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Vol. 12 (2007), Paper no. , pages 1418–1453.
Journal URL
http://www.math.washington.edu/~ejpecp/

Classical and Variational Differentiability of BSDEs
with Quadratic Growth∗
Stefan Ankirchner and Peter Imkeller and Gonçalo Dos Reis
Institut für Mathematik
Humboldt-Universität zu Berlin
Unter den Linden 6, 10099 Berlin, Germany
ankirchn@mathematik.hu-berlin.de imkeller@mathematik.hu-berlin.de gnreis@mathematik.hu-berlin.de

Abstract
We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow
quadratically in the control variable. In a more abstract setting, we first allow both the
terminal condition and the generator to depend on a vector parameter x. We give sufficient

conditions for the solution pair of the BSDE to be differentiable in x. These results can be
applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given
by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair
is differentiable with respect to the initial vector of the forward equation. Finally we prove
sufficient conditions for solutions of quadratic BSDEs to be differentiable in the variational
sense (Malliavin differentiable).
Key words: BSDE, forward-backward SDE, quadratic growth, differentiability, stochastic
calculus of variations, Malliavin calculus, Feynman-Kac formula, BMO martingale, reverse
Hölder inequality.
AMS 2000 Subject Classification: Primary 60H10; Secondary: 60H07, 65C30.
Submitted to EJP on January 31, 2007, final version accepted November 5, 2007.
∗

This research was supported by the DFG Research Center MATHEON “Mathematics for Key Technologies”
(FZT86) in Berlin.

1418

Introduction
Problems of stochastic control treated by the crucial tool of backward stochastic differential

equations (BSDEs) have been encountered in many areas of application of mathematics in recent
years. A particularly important area is focused around optimal hedging problems for contingent
claims in models of financial markets. Recently, a special class of hedging problems in incomplete
financial markets has been considered in the area where finance and insurance concepts meet.
At this interface problems of securitization arise, i.e. insurance risk is transferred to capital
markets. One particularly interesting risk source is given by climate or environmental hazards
affecting insurance companies or big branches of the economy that depend on weather such as
agriculture and fishing, transportation and tourism. The public awareness of climate hazards
such as floods or hurricanes is continually increasing with the intensity of the discussion about
irreversible changes due to human impact.
BSDEs typically appear in the following setting. On a financial market some small investors are
subject to an external risk source described for instance by weather or climate influences. There
may also be big investors such as re-insurance companies that depend in a possibly different way
on the same risk source. In this situation market incompleteness stems from the external risk not
hedgeable by the market assets. One may complete the market either by making the external
risk tradable through the introduction of an insurance asset traded among small agents, or by
introducing a risk bond issued by a big agent. In this setting, treating the utility maximization
problem for the agents under an equilibrium condition describing basically market clearing for
the additional assets, leads to the determination of the market price of external risk through a
BSDE which in case of exponential utility turns out to be quadratic in the control variable (see

[7], [3] and [4]). Alternatively, instead of maximizing utility with respect to exponential utility
functions we might minimize risk measured by the entropic risk measure. In this setting we
again encounter a BSDE with quadratic nonlinearity, of the type
Yt = ξ +

Z

T

f (s, Ys , Zs )ds −

Z

T

Zs dWs ,

0 ≤ t ≤ T,

t

t

where W is a finite-dimensional Wiener process of the same dimension as the control process Z,
with a generator f that depends at most quadratically on Z, and a bounded terminal condition ξ.
In the meantime, the big number of papers published on general BSDEs is rivalled by the number
of papers on BSDEs of this type of nonlinearity. For a more complete list of references see [5]
or [9]. In particular, there are papers in which the boundedness condition on ξ is relaxed to an
exponential integrability assumption, or where the stochastic integral process of Z is supposed
to be a BMO martingale.
In a particularly interesting case the terminal variable ξ is given by a function g(XTx ) at terminal
time T of the solution process X of a forward SDE
Z t
Z t
x
x
σ(s, Xsx )dWs , 0 ≤ t ≤ T,
b(s, Xs )ds +
Xt = x +
0

0

with initial vector x ∈ R. Similarly, the driver f may depend on the diffusion dynamics of X x .
Via the famous link given by the generalized Feynman-Kac formula, systems as the above of

1419

forward-backward stochastic differential equations are seen to yield a stochastic access to solve
nonlinear PDE in the viscosity sense, see [9].
In this context, questions related to the regularity of the solutions (X x , Y x , Z x ) of the stochastic
forward-backward system in the classical sense with respect to the initial vector x or in the sense
of the stochastic calculus of variations (Malliavin calculus) are frequently encountered. Equally,
from a more analytic point of view also questions of smoothness of the viscosity solutions of the
PDE associated via the Feynman-Kac link are seen to be very relevant.
For instance, Horst and Müller (see [7]) ask for existence, uniqueness and regularity of a global
classical solution of our PDE from the analytic point of view. Not attempting a systematic
approach of the problem, they use the natural access of the problem by asking for smoothness
of the solutions of the stochastic system in terms of the stochastic calculus of variations. But
subsequently they work under the restrictive condition that the solutions of the BSDE have

bounded variational derivatives, which is guaranteed only under very restrictive assumptions on
the coefficients.
The question of smoothness of the stochastic solutions in the parameter x arises for instance in
an approach of cross hedging of environmental risks in [1]. Here the setting is roughly the one
of an incomplete market generated by a number of big and small agents subject to an external
(e.g. climate related) risk source, and able to invest in a given capital market. The risk exposure
of different types of agents may be negatively correlated, so that typically one type profits from
the risky event, while at the same time the other type suffers. Therefore the concept of hedging
one type’s risk by transferring it to the agents of the other type in a cross hedging context makes
sense. Mathematically, in the same way as described above, it leads to a BSDE of the quadratic
type, the solution (Y x , Z x ) of which depends on the initial vector x of a forward equation with
solution X x . Under certain assumptions, the cross-hedging strategy can be explicitly given in
a formula depending crucially on x, and in which the sensitivity with respect to x describes
interesting quality properties of the strategy.
In this paper, we tackle regularity properties of the solutions (Y x , Z x ) of BSDEs of the quadratic
type such as the two previously sketched in a systematic and thorough way. Firstly, the particular
dependence on the starting vector x of the forward component of a forward-backward system
will be generalized to the setting of a terminal condition ξ(x) depending in a smooth way to be
specified on some vector x in a certain Euclidean state space. We both consider the smoothness
with respect to x in the classical sense, as well as the smoothness in the sense of Malliavin’s

