Enlargements of filtrations and applications

0.23 MB·English

by  J. M. Corcuera

Enlargements of Filtrations and Applications
José Manuel Corcuera∗† Arturo Valdivia‡§
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0 January 30, 2012
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Abstract
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Inthispaperwereviewsomeoldandnewresultsabouttheenlargementoffiltrationsproblem,aswell

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astheirapplicationstocreditriskandinsidertradingproblems. Theenlargementoffiltrationsproblem
R
P consists in the study of conditions under which a semimartingale remains a semimartingale when the
.
h
filtration is enlarged, and, in such a case, how to find the Doob-Meyer decomposition. Filtrations may
t
a
m be enlarged in different ways. In this paper we consider initial and progressive filtration enlargements
[ madeby random variables and processes.

1 Keywords: Credit Risk,Insider Trading, Enlargement of Filtrations
v
0 Mathematics Subject Classification (2010): 60-02, 60G44, 62B10, 91-02, 91G40
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.
1 1 Enlargement of filtrations
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2
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: When considering a given filtration F = ( t)t 0, accounting for the information related to a given phe-
v F ≥
i nomenon, the arrival of new information induces the consideration of an enlarged filtration G = ( ) ,
X t t 0
G ≥
r such that , for each t 0. More specifically, one considers the filtration G := ( ) defined as
a Gt ⊇ Ft ≥ Gt t≥0
:= , t 0, where H=( ) is assumed to represent the new information. Traditionally,
t t t t t t 0
G F ∨H ⊇F ≥ H ≥
whenHissuchthat =σ(L),fort 0,andforsomerandomvariableL,thenitissaidthatGisaninitial
t
H ≥
enlargement of the filtration F. Otherwise,it is saidthat G is a progressive enlargement of the filtration F.
Some natural questions arise in a filtration enlargement setting. For instance, In which cases an F-
semimartingale remains a semimartingale in the enlarged filtration G?, and How can we compute G-Doob-
Meyerdecompositions infunctionofFandH? Accordingto(1),thebeginningsofthetheoryofenlargement
∗UniversitatdeBarcelona,GranViadelesCortsCatalanes,585,E-08007Barcelona,Spain. E-mail: [email protected]
†TheworkofJ.M.Corcuera issupported bytheNILSGrant.
‡Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, E-08007 Barcelona, Spain. E-mail:
[email protected]
§Corresponding author.
1
offiltrationsmaybe tracedbacktoK.Itô,P.A.Mayer,andD.Williams whointhe lateseventies,indepen-
dentlyandseparatelyfromeachother,posedsimilarquestions. Sofar,thestudyofenlargementoffiltrations
has been devoted mainly to cases as ( =σ(τ)) and ( =1 ) , where τ is some stopping time.
t t 0 t τ t t 0
H ≥ H { ≤ } ≥
One of the drawbacks of the present approaches is that some rather restrictive or unrealistic assumptions
has to be made on the stopping time in order to apply the approach. For instance, in credit risk theory,
τ usually represents the default time of some contract. In order to preclude arbitrage, τ is assumed to be
1
eitherinitial time orhonest time . Ontheotherhand,onlyafewstudieshasbeendevelopedinthegeneral
setting. For instance, it can be considered the case2 when H = ( = σ(J )) , for (J = inf X ) ,
t t t 0 t s t s t 0
H ≥ ≥ ≥
being (X ) a 3-dimensional Bessel process. Nevertheless, this case can be reduced in fact to a case with
t t 0
≥
random times, taking into account that
J <a = t<Λ ,
t a
{ } { }
where Λ =sup t, X =a .
a t
{ }
In this paper we approach the problem of initial enlargement of filtrations by considering a general -
F
measurable random variable L (as opposed to a stopping time) and looking into its law conditioned to the
-not necessarilyBrownian- filtrationF. Regardingto the first question, we give a condition under which F-
semimartingalesremainsemimartingalesunderthe enlargedfiltrationG. Regardingtothesecondquestion,
we study conditions under which it is possible to obtain G-Doob-Meyer decompositions as well as explicit
expressions for the compensantor. For progressive enlargement of filtrations, we present the example that
motivated this paper and which represents an enlargementdone by an H that is not induced by a stopping
time. Finally, we present applications of the enlargement of filtration theory in the field of mathematical
finance, specifically in credit risk theory and insider trading.
