Quadratic Semimartingale BSDEs under an Exponential Moments Condition

0.33 MB·English

by  Markus Mocha

QUADRATIC SEMIMARTINGALE BSDES UNDER AN
EXPONENTIAL MOMENTS CONDITION
By Markus Mocha∗ and Nicholas Westray
Humboldt-Universita¨t zu Berlin
In the present article we provide existence, uniqueness and stability results
underanexponentialmomentsconditionforquadraticsemimartingalebackward
1 stochastic differential equations (BSDEs) having convex generators. We show
1 thatthemartingale partoftheBSDEsolution definesatruechangeofmeasure
0 and provide an example which demonstrates that pointwise convergence of the
2 drivers is not sufficient to guarantee a stability result within our framework.

n
a
J 1. Introduction. Since their introduction by Bismut [2] within the Pontryagin max-

3 imum principle, backward stochastic differential equations (BSDEs) have attracted much
1
attention in the mathematical literature. In a Brownian framework such equations are

usually written
]
R
P (1.1) dY = Z dW −F(t,Y ,Z )dt, Y = ξ,
t t t t t T
.
h
t where ξ is an FT-measurable random variable, the terminal value, and F is the so called
a
driver or generator. Here (F ) denotes the filtration generated by the one dimen-
m t t∈[0,T]
sional Brownian motion W. Solving such a BSDE corresponds to findinga pair of adapted
[
processes (Y,Z)such thatthe integrated version of (1.1) holds.Thepresenceof thecontrol

1
process Z stems from the requirement of adaptedness for Y together with the fact that Y
v
2 must be driven into the random variable ξ at time T. One may think of Z as arising from
8 the martingale representation theorem.
5
In the general semimartingale framework, where the main source of randomness is en-
2
. coded in a given local martingale M on a filtration (Ft)t∈[0,T] that is not necessarily
1
generated by M, we have to add an extra orthogonal component N. The corresponding
0
1 BSDE then takes the form
1
v: (1.2) dYt = ZtdMt+dNt−f(t,Yt,Zt)dhMit −gtdhNit, YT = ξ.
i
X
Solving (1.2)now correspondsto findingan adapted triple (Y,Z,N) of processes satisfying
r
a the integrated version of (1.2), where N is a local martingale orthogonal to M. We refer
to Z ·M +N as the martingale part of a solution to the BSDE (1.2).
BSDEs of type (1.1) and (1.2) have found many fields of application in mathematical
finance. The first problems to be attacked by means of such equations included pricing
and hedging, superreplication and recursive utility. The reader is directed to the survey
articles El Karoui, Peng and Quenez [10] and El Karoui, Hamad`ene and Matoussi [9] and
the references therein for further discussion. A second large focus has been on their use in
constrained utility maximization. In a Brownian setting Hu, Imkeller and Mu¨ller [15] used
the martingale optimality principle to derive a BSDE for the value process characterizing
the optimal wealth and investment strategy. Their article can be regarded as an extension
of earlier work by Rouge and El Karoui [28] as well as Sekine [29]. In related work in a
semimartingale setting Mania and Schweizer [21] used a BSDE to describe the dynamic
∗Corresponding author: [email protected]
AMS 2010 subject classification: 60H10
Keywords and phrases: Quadratic Semimartingale BSDEs, Convex Generators, Exponential Moments
1
2 M. MOCHA AND N.WESTRAY
indifference price for exponential utility. Their stochastic control approach was extended
to robust utility in Bordigoni, Matoussi and Schweizer [3] and to an infinite time horizon
in the recent article by Hu and Schweizer [16]. We also mention Becherer [1] for further
extensions to BSDEs with jumps and Mania and Tevzadze [22] to backward stochastic
partial differential equations. This list is by far not exhaustive and additional references
can be found in the stated papers.