calculus.
The common pattern of reasoning in order to tackle smoothness properties of any kind starts with
a priori estimates for difference and differential quotients, or for infinite dimensional gradients in
the sense of variational calculus. In the estimates, these quantities are related to corresponding
difference and differential quotients or Malliavin gradients of the terminal variable and the driver.
To obtain the a priori estimates, we make use to changes of probability of the Girsanov type, by
which essentially nonlinear parts of the driver are eliminated. Since terminal conditions in our
treatment are usually bounded, the exponential densities in these measure changes are related
to BM O martingales. Known results about the inverse Hölder inequality allow to show that
as a consequence the exponential densities are r-integrable for some r > 1 related to the BM O
norm. This way we are able to reduce integrability properties for the quantities to be estimated
1420

to a natural level. In a second step, the a priori inequalities are used to derive the desired
smoothness properties from corresponding properties of driver and terminal condition. To the
best of our knowledge, only Malliavin differentiability results of this type have been obtained so
far, with strong conditions on the coefficients restricting generality considerably (see [7]).
After finishing this work we found out that there exists a paper by Briand and Confortola
(see [2]) with related results based on similar techniques. The two studies were carried out
simultaneously and in a completely independent way.

The paper is organized as follows. In section 1 we fix the notation and recall some process
properties needed in the proofs of the main body of the paper. Section 2 contains the main
results on classical differentiability. In sections 3, 4 and 5 we give a priori bounds for classes of
non-linear BSDEs. Section 6 contains the proofs of the theorems stated in Section 2. Section
7 is devoted to the application of the proven results to the forward-backward SDE setting. In
Section 8 we state and prove the Malliavin differentiability results.

1

Preliminaries

Throughout this paper let (Ω, F, P ) be a complete probability space and W = (Wt )t≥0 a
d−dimensional Brownian motion. Let {Ft }t≥0 denote the natural filtration generated by W ,
augmented by the P −null sets of F.
Let T > 0, ξ be an FT -measurable random variable and f : Ω × [0, T ] × R × Rd → R. We will
consider Backward Stochastic Differential Equations (BSDEs) of the form
Z T
Z T
Zt dWt .
(1)

f (t, Yt , Zt )dt −
Yt = ξ +
t

t

As usual we will call ξ the terminal condition and the function f the generator of the BSDE
(1). A solution consists of a pair (Y, Z) of adapted processes such that (1) is satisfied. To be
RT
RT
P RT
correct we should write t hZt , dWt i or di=1 t Zsi dWsi instead of t Zt dWt , since W and Z
are d−dimensional vectors; but for simplicity we use this notation as it is without ambiguity.
It is important to know which process spaces the solution of a BSDE belongs to. We therefore
introduce the following notation for the spaces we will frequently use. Let p ∈ [1, ∞]. Then, for
m ∈ N∗
• Lp (Rm ) is the space of all progressively measurable processes (Xt )t∈[0,T ] with values in Rm
´p/2
³R
T

] < ∞.
such that kXt kpLp = E[ 0 |Xs |2 ds

• Rp (Rm ) is the space of all measurable processes (Xt )t∈[0,T ] with values in Rm such that
³
´p
kXkpRp = E[ supt∈[0,T ] |Xt | ] < ∞. Note that R∞ (Rm ) is the space of bounded measurable processes.
p

• H p (Rm ) is the class of all local martingales X such that kXkpH p = EP [hXiT2 ] < ∞.
• Lp (Rm ; P ) is the space of FT -measurable random variables X : Ω 7→ Rm such that kXkpLp =
EP [|X|p ] < ∞. We will omit reference to the space or the measure when there is no
ambiguity.
1421

Furthermore, we use the notation ∂t =

∂
∂t ,

∇ = ( ∂x∂ 1 , · · · , ∂x∂ d ) for (t, x) ∈ [0, T ] × Rd .

Suppose that the generator satisfies, for a ≥ 0 and b, c > 0
c
(2)
|f (t, x, y, z)| ≤ a(1 + b|y|) + |z|2 .
2
Kobylanski has shown in [9] that if ξ is bounded and the generator f satisfies (2), then there
exists a solution (Y, Z) ∈ R∞ × L2 . Moreover, it follows from the results in [11], that in this
case
R · the process Z is such that the stochastic integral process relative to the Brownian motion
0 ZdW is a so-called Bounded Mean Oscillation (BMO) martingale.
Since the BMO property is crucial for the proofs we present in this paper we recall its definition
and some of its basic properties. For an overview on BMO martingales see [8].
Definition 1.1 (BMO). Let M be a uniformly integrable (Ft )-martingale satisfying M0 = 0.
For 1 ≤ p < ∞ set
i´1/p
³ h
.
sup
E |M∞ − Mτ |p |Fτ
kM kBM Op =
τ stopping time

The normed linear space {M : kM kBM Op < ∞} with norm kM kBM Op is denoted by BM Op . If
we want to stress the measure P we are referring to we will write BMO(P ).
It can be shown that for any p, q ∈ [1, ∞] we have BMOp = BMOq (see [8]). Therefore we will
often omit the index and simply write BMO for the set of BMO martingales.
In the following Lemma we state the properties of BMO martingales we will frequently use.
Lemma 1.2 (Properties of BMO martingales).
1) Given a BMO martingale M with quadratic variation hM i, its stochastic exponential
1
E(M )T = exp{MT − hM iT }
2
has integral 1, and thus the measure defined by dQ = E(M )T dP is a probability measure.
2) Let M be a BMO martingale relative to the measure P . Then the process M̂ = M − hM i
is a BMO martingale relative to the measure Q (see Theorem 3.3 in [8]).
3) For any BMO Martingale, it is always possible to find a p > 1 such that E(M ) ∈ Lp ,
i.e. if kM kBM O2 < Ψ(p), then E(M ) ∈ Lp (see for example Theorem 3.1 [8]). Where
o1
n
2
2x−1
− 1 for all 1 < x < ∞ and verifies limx→1+ Ψ(x) = ∞ and
Ψ(x) = 1 + x12 log 2(x−1)
limx→∞ Ψ(x) = 0.