1.1 Initial enlargement of filtrations
Consider a stochastic basis (Ω, ,F,P), where the filtration F is assumed to satisfy the usual conditions.
F
LetLbean -measurablerandomvariablewithvaluesin(R, (R)). LetT >0denoteatime horizon,and
F B
define := ( σ(L)) and G=( ) . Notice that defining as ( σ(L)) assures
t T s>t s t t [0,T] t T s>t s
G ∩ ≥ F ∨ G ∈ G ∩ ≥ F ∨
the right-continuity of the filtration G, and therefore, that G satisfy the usual conditions.
Condition A. For all t, there exists a σ-finite measure η in (R, (R)) such that Q (ω, ) η where
t t t
B · ≪
1Seesection2.1formoredetails.
2SeeSection1.2.2in(1).
2
Q (ω,dx) is a regular version of L .
t t
|F
Notice that Condition A is satisfied in the case that L takes a countable number of values and the case
when L is independent of , just taking η the law ofL. An example where Condition Ais notsatisfied
t
F∞
is the following example:
Example 1 Let L be the n-th jump of a Poisson process (N ) with intensity λ and F the filtration
t t [0,T]
∈
generated by (N ) , then
t t [0,T]
∈
P L>x =1 +1 ∞ λe−λu(λu)n−Nt−1du,
{ |Ft} {Nx<n,Nt≥n} {Nt<n}ˆ(x−t)+ (n−Nt−1)!
then the conditional probability cannot be dominated by a non random measure.
Theorem 2 Under Condition A any X, F-semimartingale is a G-semimartingale.
Proof. The proof is based in the characterization of semimartingales of Bichteler-Dellacherie-Mokobodzki:
Let X be a càdlàg process, adapted to a filtration H define the class of predictable processes
n 1
ξ := f = − f 1 ,0=s <...s t, f H , f <ρ
ρ,t ( i (si,si+1] 0 n ≤ i ∈ si− | i| )
i=1
X
define
t n 1
−
f dX := f (X X )
ˆ s s i si+1 − si
0 i=1
X
and, for Z L1
∈
t
αX (Z,H)= sup E Z 1 f dX ,
ρ,t | | ∧ˆ s s
f∈ξρ,t (cid:20) (cid:18) 0 (cid:19)(cid:21)
then,
X S(H) limαX (1,H)=0
∈ ⇐⇒ρ 0 ρ,t
→
X S(H)= limαX (Z,H)=0.
∈ ⇒ρ 0 ρ,t
→
This result together with the fact that any f, -measurable function, can be written as f =g(ω)L(ω), where
i
G
g(ω)x is (R) -measurable, allows to get the result.
i
B ⊗F
Proposition 3 Condition A is equivalent to Q (ω,dx) η(dx) where η is the law of L.
t
≪
Proof. By Condition A we have that Q (ω,dx) = qx(ω)η (dx), where qx(ω) is (R) -measurable
t t t t B ⊗Ft
then we can write Q (ω,dx)=qˆx(ω)η(dx) with qˆx(ω)= qtx(ω) .
t t t E(qx(ω))
t
3
Proposition 4 Under Condition A there exists qx(ω) (R) -measurable such that
t B ⊗Ft
Q (ω,dx)=qx(ω)η(dx) (1)
t t
and, for fixed x, (qx) is an F-martingale.
t t [0,T]
∈
Proof. See (2) Lemma 1.8.
Theorem 5 Let (M ) be a continuous local F-martingale and kx(ω) such that
t t [0,T] t
∈
t
qx,M = kxqx d M,M
h it ˆ0 s s− h is
then
M ·kLd M,M ,
·−ˆ0 s h is
is a G-martingale.