With regards to the theory of BSDEs, existence and uniqueness results were first pro-
vided in a Brownian setting by Pardoux and Peng [26] under Lipschitz conditions. These
were extended by Lepeltier and San Mart´ın [20] to continuous drivers with linear growth
and by Kobylanski [19] to generators which are quadratic as a function of the control vari-
able Z. The corresponding results for the semimartingale case may be found in Morlais
[24] and Tevzadze [30], where in the former the main theorems of [15] are extended. In
addition a stability result for quadratic BSDEs may also be found in the recent article by
Frei [11]. In the situation when the generator has superquadraticgrowth, Delbaen, Hu and
Bao [7] show that such BSDEs are essentially ill-posed.
Astrongrequirementpresentinthearticles [19,24,30]isthat theterminalcondition be
bounded.In a Brownian setting Briand and Hu [4, 5] have replaced this by the assumption
that it need only have exponential moments but in addition the driver is convex in the Z
variable. More recently, by interpreting the Y component as the solution to a stochastic
control problem, Delbaen, Hu and Richou [8] extend their results and show that one can
reduce the order of exponential moments required.
The present article has two main contributions, the first is to extend the existence,
uniqueness and stability theorems of [5] and [24] to the unbounded semimartingale case.
The motivation here is predominantly mathematical, having results in greater generality
increases the range of applications for BSDEs. We remark however, that there are addi-
tionalpracticalapplicationsfortheresultsderivedhere,e.g.relatedtoutilitymaximization
with an unbounded mean variance tradeoff, see Nutz [25] and Mocha and Westray [23],
which provides a second motivation for the present work.
In order to prove the respective results in the unbounded semimartingale framework
technical difficulties related to an a priori estimate must be overcome. This requires an
additional assumption when compared to [5] and [24]. As a biproduct of establishing our
results we are able to show via an example that the stability theorem as stated in [5]
Proposition 7 needs a minor amendment to the mode of convergence assumed on the
drivers and we include the appropriate formulation.
Our second contribution is to address the question of measure change. It is a classical
result that when the generator has quadratic growth, the solution processes Y is bounded
if and only if the martingale part Z ·M +N is a BMO martingale. In the present setting
such a correspondence is lost, however we are able to show that whilst Z ·M +N need
not be BMO, see Frei, Mocha and Westray [12] for further discussion and some examples,
the stochastic exponential E q(Z ·M +N) is still a true martingale for q valid in some
half-line. It is not only mathematically interesting to be able to describe the properties
(cid:0) (cid:1)
of the martingale part of the BSDE but also relevant for applications in an unbounded
setting. For instance, the above result has been used in Heyne [13] to extend the results
of [15] and [24] on utility maximization. Moreover such a theorem may be used in the
partial equilibrium framework of Horst, Pirvu and dos Reis [14] where the market price of
(external) risk is given by equilibrium considerations and is typically unbounded.
The paper is organized as follows. In the next section we lay out the notation and
the assumptions and state the main results. The subsequent sections contain the proofs.
Section 3 gives the a priori estimates together with some remarks on the necessity of an
QUADRATICBSDES UNDER EXPONENTIALMOMENTS 3
additional assumption, Section 4 deals with existence and Section 5 includes the compar-
ison and uniqueness results. In Sections 6 and 7 we prove the stability property as well
as providing an interesting counterexample. In Section 8, we turn our attention to the
measure change problem and finally, in Section 9, we give interesting applications of our
results to constrained utility maximization and partial equilibrium models.
2. Model Formulation and Statement of Results. We work on a filtered prob-
ability space (Ω,F,(F ) ,P) satisfying the usual conditions of right-continuity and
t 0≤t≤T
completeness. We also assume that F is the completion of the trivial σ-algebra. The time
0
horizon T is afinitenumberin (0,∞) andall semimartingales areconsidered equaltotheir
c`adla`g modification.
Throughout this paper M = (M1,...,Md)T stands for a continuous d-dimensional local
martingale, where T denotes transposition. We refer the reader to Jacod and Shiryaev [17]
and Protter [27] for further details on the general theory of stochastic integration.
The objects of study in the present paper will be semimartingale BSDEs considered on
[0,T]. In the d-dimensional case such a BSDE may be written
(2.1) dY = ZTdM +dN −1TdhMi f(t,Y ,Z )−g dhNi , Y = ξ.
t t t t t t t t t T
Here ξ is an R-valued F -measurable randomvariable and f andg are randompredictable
T
functions[0,T]×Ω×R×Rd → Rdand[0,T]×Ω → R,respectively.Weset1 := (1,...,1)T ∈
Rd.