2

Differentiability of quadratic BSDEs in the classical sense

Suppose that the terminal condition and the generator of a quadratic BSDE depend on the
Euclidean parameter set Rn for some n ∈ N∗ . We will show that the smoothness of the terminal
condition and the generator is transferred to the solution of the BSDE
Z T
Z T
x
x
f (s, x, Ysx , Zsx )ds, x ∈ Rn ,
(3)
Zs dWs +
Yt = ξ(x) −
t

t

where terminal condition and generator are subject to the following conditions
1422

(C1) f : Ω×[0, T ]×Rn ×R×Rd → R is an adapted measurable function such that f (ω, t, x, y, z) =
l(ω, t, x, y, z) + α|z|2 , where l(ω, t, x, y, z) is globally Lipschitz in (y, z) and continuously
differentiable in (x, y, z); for all r ≥ 1 and (t, y, z) the mapping Rd → Lr , x 7→ l(ω, t, x, y, z)
is differentiable and for all x ∈ Rn
´r i
h³ Z T
′
x
x
x
x
P
|l(s,
x
,
Y
,
Z
)
−
l(s,
x,
Y
,
Z
)|ds
=0
and
lim
E
s
s
s
s
x′ →x
0
´r i
h³ Z T ∂
∂
′
x′
x′
x
x
P
l(s,
x
,
Y
,
Z
)
−
l(s,
x,
Y
,
Z
)|ds
= 0,
|
lim
E
s
s
s
s
x′ →x
∂x
∂x
0
(C2) the random variables ξ(x) are FT −adapted and for every compact set K ⊂ Rn there exists
a constant c ∈ R such that supx∈K kξ(x)k∞ ≤ c; for all p ≥ 1 the mapping Rn → Lp ,
x 7→ ξ(x) is differentiable with derivative ∇ξ.
If (C1) and (C2) are satisfied, then there exists a unique solution (Y x , Z x ) of Equation (3). This
follows from Theorems 2.3 and 2.6 in [9]. We will establish two differentiability results for the
pair (Y x , Z x ) in the variable x. We first consider differentiability of the vector valued map
x 7→ (Y x , Z x )
with respect to the Banach space topology defined on Rp (R1 ) × Lp (Rd ). This will be stated in
Theorem 2.1. A slightly more stringent result will be obtained in the subsequent Theorem 2.2.
Here, we consider pathwise differentiability of the maps
x 7→ (Ytx (ω), Ztx (ω))
in the usual sense, for almost all pairs (ω, t). In both cases, the derivatives will be identified
with (∇Y x , ∇Z x ) solving the BSDE
RT
∇Ytx = ∇ξ(x) − t ∇Zsx dWs
RT
+ t [∂x l(s, x, Ysx , Zsx ) + ∂y l(s, x, Ysx , Zsx )∇Ysx + ∂z l(s, x, Ysx , Zsx )∇Zsx + 2αZsx ∇Zsx ] ds.
(4)
We emphasize at this place that it is not immediate that this BSDE possesses a solution. In
fact, without considering it as a component of a system of BSDEs also containing the original
quadratic one, it can only be seen as a linear BSDE with global, but random (and not bounded)
Lipschitz constants.
Theorem 2.1. Assume (C1) and (C2). Then for all p ≥ 1, the function Rn → Rp (R1 )×Lp (Rd ),
x 7→ (Y x , Z x ), is differentiable, and the derivative is a solution of the BSDE (4).
Under slightly stronger conditions one can show the existence of a modification of Y x which is
P -a.s. differentiable as a mapping from Rn to R. Let ei = (0, . . . , 1, . . . , 0) be the unit vector
in Rn where the ith component is 1 and all the other components 0. For x ∈ Rn and h 6= 0 let
ζ(x, h, ei ) = h1 [ξ(x + hei ) − ξ(x)]. For the existence of differentiable modifications we will assume
that
(C3) for all p ≥ 1 there exists a constant C > 0 such that for all i ∈ {1, . . . , n}, x, x′ ∈ Rn and
h, h′ ∈ R \ {0}
h
i
E |ξ(x + hei ) − ξ(x′ + h′ ei )|2p + |ζ(x, h, ei ) − ζ(x′ , h′ , ei )|2p ≤ C(|x − x′ |2 + |h − h′ |2 )p .
1423

Theorem 2.2. Suppose, in addition to the assumptions of Theorem 2.1, that (C3) is satisfied
and that l(t, x, y, z) and its derivatives are globally Lipschitz continuous in (x, y, z). Then there
exists a function Ω × [0, T ] × Rn → R1+d , (ω, t, x) 7→ (Ytx , Ztx )(ω), such that for almost all ω,
Ytx is continuous in t and continuously differentiable in x, and for all x, (Ytx , Ztx ) is a solution
of (3).

3

Moment estimates for linear BSDEs with stochastic Lipschitz
generators

By formally deriving a quadratic BSDE with generator satisfying (C1) and (C2) we obtain a
linear BSDE with a stochastic Lipschitz continuous generator. The Lipschitz constant depends
on the second component of the solution of the original BSDE. In order to show differentiability, we start deriving a priori estimates for this type of linear BSDE with stochastic Lipschitz
continuous generator. For this purpose, we first need to show that the moments of the solution
can be effectively controlled. Therefore this section is devoted to moment estimates of solutions
of BSDEs of the form
Z T
Z T
[l(s, Us , Vs ) + Hs Vs + As ] ds.
(5)
Vs dWs +
Ut = ζ −
t

t

We will make the following assumptions concerning the drivers:
(A1) For all p ≥ 1, ζ is FT −adapted and we have ζ ∈ Lp (R1 ),

(A2) H is a predictable Rd −valued process, integrable with respect to W , such that
a BMO-martingale,

R

HdW is

(A3) l : Ω × [0, T ] × R × Rd → R is such that for all (u, v), the process l(ω, t, u, v) is (Ft )predictable and there exists a constant M > 0 such that for all (ω, t, u, v),
|l(ω, t, u, v)| ≤ M (|u| + |v|),
(A4) A is a measurable adapted process such that for all p ≥ 1 we have E[
Moreover, we assume that (U, V ) is a solution of (5) satisfying
RT
RT
1
(A5) [ 0 Us2 |Vs |2 ds] 2 and 0 |Us As |ds are p-integrable for all p ≥ 1.