Proof. Except for a localization procedure (see details in (2) Theorem 2.1) the proof is the following: let
s<t, Z , and g be abounded Borel function, then
s
∈F
E[Zg(L)(M M )] = E[E[Zg(L)(M M ) ]]
t s t s t
− − |F
= E[Z(M M )E[g(L) ]]
t s t
− |F
= g(x)η(dx)E[Z(M M )qx]
ˆR t− s t
= g(x)η(dx)E[Z(M qx M qx)]
ˆR t t − s s
= g(x)η(dx)E[Z( M,qx M,qx )]
ˆR h it−h is
t
= g(x)η(dx)E Z kxqx d M,M
ˆR (cid:20) (cid:18)ˆs u u− h iu(cid:19)(cid:21)
t
= E Zg(L) kxqx d M,M
(cid:20) (cid:18)ˆs u u− h iu(cid:19)(cid:21)
Example 6 Let T = 1, and take (M := B , t [0,T]) where (B ) is a standard Brownian motion,
t t t t [0,T]
∈ ∈
and take L:=B . It follows easily from (1) that
T
1 1 x2
qx(ω)∼ exp (B (ω) x)2+ .
t (T t)1/2 −2(T t) t − 2
− (cid:18) − (cid:19)
4
Applying Itô’s formula we get
x B
dqx =qx − tdB ,
t t T t t
−
hence ktx = xT−Btt and
− B B
B · T − sds
·−ˆ0 T −s
is a G := FB ∨ σ(BT) martingale. Note that, by the Lévy theorem, B· − ´0· BTT−−Bssds is a (standard)
G-Brownian motion and, since B is -measurable, it is independent of (W ) .
T 0 t t [0,T]
G ∈
Example 7 Note that if the filtration F is that generated by a Brownian motion, (B ) , then for any
t t [0,T]
∈
F-martingale (M ) we have
t t [0,T]
∈
dM =σ dB
t t t
and so
d M,M =σ2dt.
h it t
Also, assuming that
qx(ω)=hx(B )
t t t
and h C1,2 we will have that
∈
dqx =∂hx(B )dB ,
t t t t
and
∂loghx(B )
kx = t t .
t σ
t
Example 8 The previous example can be generalized as follows: let (Y ) be the following Brownian
t t [0,T]
∈
semimartingale
t t
Y =Y + σ(Y )dB + b(Y )ds,
t 0 ˆ s s ˆ s
0 0
and assume that
Y ∼π(T t,Y ,x)dx,
T t t
|F −
with π smooth. We know that (π(T t,Y ,x)) is an F-martingale, then
− t t
∂π
dπ(T t,Y ,x)= (T t,Y ,x)σ(Y )dB
− t ∂y − t t t
5
and by the Jacod theorem
·σ(Y )dB · ∂logπ(T s,Y ,Y )σ2(Y )ds
ˆ s s−ˆ ∂y − s T s
0 0
is an F σ(Y )-martingale, and we can write
T
∨
Y =Y + ·σ(Y )dB˜ + ·b(Y )ds+ · ∂logπ(T s,Y ,Y )σ2(Y )ds,
· 0 ˆ0 s s ˆ0 s ˆ0 ∂y − s T s
where (B˜ ) is an F σ(Y )-Brownian motion.
t t [0,T] T
∈ ∨
We have also a similar result for locally bounded martingales.
Theorem 9 Let (M ) be an F-local martingale locally bounded. Then, there exist kx(ω) such that
t t [0,T] t
∈
qx,M = ·kxqx d M,M ,
h i· ˆ0 s s− h is
and
M ·kLd M,M
·−ˆ0 s h is
is a G-martingale.
Suppose that any F-local martingale admits a representationof the form
t t
M =M + KndXn+ W(ω,x,s)(Q(ω,dx,ds) ν(ω,dx,ds))
t 0 ˆ0 s s ˆ0 ˆR −
X(n)
where(Xn)arecontinuouslocalmartingalespairwiseorthogonal,assumethatqx admitstherepresentation
t t
qx =qx+ kn,xqx dXn+ qx Ux(Q(,dx,ds) ν(,dx,ds)),
t 0 ˆ0 s s− s ˆ0 ˆR s− s · − ·
X(n)
then
t t
M Knkn,Ld Xn,Xn W(,x,s)ULν(,dx,ds)
t− ˆ0 s s h is−ˆ0 ˆR · s ·
X(n)
is a G-martingale with continuous part,
t t
M + KndXn Knkn,Ld Xn,Xn
0 ˆ s s − ˆ s s h is
0 0
X(n) X(n)
6
and jump part
t
W(ω,x,s) Q(ω,dx,ds) (1+UL)ν(ω,dx,ds) .