The format in which the BSDE (2.1) encodes its finitevariation parts is not so tractable
from the point of view of analysis. Therefore we write semimartingale BSDEs by factoriz-
ing the matrix-valued process hMi =hMi,Mji . This separates its matrix property
i,j=1,...,d
from its nature as measure. This step could also be regarded as a reduction of dimension-
ality.
For i,j ∈ {1,...,d} wemay writehMi,Mji= Cij·AwhereCij arethe components of a
predictable process C valued in the space of symmetric positive semidefinite d×dmatrices
and A is a predictable increasing process. There are many such factorizations (cf. [17]
Section III.4a). We may choose A:= arctan d Mi so that A is uniformly bounded
i=1
by K = π/2 and derive the absolute cont(cid:16)inuity of all(cid:17)the hMi,Mji with respect to A
A P (cid:10) (cid:11)
from the Kunita-Watanabe inequality. This together with the Radon-Nikody´m theorem
provides C. Furthermore, we can factorize C as C = BTB for a predictable process B
valued in the space of d×d matrices. We note that all the results below do not rely on the
specificchoiceofA,butonlyonitsboundedness.Inparticular,ifM = W isad-dimensional
Brownian motion we may choose A = t, t ∈ [0,T], and B the identity matrix. Then A is
t
bounded by K = T.
A
We let P denote the predictable σ-algebra on [0,T] × Ω generated by all the left-
continuous processes. The process A induces a measure µA on P, the Dol´eans measure,
defined for E ∈ P by
T
µA(E) := E 1 (t)dA .
E t
(cid:20)Z0 (cid:21)
Given the above discussion the equation (2.1) may be rewritten as
(2.2) dY = ZTdM +dN −F(t,Y ,Z )dA −g dhNi , Y = ξ,
t t t t t t t t t T
where again ξ is an R-valued F -measurable random variable, the terminal condition, and
T
F and g are random predictable functions [0,T]×Ω×R×Rd → R and [0,T]×Ω → R
4 M. MOCHA AND N.WESTRAY
respectively, called generators or drivers. This formulation of the BSDE is very flexible,
allowing for various applications and being amenable to analysis.
Starting with (2.1) and setting F(t,y,z) := 1TC f(t,y,z) = 1TBTB f(t,y,z) we get
t t t
(2.2).Thereversionofthisprocedureisnotsoclear,howeverisnotrelevantinapplications.
Under boundedness assumptions, existence of solutions to (2.2) is provided in [24] via
an exponentialtransformation thatmakes thedhNi term disappear.Anecessary condition
for this kind of transformation to work properly is dg = 0. In the sequel we thus consider
the above BSDE to be given in the form
1
(2.3) dY = ZTdM +dN −F(t,Y ,Z )dA − dhNi , Y = ξ,
t t t t t t t 2 t T
except in specific situations where a solution is assumed to exist.
Definition 2.1. A solution to the BSDE (2.2), or (2.3), is a triple (Y,Z,N) of pro-
cesses valued in R×Rd×R satisfying (2.2), or (2.3), P-a.s. such that:
(i) The function t 7→ Y is continuous P-a.s.
t
(ii) The process Z is predictable and M-integrable, in particular T ZTdhMi Z < +∞
0 t t t
P-a.s.
R
(iii) The local martingale N is continuous and orthogonal to each component of M, i.e.
hN,Mii = 0 for all i= 1,...,d.
(iv) We have that P-a.s.
T
|F(t,Y ,Z )|dA +hNi < +∞.
t t t T
Z0
As in the introduction we call Z ·M +N the martingale part of a solution.
In what follows we collect together the assumptions that allow for all the assertions of
this paper to hold simultaneously. However we want to point out that not all of our results
require that every item of Assumption 2.2 be satisfied, as will be indicated in appropriate
remarks.
Assumption 2.2. There exist nonnegative constants β and β, positive numbers β
f
and γ ≥ max(1,β) together with an M-integrable (predictable) Rd-valued process λ so that
writing
T T
α := kBλk2 and |α| := α dA = λTdhMi λ
1 t t t t t
Z0 Z0
we have P-a.s.