¡RT
0

¢p
|As |ds ] < ∞.

Under the assumptions (A1), (A2), (A3), (A4) and (A5) one obtains the following estimates.

Theorem 3.1
Assume that (A1)-(A5) are satisfied. Let p > 1 and r > 1
R (Moment estimates).
r
such that E( HdW )T ∈ LR (P ). Then there exists a constant C > 0, depending only on p, T ,
M and the BMO-norm of HdW ), such that with the conjugate exponent q of r we have
h
´2pq2 i 12
´p i
i
h
³Z T
h³ Z T
q
P
2p
2
P
P
2pq 2
.
(6)
E
sup |Ut |
|As |ds
|Vs | ds
≤ CE |ζ|
+E
+
t∈[0,T ]

Moreover we have
Z
Z T
|Us |2 ds] + EP [
EP [
0

0

0

T
0

h
³Z
2
|Vs |2 ds] ≤ CEP |ζ|2q +
1424

T
0

´q2 i 12
q
|As |2 ds
.

(7)

The proof is divided into several steps. First let β > 0 and observe that by applying Itô’s formula
to eβt Ut2 we obtain
Z T
βt 2
βT 2
eβs Us Vs dWs
e U t = e UT − 2
t
Z T
¢
¤
¡
£
eβs − βUs2 + 2Us l(s, Us , Vs ) + Hs Vs + As − |Vs |2 ds.
+
t

By (A2), the auxiliary
R t measure defined by Q = E(H · W )T · P is in fact a probability measure.
Then Ŵt = Wt − 0 Hs ds is a Q-Brownian motion, and
Z T
eβs Us Vs dŴs
eβt Ut2 ≤ eβT UT2 − 2
t
Z T
£
¤
eβs (−β + 2M )Us2 + 2M |Us ||Vs | − |Vs |2 + |Us As | ds
+
t

M2

By choosing β =
+ 2M , we obtain
Z T
Z
eβt Ut2 +
eβs (M |Us | − |Vs |)2 ds ≤ eβT UT2 − 2
t

T

eβs Us Vs dŴs +

t

Z

T

eβs |Us As |ds. (8)

t

We therefore first prove moment estimates under the measure Q.
Lemma 3.2. For all p > 1 there exists a constant C, depending only on p, T and M , such that
"µZ
¶p #
h
´2p i
i
h
³Z T
T
Q
2p
Q
2
Q
2p
E
sup |Ut |
|As |ds
.
(9)
≤ CE |ζ| +
+E
|Vs | ds
0

0

t∈[0,T ]

Moreover we have
hZ
EQ

T

·Z
i
|Us |2 ds + EQ

T

0

0

¸
Z
h
|Vs |2 ds ≤ CEQ |ζ|2 +

0

T

i
|As |2 ds .

(10)

Proof. Throughout this proof let C1 , C2 , . . ., be constants depending only on p, T and M .
Inequality (8) implies
βt

e

Ut2

βT

≤ e

UT2

−2

Z

T

βs

e Us Vs dŴs +

Z

T

eβs |Us As |ds,

(11)

t

t

R
RT
and (A5) together with the existence of the rth moment for E( HdW )T yield 0 Us2 |Vs |2 ds ∈
L1 (Q). Hence, since eβt Ut2 is (Ft )-adapted,
Z T
i
h
βt 2
βT Q
2
(12)
e Ut ≤ e E |ζ| +
eβs |Us As |ds|Ft .
t

Integrating both sides and using Young’s inequality, we obtain
Z T
Z T
2
Q
2
Q
|Us As |ds]
Us ds] ≤ C1 E [|ζ| +
E [
0
0
Z T
Z
1 Q T 2
2
Q 2
|As | ds] + E [
≤ C1 E [ζ + 2C1
Us ds],
2
0
0
1425

and hence
Q

E [

T

Z

0

Us2 ds]

Q

2

≤ C2 E [|ζ| +

Z

T

|As |2 ds].

(13)

0

Inequality (12), (A5) and Doob’s Lp inequality imply for p > 1
Z T
´p i
h³
Q
2p
Q
2
E [ sup |Ut | ] ≤ C3 E
|Us As |ds
|ζ| +
0

t∈[0,T ]

Q

≤ C4 E

h

2p

|ζ|

+

³

sup |Ut |

Z

T

´p i
|As |ds
.

0

t∈[0,T ]

RT
RT
By Young’s inequality, (supt∈[0,T ] |Ut |p )( 0 |As |ds)p ≤ 2C1 4 supt∈[0,T ] |Ut |2p + 2C4 ( 0 |As |ds)2p ,
and hence
´2p i
h
³Z T
|As |ds
.
(14)
EQ [ sup |Ut |2p ] ≤ C5 EQ |ζ|2p +
0

t∈[0,T ]

In order to complete the proof, note that (8) implies
Z T
eβs |Vs |2 ds
t
Z
Z T
Z T
βs
βs
βT 2
e M |Us ||Vs |ds +
e Us Vs dŴs + 2
≤ e UT − 2

eβs |Us ||As |ds.