ˆ0 ˆR − s
(cid:0) (cid:1)
Example 10 Consider the Poisson process (N ) of intensity λ, as well as the filtration F := FN
t t [0,1]
∈
generated by it. Let
M =N λt, t [0,1],
t t
− ∈
and L=N , then
1
Q (,dk) = P N =k =P N =k N
t 1 t 1 t t
· { |F } { − − }
= e−λ(1−t)(λ(1−t))k−Nt,
(k N )!
t
−
and
qk = e−λ(1−t)(λ((1k−−tN))tk)−!Nt
t e λλk
− k!
= eλtλ−Nt(1−t)k−Ntk!.
(k N )!
t
−
Now, if there is a jump at t:
qk qk k N
Utk = t q−k t− = λ(−1 tt−) −1,
t− −
so
N N
1+UtL = λ1(1− tt)−,
−
and
t N N
N 1− sds,0 t<1
t−ˆ 1 s ≤
0 −
is a G-martingale.
Remark 11 A more general result, concerning Lévy processes, can be obtained by using the characteristic
function instead of the conditional density. Let (Z ) be a Lévy process with characteristic function
t t 0
≥
E[eiθZt]=etψ(θ).
Let 0 s u t T. Then, in virtue of the independence of the increments of (Z ) , we have that the
t t 0
≤ ≤ ≤ ≤ ≥
7
following chain of equations
E[eiθZT(Zt Zu)hs] = E[eiθ(ZT−Zt)]E[eiθ(Zt−Zu)(Zt Zu)]E[eiθZuhs]
− −
1
= E[eiθ(ZT−Zt)] iE[eiθ(Zt−Zu)]∂θlogE[eiθ(Zt−Zu)] E[eiθZuhs]
(cid:18) (cid:19)
1
= iE[eiθZThs]∂θlogE[eiθ(Zt−Zu)]
1
= (t u)E[eiθZTh ]∂ ψ(θ).
i − s θ
And, consequently,
Z Z 1
E eiθZTh t− u = E[eiθZTh ]∂ ψ(θ).
s t u i s θ
(cid:20) − (cid:21)
Since the right hand side of previous equation does not depend on t, we have
Z Z 1
E eiθZTh T − u = E[eiθZTh ]∂ ψ(θ).
s T u i s θ
(cid:20) − (cid:21)
Thus, by integrating with respect u and applying Fubini we obtain
t Z Z 1
E eiθZTh T − udu = (t s)E[eiθZTh ]∂ ψ(θ)
sˆ T u i − s θ
(cid:20) s − (cid:21)
= E[eiθZTh (Z Z )].
s t s
−
In certain non-homogeneous cases we can use a similar argument. Assume that
E[eiθ(Zt−Zu)]=eg(t,u)ψ(θ),
then
t Z Z
E eiθZTh T − u( 1)∂ g(t,u)du =E[eiθZTh (Z Z )],
sˆ g(T,u) − u s t− s
(cid:20) s (cid:21)
provided that st ZgT(T−,Zu)u(−1)∂ug(t,u)du is well defined.
´
ConditionAisasufficientconditionthatallowustofindtheDoobdecompositionintheenlargedfiltration.
Howeverwehavethefollowingpropositionthatallowsustoobtainthe Doob-Meyerdecompositionwithout
requiring Condition A:
Proposition 12 With the notations above, assume that there exist αx(ω) such that
t
t
∞Q (,dx),M = ∞αxQ (,dx)d M,M , for all a R,
hˆa t · i ˆ0 ˆa s s− · h is ∈
8
then
M ·αxd M,M ,
·−ˆ0 s h is
is a G-martingale.