(i) The random variable |ξ| + |α| has exponential moments of all orders, i.e. for all
1
p > 1
(2.4) E exp p |ξ|+|α| < +∞.
1
h (cid:16) (cid:17)i
(cid:2) (cid:3)
(ii) For all t ∈ [0,T] the driver (y,z) 7→ F(t,y,z) is continuous in (y,z), convex in z and
Lipschitz continuous in y with Lipschitz constant β, i.e. for all y , y and z we have
1 2
(2.5) |F(t,y ,z)−F(t,y ,z)| ≤ β|y −y |.
1 2 1 2
QUADRATICBSDES UNDER EXPONENTIALMOMENTS 5
(iii) The generator F satisfies a quadratic growth condition in z, i.e. for all t,y and z we
have
γ
(2.6) |F(t,y,z)| ≤ α +α β|y|+ kB zk2.
t t t
2
(iv) The function F is locally Lipschitz in z, i.e. for all t,y,z and z
1 2
|F(t,y,z )−F(t,y,z )| ≤ β kB λ k+kB z k+kB z k kB (z −z )k.
1 2 f t t t 1 t 2 t 1 2
(cid:16) (cid:17)
(v) The constant β in (iii) equals zero and then we set c := 0. Alternatively, β > 0, but
A
additionally assume that for all y and z we have
|F(t,y,z)−F(t,0,z)| ≤ β|y|
and that there is a positive constant c such that A ≤ c ·t for all t ∈ [0,T].
A t A
If this assumption is satisfied we refer to (2.3) as BSDE(F,ξ) with the set of parameters
(α,β,β,β ,γ).
f
Remark 2.3. The above items (i)-(iv) correspond to the assumptions made in [5] and
[24]. In particular, the BSDEs under consideration are of quadratic type (in the control
variable z) and of Lipschitz type in y. Item (v) is new and arises from the fact that the
methods used in [24] to derive an a priori estimate may no longer be directly applied so
that an additional assumption is required. We elaborate further on this topic in Section 3.
Observe that in the key application of utility maximization, cf. [23], the associated driver
is independent of y and hence β = 0 applies.
Notice that items (ii) and (iii) from above provide
γ
(2.7) |F(t,y,z)| ≤ α +β|y|+ kB zk2,
t t
2
for all t, y and z, P-a.s. This is an inequality which does not involve α in the |y| term on
the right hand side and which is used repeatedly throughout the proofs. We also define
the constant
β∗ := c ·β.
A
Before giving the main results of the paper let us introduce some notation. For p ≥ 1,
Sp denotes the set of R-valued, adapted and continuous processes Y on [0,T] such that
1/p
E sup |Y |p < +∞.
t
"0≤t≤T #
The space S∞ consists of the continuous bounded processes. An R-valued, adapted and
continuous process Y belongs to E if the random variable
Y∗ := sup |Y |
t
t∈[0,T]
has exponential moments of all orders. We also recall that Y is called of class D if the
family{Y |τ ∈ [0,T] stopping time}isuniformlyintegrable.Thesetof(equivalenceclasses
τ
of) Rd-valued predictable processes Z on [0,T]×Ω satisfying
T p/2 1/p
E ZTdhMi Z < +∞
t t t
"(cid:18)Z0 (cid:19) #
6 M. MOCHA AND N.WESTRAY
is denoted by Mp. Finally, Mp stands for the set of R-valued martingales N on [0,T], such
that
1/p
kNkMp := E hNipT/2 < +∞.
Notice that if the following assumption ohn the fiiltration holds the elements of Mp are
continuous.
Assumption 2.4. The filtration (F ) is a continuous filtration, in the sense that
t t∈[0,T]
all local (F ) -martingales are continuous.
t t∈[0,T]
The following four theorems constitute the main results of the paper. We mention that
only the existence result requires the assumption of the continuity of the filtration.
Theorem 2.5 (Existence). If Assumptions 2.2 and 2.4 hold there exists a solution
(Y,Z,N) to the BSDE (2.3) such that Y ∈ E and Z ·M +N ∈ Mp for all p ≥ 1.