(15)

t

t

t

T

RT
RT
RT
By Young’s inequality, 2 t eβs M |Us ||Vs |ds ≤ 21 t eβs |Vs |2 ds + 8M 2 t eβs Us2 ds, and hence
Z T
Z
Z T
i
i
1 Q h T βs
2
Q βT 2
2
βs 2
e |Vs | ds ≤ E [e UT + 8M
E
eβs Us2 + eβs |As |2 ds
e Us ds +
2
0
0
0
Z T
|As |2 ds]
≤ C6 EQ [ζ 2 +
0

which, combined with (13) leads to the desired Inequality (10).
Equation (15), Young’s inequality, Doob’s Lp -inequality and the Burkholder-Davis-Gundy inequality imply
"µZ
¶p #
T
eβs |Vs |2 ds
EQ
0

´p ³ Z
h
³
≤ C7 EQ |ζ|2p + T sup eβt Ut2 +

T

eβs Us Vs dŴs

0

t∈[0,T ]

h
³Z
Q
2p
βtp
2p
≤ C8 E |ζ| + sup e |Ut | +
t∈[0,T ]

T

0

+ sup |Ut |2p +
t∈[0,T ]

0

T

³Z

T

0

´p i
eβs |Us ||As |ds

t∈[0,T ]

t∈[0,T ]

³Z

+

´p
³Z
2
2βs 2
2
p
e Us |Vs | ds + sup |Ut |

h
³
´p ³ Z
2
≤ C8 EQ |ζ|2p + sup eβtp |Ut |2p + sup eβt Ut2
t∈[0,T ]

´p

´2p i
eβs |As |ds
1426

T

0

´p
2
eβs |Vs |2 ds

0

T

´p i
eβs |As |ds

By Young’s inequality,
³
´p ³ Z
βt 2 2
sup e Ut

0

t∈[0,T ]

T

Z
´p
´p
³
´p
1 ³ T βs
2
βt 2
+
e |Vs |2 ds ,
e |Vs | ds ≤ 2C8 sup e Ut
2C8 0
t∈[0,T ]
βs

2

which implies
EQ

"µZ

T
0

¶p #
h
³Z
Q
2p
2p
βs
2
≤ C9 E |ζ| + sup |Ut | +
e |Vs | ds
t∈[0,T ]

0

T

´2p i
|As |ds
.

h
³Z
Q
2p
≤ C10 E |ζ| +

Thus, with Inequality (14), the proof is complete.

0

T

´2p i
|As |ds

R
Proof of Theorem 3.1. Notice that by the second statement of Lemma 1.2, the process HdŴ =
R
R
R
R·
HdW − 0 Hs2 ds belongs to BMO(Q), and hence − HdŴ also. Moreover, E( HdW )−1 =
R
E(− HdŴ ). Consequently, by the third statement of Lemma 1.2, there exists an r > 1
r
such that E(H · W )T ∈ Lr (P ) and E(H · W )−1
T ∈ L (Q). Throughout let D = max{kE(H ·
W )T kLr (P ) , kE(H ·W )−1
T kLr (Q) }. Hölder’s inequality and Lemma 3.2 imply that for the conjugate
exponent q of r we have
1

EP [ sup |Us |2p ] = EQ [E(H · W )−1
sup |Us |2p ] ≤ DEQ [ sup |Us |2pq ] q
T
s∈[0,T ]

s∈[0,T ]

s∈[0,T ]

´2pq i 1
T
q
Q
2pq
|As |ds
≤ C1 D E |ζ| +
0
´2pq ´ 1
³
³Z T
|As |ds
]q
= C1 D EP [E(H · W )T |ζ|2pq +
0
´2pq2 1
³Z T
1+q
P
2pq 2
q
] q2 ,
|As |ds
+
≤ C2 D E [|ζ|
³Z

h

0

R
where C1 , C2 represent constants depending on p, M, T and the BM O norm of HdW . Sim´2pq2 ´ 1
³R
1+q
RT
2
T
ilarly, with another constant C3 , EP [ 0 |Vs |2p ds] ≤ C3 D q EP [|ζ|2pq +
|A
|ds
] q2 ,
s
0
and hence (6). By applying the same arguments to (10) we finally get (7).

4

A priori estimates for linear BSDEs with stochastic Lipschitz
constants

In this section we shall derive a priori estimates for the variation of the linear BSDEs that play
the role of good candidates for the derivatives of our original BSDE. These will be used to prove
continuous differentiability of the smoothly parametrized solution in subsequent sections. Let
(ζ, H, l1 , A) and (ζ ′ , H ′ , l2 , A′ ) be parameters satisfying the properties (A1), (A2), (A3) and (A4)
of Section 3 and suppose that l1 and l2 are globally Lipschitz continuous and differentiable in
(u, v). Let (U, V ) resp. (U ′ , V ′ ) be solutions of the linear BSDE
Z T
Z T
[l1 (s, Us , Vs ) + Hs Vs + As ]ds
(16)
Vs dWs +
Ut = ζ −
t

t

1427

resp.
Ut′ = ζ ′ −

Z

t

T

Vs′ dWs +

Z

T

[l2 (s, Us′ , Vs′ ) + Hs′ Vs′ + A′s ]ds

t

both satisfying property (A5). Throughout let δUt = Ut − Ut′ , δVt = Vt − Vt′ , δζ = ζ − ζ ′ ,
δAt = At − A′t and δl(t, u, v) = l1 (t, u, v) − l2 (t, u, v).
RT
Theorem 4.1 (A priori estimates). Suppose we have for all β ≥ 1, 0 δUs2 |δVs |2 ds ∈ Lβ (P )
RT
R
and 0 |δUs δAs |ds ∈ Lβ (P ). Let p ≥ 1 and r > 1 such that E( H ′ dW )T ∈ RLr (P ). Then there
exists a constant C > 0, depending only on p, T , M and the BMO-norm of H ′ dW , such that
with the conjugate exponent q of r we have
´p i
h
i
h³ Z T
P
2p
P
E
|δVs |2 ds
sup |δUt |
+E
0

t∈[0,T ]

´2pq2 i 12
n h
³Z T
q
P
2pq 2
|δl(s, Us′ , Vs′ ) + δAs |ds
≤ C E |δζ|
+
0
Z T
Z
´2pq2 i 12 o
¢2pq2 12 P h³ T
¡
2
2q
|As |ds
]) 2q E
|Hs − Hs′ |2 ds
+ (EP [|ζ|2pq +
0

0

We proceed in the same spirit as in the preceding section. Before proving Theorem 4.1 we
will show
auxiliary probability measure Q defined by
R a priori estimates with respect toR the
t
Q = E( H ′ dW )T · P . Note that Ŵt = Wt − 0 Hs′ ds is a Q-Brownian motion.
Lemma 4.2. Let p > 1. There exists a constant C > 0, depending only on p, T and M , such
that

