v, F measurable nonnegative random variable
0 m0−1
∆v , P1(∆v >0)>0, such that
m0 0 m0
fm0−1−EP0i{fm0|Fm0−1}≥∆vm0, i=1,v. (12)
Due to Theorem 1, we have
l
f −EQ{f |F }≥ ∆v =ϕv , Q∈M. (13)
m0−1 m0 m0−1 1+L m0 m0
The last inequality imply
EQ{f |F }−EQ{f |F }≥EQ{ϕv |F }, Q∈M, s<m . (14)
m0−1 s m0 s m0 s 0
But EQ{f |F }≤f , s<m . Therefore,
m0−1 s s 0
f −EQ{f |F }≥EQ{ϕv |F }, Q∈M, s<m . (15)
s m0 s m0 s 0
Since
f −EQ{f |F }≥0, Q∈M, m≥m , (16)
m0 m m0 0
we have
EQ{f |F }−EQ{f |F }≥0, Q∈M, s<m , m≥m . (17)
m0 s m s 0 0
Adding (17) to (15), we obtain
f −EQ{f |F }≥EQ{ϕv |F }, Q∈M, s<m , m≥m , (18)
s m s m0 s 0 0
or
f −EQ{f |F }≥EQ{ϕv |F }χ (m)−ϕv χ (s), (19)
s m s m0 s [m0,∞) m0 [m0,∞)
Q∈M, s≤m , m≥m .
0 0
Introduce an adapted non-decreasing process
gm0 ={gm0}∞ , gm0 =ϕv χ (m),
m m=0 m m0 [m0,∞)
whereχ (m)isanindicatorfunctionoftheset[m ,∞).Then(19)implies
[m0,∞) 0
that
EQ{f +gm0|F }≤f +gm0, 0≤k ≤m, Q∈M.
m m k k k
A sample document 9
In the Theorem 2 a convex set of equivalent measures
n n
M ={Q, Q= α P , α ≥0, i=1,n, α =1} (20)
i i i i
i=1 i=1
X X
satisfies conditions
dP
i
0<l≤ ≤L<∞, i,j =1,n, (21)
dP
j
where l, L are real numbers.
DenotebyGthesetofalladaptednon-decreasingprocessesg ={g }∞ ,
m m=0
g =0, such that {f +g }∞ is a supermartingale relative to all measures
0 m m m=0
from M.
Theorem 2. Let a supermartingale {f ,F }∞ relative to the set of mea-
m m m=0
sures (20) satisfy the conditions (4), and let there exist a natural number 1≤
m < ∞, and F measurable nonnegative random value ϕn , P (ϕn >
0 m0−1 m0 1 m0
0)>0, such that
fm0−1−EPi{fm0|Fm0−1}≥ϕnm0, i=1,n. (22)
If for the maximal element g0 ={g0 }∞ in a certain maximal chain G˜ ⊆G
m m=0
the equalities
EPi(f +g0 )=f , P ∈M, i=1,n, (23)
∞ ∞ 0 i
are valid, where f = lim f , g0 = lim g0 , then there hold equalities
∞ m ∞ m
m→∞ m→∞
EP{f +g0 |F }=f +g0, 0≤k ≤m, m=1,∞, P ∈M. (24)
m m k k k
Proof. The set M is compact one in the introduced metric topology. From
the inequalities (22) and the formula
n
αiEP1{ϕi|Fm0−1}EPi{fm0|Fm0−1}
EQ{f |F }= i=1 , Q∈M, (25)
m0 m0−1 P n
αiEP1{ϕi|Fm0−1}
i=1
P
where ϕi = ddPP1i, we obtain
f −EQ{f |F }≥ϕn , Q∈M. (26)
m0−1 m0 m0−1 m0
The inequalities (21) lead to inequalities
1 dQ
≤ ≤nL, P,Q∈M. (27)
nL dP
Inequalities (26) and (27) imply that conditions of Lemma 4 are satisfied for
any set of measuresQ ,...,Q ∈M. Hence, it follows that the set G contains
1 s
10 N. Gonchar
nonzero element. Let G˜ ⊆ G be a maximal chain in G satisfying condition of
Theorem 2. Denote by g0 ={g0 }∞ , g0 = 0, a maximal element in G˜ ⊆ G.
m m=0 0
Theorem 2 and Lemma 3 yield that as {f }∞ and {g0 }∞ are uniformly
m m=0 m m=0
integrable relative to each measure from M. There exist therefore limits
lim f =f , lim g0 =g0
m ∞ m ∞
m→∞ m→∞
with probability 1. Due to Theorem 2 condition, in this maximal chain
EPi(f∞+g∞0 )=f0, Pi ∈M, i=1,n.
Since{f +g0 }∞ isasupermartingaleconcerningallmeasuresfromM,we
m m m=0
have
EPi(fm+gm0 )≤EPi(fk+gk0)≤f0, k <m, m=1,∞, Pi ∈M. (28)
By passing to the limit in (28), as m→∞, we obtain
f0 =EPi(f∞+g∞0 )≤EPi(fk+gk0)≤f0, k =1,∞, Pi ∈M. (29)
So, EPi(fk +gk0) = f0, k = 1,∞, Pi ∈ M, i = 1,n. Taking into account
Remark 1 we have
EPi{fm+gm0 |Fk}=fk+gk0, 0≤k ≤m, m=1,∞, Pi ∈M, i=1,n. (30)
Hence,
EP{f +g0 |F }=
m m k
n
αiEP1{ϕi|Fk}EPi{fm+gm0 |Fk}
i=1 =f +g0, 0≤k ≤m, P ∈M, (31)
P n k k
αiEP1{ϕi|Fk}
i=1
P
where ϕi = ddPP1i, i=1,n. Theorem 2 is proven.
LetM beaconvexsetofequivalentmeasures.Bellow,G isasetofadapted
s
non-decreasing processes {g }∞ , g = 0, for which {f + g }∞ is a
m m=0 0 m m m=0
supermartingale relative to all measures from
s s
Mˆ ={Q,Q= γ Pˆ, γ ≥0, i=1,s, γ =1}, (32)
s i i i i
i=1 i=1
X X
where Pˆ ,...,Pˆ ∈M and satisfy conditions
1 s
dPˆ
i
0<l ≤ ≤L<∞, i,j =1,s, (33)
dPˆ
j
l,L are real numbers depending on the set of measures Pˆ ,...,Pˆ ∈M.
1 s