Theorem 2.6 (Uniqueness). Suppose that Assumption 2.2 holds. Then any two solu-
tions (Y,Z,N) and (Y′,Z′,N′) in E×M2×M2 to the BSDE (2.3) coincide in the sense
that Y and Y′, Z ·M and Z′·M, and N and N′ are indistinguishable.
Theorem 2.7 (Stability). Consider a family of BSDEs(Fn,ξn) indexed by the ex-
tended natural numbers n ≥ 0 for which Assumption 2.2 holds true with parameters
(αn,βn,β,β ,γ). Assume that the exponential moments assumption (2.4) holds uniformly
f
in n, i.e. for all p > 1,
supE ep(|ξn|+|αn|1) < +∞.
n≥0
h i
If for n≥ 0 (Yn,Zn,Nn) is the solution in E×M2×M2 to the BSDE(Fn,ξn) and if
T
(2.8) |ξn −ξ0|+ Fn−F0 (s,Y0,Z0)dA −→ 0 in probability, as n → +∞,
s s s
Z0
(cid:12) (cid:12)
then for each p ≥ 1 as n(cid:12)→ +∞ (cid:12)
p
E exp sup Yn−Y0 −→ 1 and Zn·M +Nn −→ Z0·M +N0 in Mp.
t t
" 0≤t≤T !! #
(cid:12) (cid:12)
(cid:12) (cid:12)
Theorem 2.8 (ExponentialMartingales). Suppose that Assumption 2.2holds, let|q| >
γ/2andlet(Y,Z,N) ∈ E×M2×M2 beasolution tothe BSDE (2.3).ThenE q(Z·M+N)
is a true martingale on [0,T].
(cid:0) (cid:1)
Remark 2.9. The preceding theorems generalize the results of [5] and [24] and the
method of proof is therefore similar. We combine the localization and θ-technique from [5]
together with the existence and stability results for BSDEs with bounded solutions found
in[24].Similarideasareusedin[16]onaspecificquadraticBSDEarisinginarobustutility
maximization problem where the authors also investigate the measure change problem for
their special BSDE, however here we pursue the general theory. We point out that when
the BSDE is of quadratic type and |ξ|+|α| does not have sufficiently large exponential
1
momentsthereareexampleswheretheBSDEadmitsnosolution.Thustheresultsherecan
beconsidered, in some sense, the best possible.In particular, we present all the theoretical
background for the study of utility maximization under exponential moments, see [13] and
[23], as well as partial equilibrium, see [14].
QUADRATICBSDES UNDER EXPONENTIALMOMENTS 7
3. A Priori Estimates. In this section we show that, under appropriate conditions,
solutions to the BSDE (2.2) satisfy some a priori norm bounds. After giving an impor-
tant result used in the subsequent sections we motivate Assumption 2.2 (v) by showing
that without such an assumption the method utilized in [24] for the purpose of deriving
appropriate a priori bounds fails in the present unbounded case.
Let (Y,Z,N) be a solution to (2.2), suppose that Assumption 2.2 (iii) and (v) hold and
that g is uniformly bounded by γ/2. Fix s ∈ [0,T] and set, for t ∈ [s,T],
t
H := exp γeβ∗(t−s)|Y |+γ eβ∗(r−s)dhλ·Mi .
t t r
(cid:18) Zs (cid:19)
where we have writteen hλ·Mi := tλTdhMi λ = tα dA . First we show that H is, up
t 0 r r r 0 r r
to integrability, a local submartingale.
R R
From Tanaka’s formula, e
(3.1) d|Y | = sgn(Y )(ZTdM + dN ) − sgn(Y ) F(t,Y ,Z )dA + g dhNi + dL ,
t t t t t t t t t t t t
(cid:0) (cid:1)
where L is the local time of Y at 0. Itˆo’s formula then yields
dH =γH eβ∗(t−s) sgn(Y )(ZTdM +dN )+β|Y |(c dt−dA )
t t t t t t t A t
"
e e + −sgn(Y )F(t,Y ,Z )+α +β|Y |+ γ eβ∗(t−s)kB Z k2 dA
t t t t t t t t
2
(cid:16) (cid:17)
+ −sgn(Y )g + γ eβ∗(t−s) dhNi +dL .