EQ

h

sup |δUt |2p
t∈[0,T ]

i

T

n h
³Z
≤ C EQ |δζ|2p +

0

µ h
Z
¡
Q
2p
+ E |ζ| +

¶1
Z
¢2p i 2 Q h³
|As |ds
E

T

0

Q

E [

³Z

T
0

´2p i
|δl(s, Us′ , Vs′ ) + δAs |ds
T

0

n h
´p
³Z
Q
2p
|δVs | ds ] ≤ C E |δζ| +

T

2

0

µ h
Z
¡
+ EQ |ζ|2p +

0

T

´2p i 1 o
2
|Hs − Hs′ |2 ds
,

´2p i
|δl(s, Us′ , Vs′ ) + δAs |ds

¶1
Z
¢2p i 2 Q h³
|As |ds
E

0

T

1428

(18)

´2p i 1 o
2
|Hs − Hs′ |2 ds
.

Proof. The difference δU satisfies
Z T
Z T
[(Hs Vs − Hs′ Vs′ ) + l1 (s, Us , Vs ) − l2 (s, Us′ , Vs′ ) + δAs ]ds
δVs dWs +
δUt = δζ −
t
t
Z T
Z T
[l1 (s, Us′ , Vs′ ) − l2 (s, Us′ , Vs′ ) + Hs′ δVs + δAs ]ds
δVs dWs +
= δζ −
t
t
Z T
[(Hs − Hs′ )Vs + l1 (s, Us , Vs ) − l1 (s, Us′ , Vs′ )]ds.
+
t

(17)

Let β > 0. Applying Itô’s formula to eβt δUt2 , t ≥ 0, yields the equation
βt

e

δUt2

βT

δUT2

= e

Z

−2

T

Z

T

βs

e δUs δVs dWs + 2
t

Z

T
t

eβs δUs Hs′ δVs ds

i
h
¡
¢
eβs − βδUs2 − |δVs |2 + 2 l1 (s, Us , Vs ) − l1 (s, Us′ , Vs′ ) δUs ds
t
Z T
Z T
eβs δUs (δls + δAs )ds,
eβs δUs (Hs − Hs′ )Vs ds + 2
+2

+

(19)

t

t

where δls = l1 (s, Us′ , Vs′ ) − l2 (s, Us′ , Vs′ ). Using the Lipschitz property of l1 we obtain
βt

e

δUt2

βT

≤ e

+2

δUT2
Z

t

T

Z

T

h
i
eβs (−β + 2M )δUs2 − |δVs |2 + 2M |δUs | |δVs | ds
t
Z T
βs
′
e δUs [(Hs − Hs )Vs + δls + δAs ]ds − 2
eβs δUs δVs dŴs .
+

t

If β = (M 2 + 2M ), then
eβt δUt2 +

Z

T
t

Z T
eβs (M |δUs | − |δVs |)2 ds ≤ eβT δUT2 + 2
eβs δUs [(Hs − Hs′ )Vs + δls + δAs ]ds
t
Z T
eβs δUs δVs dŴs .
(20)
−2
t

We will now derive the desired estimates from Equation (20). First observe that by taking
conditional expectations, we get
·
¸
Z T
¯
βs
′
βt
2
βT Q
2
¯
e δUs [(Hs − Hs )Vs + δls + δAs ]ds Ft .
e δUt ≤ e E δUT + 2
t

Let p > 1. Then for some constants C1 , C2 , . . ., depending on p, T and M , we obtain
¾
½³
Z
¯ ¤´p
£
¤
£ T
2p
′
Q
2
¯
sup |δUt |
≤ C1 sup
|δUs [(Hs − Hs )Vs + δls + δAs ]|ds Ft
E |δUT | |Ft + E
t∈[0,T ]

0

t∈[0,T ]

and by Doob’s Lp inequality we get
½ h
³Z
Q
2p
Q
2p
E [ sup |δUt | ] ≤ C2 E |δUT | ] + E[

0

t∈[0,T ]

1429

T

|δUs [(Hs −

Hs′ )Vs

´p ¾
+ δls + δAs ]|ds ] .

By using Young’s and Hölder’s inequalities we have
´p i
h³ Z T
|δUs [(Hs − Hs′ )Vs + δls + δAs ]|ds
EQ
0
(
·³ Z T
´p ³ Z
2
Q
p
′ 2
≤ C3 E
sup |δUt |
|Hs − Hs | ds
0

t∈[0,T ]

≤

1 Q
E
2C4

h

sup |δUt |2p
t∈[0,T ]

T

´p ³ Z
2
2
|Vs | ds +

0

i

T
0

)
´p ¸
|δls + δAs |ds

´2p i
´p ³ Z T
´p ³ Z T
|δls + δAs |ds
|Vs |2 ds) +
|Hs − Hs′ |2 ds
0
0
0
´2p
n ³Z T
i
1 Qh
≤
|δls + δAs |ds
E
sup |δUt |2p + C5 EQ
2C4
0
t∈[0,T ]
´2p i 1 h³ Z T
h³ Z T
´2p i 1 o
2 Q
2
′ 2
Q
2
(Hs − Hs ) ds
+E
|Vs | ds
E
.
+4C4 EQ

h³ Z

T

0

(21)

0

Therefore, we may further estimate

Z
n
¡
Q
2p
Q
E [ sup |δUt | ] ≤ C6 E [ |δζ| ] + E [
Q

T

2p

t∈[0,T ]

Q

+E

h³ Z

T

|Hs −
0

0

¢2p
|δls + δAs |ds ]

´2p i 1

Hs′ |2 ds

2

Q

E

h³ Z

T

0

´2p i 1 o
2
|Vs |2 ds
.