Proof. For every Z we have that
s
∈F
E[Z1 (M M )] = E E[Z1 (M M ) ]
L>a t s L>a t s t
{ } − { } − |F
= E(cid:2)Z(M M )E[1 ](cid:3)
t s L>a t
− { }|F
= E(cid:2)Z(M M ) ∞Q (ω,dx(cid:3))
t− s ˆ t
(cid:20) a (cid:21)
= E Z M ∞Q (ω,dx) M ∞Q (ω,dx)
tˆ t − sˆ s
(cid:20) (cid:18) a a (cid:19)(cid:21)
= E Z M, ∞Q(ω,dx) M, ∞Q(ω,dx)
(cid:20) (cid:18)h ˆa · it−h ˆa · is(cid:19)(cid:21)
t
= E Z ∞αxQ (ω,dx)d M,M
(cid:20) ˆs ˆa u u− h iu(cid:21)
t
= E Z1 αLd M,M
(cid:20) {L>a}ˆs u h iu(cid:21)
Example 13 Let (B ) be a Brownian motion and take τ =inf t>0,B = 1 . It is well known that
t t 0 t
≥ { − }
1+B
P τ s =2Φ t 1 +1 ,
{ ≤ |Ft} −√s t {τ∧s>t} {s<τ∧t}
(cid:18) − (cid:19)
where Φ is the c.d.f. of a standard normal distribution. Then in t<s τ we have, by Itô’s formula,
∧
P{τ ≤s|Ft}=2Φ −√1s + π2 ˆ t √s1 ue−(12+(sB−uu))2dBu,
(cid:18) (cid:19) r 0 −
so
dhP{τ ≤s|F·},Bit =rπ2√s1 te−(12+(sB−tt))2dt,
−
and
αsQ (,ds)
t t ·
= ∂ 2 1 e−(12+(sB−tt))2 = 1 1 (1+Bt)2 e−(12+(sB−tt))2,
∂s rπ√s−t ! √2π  (s t)3 − (s t)5
− −
q q 
9
finally
(1+Bt)2
Q (,ds)= ∂ P τ s = e− 2(s−t) (1+B ),
t · ∂s { ≤ |Ft} √2π (s t)3 t
−
q
and
∂ 2 1 e−(12+(sB−tt))2
∂s π√s t 1 1+B
αst = (cid:18)q∂ P τ− s (cid:19) = 1+B − s tt.
∂s { ≤ |Ft} t −
Consequently
B t∧τ 1 1+Bs ds, t 0,
t−ˆ 1+B − τ s ≥
0 (cid:18) s − (cid:19)
is a G-martingale.
Proposition 14 If we have a random measure P(1)(ω,dx) and a finite deterministic measure m(dt) such
t
that
t
E[Zg(L)(M M )]=E Zg(x)P(1)(ω,dx)m(dt)
t− s (cid:20)ˆs ˆR t (cid:21)
and P(1)(ω,dx)=αx(ω)Q (ω,dx) then
t t t
M ·αLm(ds)
·−ˆ0 s
is a G-martingale.
Proof.
t
E[Zg(L)(M M )] = E Zg(x)P(1)(ω,dx)m(du)
t− s (cid:20)ˆs ˆR u (cid:21)
t
= E Zg(x)αx(ω)Q (ω,dx)m(du)
ˆ ˆ u u
(cid:20) s R (cid:21)
t
= E ZE[g(L)αL ]m(du)
ˆ u|Ft
(cid:20) s (cid:21)
t
= E Zg(L)αLm(du)
ˆ u
(cid:20) s (cid:21)
This expression is more appropriate to treat cases like Example 1. Suppose that L is a random time and
that there exist P(1)(ω,dx) and P(2)(ω,dx) such that
u u
t s
E[Zg(L)1 (M M )] = E Zg(x)P(1)(ω,dx)m (du)
{L<s} t− s (cid:20)ˆs ˆ0 u 1 (cid:21)
t
E[Zg(L)1 (M M )] = E ∞Zg(x)P(2)(ω,dx)m (du) ,
{L>t} t− s (cid:20)ˆs ˆt u 2 (cid:21)
10