t t t t
2
#
(cid:16) (cid:17)
An inspection of the finite variation parts shows that under the present assumptions they
are nonnegative. In particular, the semimartingale H is a local submartingale, which leads
to the following result.
e
Proposition 3.1 (A Priori Estimate). Suppose Assumption 2.2 (iii) and (v) hold and
assume that the function g is uniformly bounded by γ/2, P-a.s. Let (Y,Z,N) be a solution
to the BSDE (2.2) and let the process
T
exp γeβ∗T|Y|+γ eβ∗rdhλ·Mi
r
(cid:18) Z0 (cid:19)
be of class D. Then P-a.s. for all s ∈ [0,T],
(3.2) |Y | ≤ 1 logE exp γeβ∗(T−s)|ξ|+γ T eβ∗(r−s)dhλ·Mi F .
s r s
γ " (cid:18) Zs (cid:19)(cid:12)(cid:12) #
(cid:12)
(cid:12)
Proof. Fix s ∈ [0,T] and set H as above. Since H is a local submart(cid:12)ingale there exists
a sequence of stopping times (τ ) valued in [s,T], which converges P-a.s. to T, such
n n≥1
that Hτn is a submartingale for eaech n≥ 1. We thenederive
T
exp(eγ|Y |) ≤ E[H |F ]≤ E exp γeβ∗(T−s)|Y |+γ eβ∗(r−s)dhλ·Mi F .
s T∧τn s T∧τn r s
" (cid:18) Zs (cid:19)(cid:12)(cid:12) #
(cid:12)
e
Letting n → +∞ the claim follows from the class D assumption. (cid:12)
(cid:12)
8 M. MOCHA AND N.WESTRAY
Proposition 3.1 provides the appropriate a priori estimate, indeed suppose that |ξ| and
|α| are bounded random variables and (Y,Z,N) is a solution to (2.3). If the current
1
assumptions hold and exp(γeβ∗T|Y|) is of class D, then Y satisfies
(3.3) |Y| ≤ eβ∗T(|ξ|+|α| ) .
1
∞
(cid:13) (cid:13)
Comparing with (3.2) this indicates th(cid:13)at the inclusion(cid:13)of Assumption 2.2 (v) allows us to
(cid:13) (cid:13)
prove similar estimates to the bounded case which enables us to establish existence for the
BSDE (2.3) when |ξ|+|α| has exponential moments of all orders, to be more precise, an
1
order of at least γeβ∗T.
Contrary to the above let us investigate the method utilized in [24] under Assumption
2.2 (iii) only, supposing that g be bounded by γ/2. We set
t
(3.4) H := exp γeβhλ·Mis,t|Y |+γ eβhλ·Mis,r dhλ·Mi ,
t t r
(cid:18) Zs (cid:19)
t
where hλ·Mi := hλ·Mi −hλ·Mi = α dA . We derive from Itˆo’s formula
s,t t s s r r
R
dH = γH eβhλ·Mis,t sgn(Y )(ZTdM +dN )
t t t t t t
"
γ
+ −sgn(Y )F(t,Y ,Z )+α +α β|Y |+ eβhλ·Mis,tkB Z k2 dA
t t t t t t t t t
2
(cid:16) (cid:17)
γ
+ −sgn(Y )g + eβhλ·Mis,t dhNi +dL .
t t t t
2
#
(cid:16) (cid:17)
Once again, the finite variation parts are nonnegative. We conclude in the same way as for
Proposition 3.1 that the corresponding a priori result holds for H as well. To sum up, we
have that under a similar class D assumption, now on
T
exp γeβhλ·MiT|Y|+γ eβhλ·Mir dhλ·Mi ,
r
(cid:18) Z0 (cid:19)
P-a.s. for all s ∈ [0,T],
1 T
(3.5) |Y | ≤ logE exp γeβhλ·Mis,T|ξ|+γ eβhλ·Mis,r dhλ·Mi F .
s r s
γ " (cid:18) Zs (cid:19)(cid:12)(cid:12) #
(cid:12)
(cid:12)
If β = 0, then H from above equals H and there is no difference with the statement of
(cid:12)
Proposition 3.1. However when β > 0 the estimate (3.5) is not sufficient for our purposes.