´2p i 1
h³ R
h
i1
R
2
T
2 ds
Q |ζ|2p + ( T |A |ds)2p 2 < ∞, which implies
Due to Lemma 3.2, EQ
|V
|
≤
C
E
s
7
s
0
0
the δUs part of Inequality (17).
In order to prove the second inequality, note that (20) also implies
Z T
Z T
βs
2
βT
2
eβs δUs [(Hs − Hs′ )Vs + δls + δAs ]ds
e |δVs | ds ≤ e δUT + 2
t
t
Z T
Z T
βs
eβs δUs δVs dŴs .
e |δUs | |δVs |ds − 2
+2M
t

t

Equation (22), Doob’s Lp -inequality and the Burkholder-Davis-Gundy inequality imply
½
Z T
´p
i
³Z T
h
´p
³Z T
2
2
2
Q
2p
Q
2p
2
Q
δUs δ|Vs | ds ]
|δUs | ds + E [
|δVs | ds ] ≤ C8 (E |δζ| +
E [
0
0
0
¾
Z T
¢p
¡
|δUs [(Hs − Hs′ )Vs + δls + δAs ]|ds ] .
+EQ [
0

Consequently, Young’s inequality allows to deduce
½ h
Z
´p
³Z T
Q
2p
2
Q
|δVs | ds ] ≤ C9 E |δζ| +
E [

0

0

Q

+E [

¡

Z

T

T

i
|δUs |2p ds + EQ [ sup |δUt |2p ]

|δUs [(Hs −

0

1430

t∈[0,T ]

Hs′ )Vs

¾
¢p
+ δls + δAs ]|ds ] .

(22)

Finally, (17) and (21) imply
Q

E [

³Z

T
0

T

´2p i
|δls + δAs |ds
0
Z T
h³ Z
2p 12 Q
Q
2p
|As |ds) ] E
+C10 E [|ζ| + (

´p
h
³Z
Q
2p
|δVs | ds ] ≤ C10 E |δζ| +
2

0

T

0

´2p i 1
2
|Hs − Hs′ |2 ds

and hence the proof is complete.
Proof of Theorem 4.1. This can be deduced from Lemma 4.2 with arguments similar to those
of Theorem 3.1. We just have to invoke Lemma 1.2.

5

A priori estimates for quadratic BSDEs

Consider the two quadratic BSDEs
Yt = ξ −

T

Z

Zs dWs +

t

and
Yt′ = ξ ′ −

Z

T
t

Zs′ dWs +

Z

Z

T
t

T

t

[l1 (s, Ys , Zs ) + αZs2 ]ds

(23)

[l2 (s, Ys′ , Zs′ ) + α(Zs′ )2 ]ds,

(24)

where ξ and ξ ′ are two bounded FT -measurable random variables, and l1 and l2 are globally
Lipschitz and differentiable in (y, z). Put now δYt = Yt − Yt′ , δZt = Zt − Zt′ , δξ = ξ − ξ ′ and
δl = l1 − l2 . The a priori estimates we shall prove next will serve for establishing (moment)
smoothness of the solution of the quadratic BSDE with respect to a parameter on which the
terminal
smoothly. Note first that by boundedness of ξ and ξ ′ we have that
R variable Rdepends
both ZdW and Z ′ dW are BM O martingales, so that we may again invoke the key Lemma
1.2.
RT
Theorem 5.1. Suppose that for all β ≥ 1 we have 0 |δl(s, Ys , Zs )|ds ∈ Lβ (P ). Let p > 1
and choose r > 1 such that E(α(Zs + Zs′ ) · W )T ∈ LRr (P ). Then there exists a constant C > 0,
depending only on p, T , M and the BMO-norm of (α (Zs +Zs′ )dW ), such that with the conjugate
exponent q of r we have
"µ Z
¶p #
h
i
T
EP sup |δYt |2p + EP
|δZs |2 ds
0

t∈[0,T ]

µ h
Z
P
2pq 2
≤ C E |δξ|
+(

T

2pq 2

|δl(s, Ys , Zs )|ds)
0

i¶ q12

.

Moreover we have
P

E [

Z

0

T

2

P

|δYs | ds] + E

£

Z

T
0

µ ·
³Z
¤
2
|δZs | ds ≤ C EP |δξ|2q +

T

2

1431

0

´2q2 ¸¶ q12
|δl(s, Ys , Zs )|ds
.

We give only a sketch of the proof since the arguments are very similar to the ones used in the
proofs in Sections 3 and 4.
First observe that
Z
Z T
δZs dWs +
δYt = δξ −

T

t

t

[l1 (s, Ys , Zs ) − l1 (s, Ys′ , Zs′ ) + δl(s, Ys′ , Zs′ ) + α(Zs + Zs′ )δZs ]ds.

By applying Itô’s formula to eβt |δYt |2 we obtain
eβt |δYt |2 − eβT |δYT |2
Z T
Z T
³
´
eβs (β|δYs |2 + |δZs |2 )ds
=2
eβs δYs l1 (s, Ys , Zs ) − l1 (s, Ys′ , Zs′ ) + δl(s, Ys′ , Zs′ ) ds − 2
t
t
Z T
Z T
eβs δYs δZs dWs .
(25)
eβs δYs α(Zs + Zs′ )δZs ds − 2
+2
t

t

We Rstart with a priori estimates under the auxiliary
probability measure Q defined by Q =
Rt
E(α (Zs + Zs′ )dW ) · P . Note that W̃t = Wt − 0 α(Zs + Zs′ )ds is a Q-Brownian motion.
Let β > 0. Equality (25) and the Lipschitz property of l1 yield
Z T
Z T
eβt |δYt |2 ≤ eβT |δξ|2 − 2
eβs δYs δZs dW̃s + 2
eβs δYs δl(s, Ys′ , Zs′ )ds
t
t
Z T
³
´
eβs (−β + 2M )|δYs |2 − |δZs |2 + 2M |δYs ||δZs | ds.
+
t

By choosing β =

M2

+ 2M we obtain the general inequality
Z T
βt
2
eβs (|δZs | − M |δYs |)2 ds
e |δYt | +
t
Z
Z T
βs
′
′
βT
2
e δYs δl(s, Ys , Zs )ds − 2
≤ e |δξ| +
t

T

eβs δYs δZs dW̃s .

(26)

t

Rt
Note that the process 0 eβs δYs δZs dW̃s is a strict martingale because δYs is bounded and
(δZ · W̃ ) is BMO relative to Q.