We aim at usingethe a priori estimate to show the existence of solutions to the BSDE (2.3)
inE×M2×M2 usinganappropriateapproximatingprocedure.If|ξ|and|α| arebounded
1
random variables there exists a solution (Y,Z,N) to (2.3) with Y bounded, cf. [24]. With
(3.5) at our disposal we then have the estimate
(3.6) |Y|≤ eβ|α|1 |ξ|+|α| .
1
∞
(cid:13) (cid:13)
Our goal is to remove the boundedne(cid:13)ss assu(cid:0)mption an(cid:1)d(cid:13)to replace it with the assumption
(cid:13) (cid:13)
on the existence of exponential moments of |ξ|+|α| in the spirit of [5]. However a closer
1
inspectionoftheaprioriestimatefrom(3.5)togetherwith(3.6)alreadyindicatesthatmore
restrictive assumptions are necessary. More specifically, when β > 0 we cannot deduce any
integrability of exp γeβ|α|1 |ξ|+|α| when |ξ| and |α| have only exponential moments,
1 1
this motivates Assumption 2.2 (v).
(cid:0) (cid:0) (cid:1)(cid:1)
QUADRATICBSDES UNDER EXPONENTIALMOMENTS 9
4. Existence. In the present section we establish Theorem 2.5 together with some
related results on norm bounds of the solution. The proof of existence follows the follow-
ing recipe. Firstly we truncate hλ ·Mi to get approximate solutions. Then by using the
estimate from Proposition 3.1 we localize and work on a random time interval so that the
approximations are uniformlyboundedandwe can apply a stability result. Finally we glue
together on [0,T] to construct a solution. The a priori estimates ensure that we may take
all limits in the described procedure.
Theorem 4.1 (Existence). Let Assumptions 2.2 (ii)-(v) and 2.4 hold and |ξ|+ |α|
1
have an exponential moment of order γeβ∗T. Then the BSDE (2.3) has a solution (Y,Z,N)
such that
(4.1) |Y | ≤ 1 logE exp γeβ∗(T−t)|ξ|+γ T eβ∗(r−t)dhλ·Mi F .
t r t
γ
" (cid:18) Zt (cid:19)(cid:12)(cid:12) #
(cid:12)
(cid:12)
Proof. Exactly as in[5]wefirstassumethatF andξ arenonnegative(cid:12). For each integer
n ≥ 1, set
t
σ := inf t ∈ [0,T] hλ·Mi := α dA ≥ n ∧T,
n t s s
(cid:26) (cid:12) Z0 (cid:27)
(cid:12)
ξn := ξ∧n, λnt := 1{t≤σn}λt and Fn(t,y(cid:12)(cid:12),z) := 1{t≤σn}F(t,y,z). Then Fn satisfies Assump-
tion 2.2 (ii)-(v) with the same constants, but with the processes λn and αn where
αn := kB λnk2 = 1 kB λ k2 = 1 α .
t t t {t≤σn} t t {t≤σn} t
In particular, |αn| = σnα dA ≤ n and
1 0 s s
TR T
(λn)TdhMi λn = kB λnk2dA = |αn| ≤ n,
t t t t t t 1
Z0 Z0
so we may apply [24] Theorem 2.5 and Theorem 2.6 to conclude that there exists a unique
solution (Yn,Zn,Nn) ∈ S∞ ×M2 ×M2 to the BSDE (2.3), where F is replaced by Fn
and ξ by ξn. From Proposition 3.1 we derive
|Yn| ≤ 1 logE exp γeβ∗(T−t)|ξn|+γ T eβ∗(r−t)dhλn ·Mi F
t γ r t
" (cid:18) Zt (cid:19)(cid:12)(cid:12) #
(cid:12)
≤ 1 logE exp γeβ∗(T−t)|ξ|+γ T eβ∗(r−t)dhλ·Mi F(cid:12)(cid:12)
r t
γ
" (cid:18) Zt (cid:19)(cid:12)(cid:12) #
(cid:12)
(4.2) ≤ 1 logE exp γeβ∗T |ξ|+|α| F =:X . (cid:12)(cid:12)
1 t t
γ
" (cid:12) #
(cid:16) (cid:17)(cid:12)
(cid:0) (cid:1) (cid:12)
Let n ≤ m so that then we have σ ≤ σ and 1 (cid:12) ≤ 1 . In particular, ξn ≤ ξm
n m {(cid:12)t≤σn} {t≤σm}
and Fn ≤ Fm, from which we deduce that the current assumptions, hence the corre-
sponding assumptions in [24], hold for both Fn and Fm with the same set of parameters
(αm,β,β,β ,γ) where the additional c in [24] is equal to m. An application of Theorem
f θ
2.7thereinnowshowsthatYn ≤ Ym sothat(Yn) isanincreasingsequenceof bounded
n≥1
continuous processes.
Let k ≥ 1 be a fixed integer and
τ := inf t ∈ [0,T] X ≥ k or hλ·Mi ≥ k ∧T.
k t t
n (cid:12) o
(cid:12)
(cid:12)
10 M. MOCHA AND N.WESTRAY
Thanks to the continuity of the filtration the martingale exp(γX) is continuous so that
the random variable
V := max X ∨hλ·Mi
t T
t∈[0,T]
(cid:0) (cid:1)
is finiteP-a.s. Wederive thatP-a.s.τ = T for large k.Dueto(4.2)thesequence(Yn,k)
k n≥1
given by
Yn,k := Yn ,
t t∧τk
is uniformly bounded by k. For the martingale parts we define
Zn,k := 1 Zn and Nn,k := 1 Nn.
t {t≤τk} t t {t≤τk} t
An inspection of the respective cases shows that
T T
Yn,k = Yn − Zn,k TdM − dNn,k
t τk s s s
Zt Zt
(cid:0) (cid:1) T 1 T
+ 1 F(s,Yn,k,Zn,k)dA + dhNn,ki .
{s≤τk∧σn} s s s 2 s
Zt Zt
Moreover, Yn −n−↑−+−∞→ sup Yn =: ξ , where ξ is bounded by k. Next we appeal to the
τk n≥1 τk k k
stability result stated in [24] Lemma 3.3, noting Remark 3.4 therein. Note that this result
requires estimates that are uniformin n which is accomplished by the specificchoice of the
stopping time τ . Hence (Yn,k,Zn,k,Nn,k) converges to (Y∞,k,Z∞,k,N∞,k) in the sense
k
that
lim E sup Yn,k −Y∞,k = 0,
t t
n→+∞ 0≤t≤T !
(cid:12) (cid:12)
T (cid:12) T (cid:12)
lim E Zn,k −Z∞,k dhMi Zn,k −Z∞,k = 0
s s s s s
n→+∞
(cid:18)Z0 (cid:16) (cid:17) (cid:16) (cid:17)(cid:19)
and
2
lim E Nn,k −N∞,k = 0,
T T
n→+∞
(cid:18)(cid:12) (cid:12) (cid:19)
where the triples (Y∞,k,Z∞,k,N∞,k) s(cid:12)(cid:12)olve the BSDE(cid:12)(cid:12)
T
∞,k ∞,k ∞,k
dY = Z dM +dN
t t t t
(cid:16) (cid:17) 1
−1 F(t,Y∞,k,Z∞,k)dA − dhN∞,ki , Y∞,k = ξ ,
{t≤τk} t t t 2 t τk k
on the random horizon [[0,τ ]] ⊂ [0,T]. The stopping times τ are monotone in k and
k k
therefore it follows that
Yn,k+1 = Yn,k, 1 Zn,k+1 = Zn,k and 1 Nn,k+1 = Nn,k,
t∧τk t {t≤τk} t t {t≤τk} t t
so that the above convergence yields (for the two last objects in M2)
Y∞,k+1 = Y∞,k, 1 Z∞,k+1 ·M = Z∞,k ·M and 1 N∞,k+1 = N∞,k.
t∧τk t {t≤τk} t t {t≤τk} t t
(cid:16)(cid:16) (cid:17) (cid:17) (cid:16) (cid:17)