Notice that Equation (26) is of similar but simpler form than Equation (20). This is because
the (Hs − Hs′ ) term in (20) has been completely absorbed by the Girsanov measure change. As
a consequence, following the proof of Lemma 4.2, we obtain the following estimates:

Lemma 5.2. For all p > 1 there exists a constant C > 0, depending only on p, M and T , such
that
"µZ
¶p #
Z T
i
h
h
i
T
Q
2p
Q
2p
2
Q
|δl(s, Ys , Zs )|ds)2p .
≤ C E |δξ| + (
E
sup |δYt |
|δZs | ds
+E
Moreover we have
Z
Z T
£
2
Q
Q
|δYt | ds] + E
E [
0

0

0

t∈[0,T ]

0

T

·
³Z
¤
2
Q
|δξ| +
|δZs | ds ≤ CE

T

2

0

´2 ¸
.
|δl(s, Ys , Zs )|ds

(27)

Proof of Theorem 5.1. The arguments are similar to those of the proof of Theorem 3.1. Just
make use of Lemma 1.2.
1432

6

Proof of the differentiability

We now approach the problem of differentiability of the solutions of a quadratic BSDEs with
respect to a vector parameter on which the terminal condition depends differentiably. We start
with the proof of the weaker property of Theorem 2.1. Our line of reasoning will be somewhat
different from the one used for instance by Kunita [10] in the proof of the diffeomorphism property of smooth flows of solutions of stochastic differential equations. He starts with formally
differentiating the stochastic differential equation, and showing that the resulting equation possesses a solution. The latter is then used explicitly in moment estimates for its deviation from
difference quotients of the original equation. The estimates are then used to prove pathwise
convergence of the difference quotients to the solution of the differentiated SDE. We emphasize
that in our proofs, we will have to derive moment estimates for differences of difference quotients
instead. They will allow us to show the existence of a derivative process in a Cauchy sequence
type argument using the completeness of underlying vector spaces, which of course will be the
solution process of the formally differentiated BSDE. So our procedure contains the statement
of the existence of a solution of the latter as a by-product of the proof of the Theorem 2.1. It is
not already available as a good candidate for the derivative process, since, as we stated earlier,
the formally differentiated BSDE is a globally Lipschitz one with random Lipschitz constants
for which the classical existence theorems do not immediately apply. Throughout assume that
f (t, x, y, z) = l(t, x, y, z) + α|z|2 and ξ(x) satisfy (C1) and (C2) respectively.
For all x ∈ Rn let (Ytx , Ztx ) be a solution of the BSDE (3). It is known that the solution is
unique and that (Y x , Z x ) ∈ R∞ (R1 ) × L2 (Rd ) (see [9]).
It follows from Lemma 1 in [11] that there exists a constant D > 0 such that for all x ∈ Rn
we have k(Z x · W )T kBM O2 ≤ D. Now let r > 1 be such that Ψ(r) > 2αD (see property 3) of
Lemma 1.2), and denote as before by q the conjugate exponent of r.
Proof of Theorem 2.1. To simplify notation we assume that M > 0 is a constant such that ξ(x),
x ∈ Rn , and the derivatives of l in (y, z) are all bounded by M . We first show that all the partial
derivatives of Y and Z exist. Let x ∈ Rn and ei = (0, . . . , 1, . . . 0) be the unit vector in Rn
the ith component of which is 1 and all the others 0. For all h 6= 0, let Uth = h1 (Ytx+ei h − Ytx ),
Vth = h1 (Ztx+hei − Ztx ) and ζ h = h1 (ξ(x + hei ) − ξ(x)).
Let p > 1. Note that for all h 6= 0
Z T
Z T
1
h
h
h
Vs dWs +
Ut = ζ −
[f (s, x + hei , Ysx+hei , Zsx+hei ) − f (s, x, Ysx , Zsx )]ds.
t h
t
To simplify the last term we use a line integral transformation. For all (ω, t) ∈ Ω × R+ let sx,h =
sx,h (ω, t) : [0, 1] → Rn+1+d be defined by sx,h (θ) = (x+θhei , Ytx +θ(Ytx+hei −Ytx ), Ztx +θ(Ztx+hei −
Ztx )). Though sx,h depends on i we omit to indicate this dependence for ease of notation. Note
R 1 ∂l
R 1 ∂l
x,h
that h1 s′x,h (θ) = (ei , Uth , Vth ). Moreover, Ax,h
= 0 ∂x
(s
(θ))dθ,
G
=
x,h
t
t
0 ∂y (sx,h (θ))dθ and
i
R 1 ∂l
x,h
It = 0 ∂z (sx,h (θ))dθ are (Ft )-adapted processes satisfying
1
[l(t, x + hei , Ytx+hei , Ztx+hei ) − l(t, x, Ytx , Ztx )] =
h
=
1433

Z

1

h∇l(sx,h (θ)), s′x,h (θ)idθ

0
Ax,h
t

x,h h
h
+ Gx,h
t Ut + It Vt .

and Itx,h are bounded by M as well. However,
Since the derivatives of l are bounded by M , Gx,h
t
x,h
we stress that At is not necessarily bounded. We define two random functions mx,h
s (u, v)
x,h
x,h
v)
and
m
u
+
I
(u,
v)
=
(G
and ms (u, v) from R1+d to R such that (u, v) 7→ mx,h
s
s
s
s (u, v) =
x
x
x
x
[∂y l(s, Ys , Zs )u + ∂z l(s, Ys , Zs )v]. Observe that these functions satisfy (A3) and that they are
Lipschitz continuous and differentiable in (u, v). In these terms,
Uth

h

=ζ −

Z

T
t

Vsh dWs

+

Z

T
t

h
h
x,h
x+hei
[mx,h
+ Zsx )Vsh ]ds,
s (Us , Vs ) + As + α(Zs

and thus we obtain an equation as modelled by (5). Notice that for all h, h′ 6= 0 the pairs
′
′
(U h , V h ) and (U h − U h , V h − V h ) satisfy assumptions (A4) and (A5). Therefore Theorem 4.1
x,h′
implies with δAt = Ax,h
t − At
h
i
′
E sup |Uth − Uth |2p
t∈[0,T ]

µZ
n h
h
h′ 2pq 2
≤ C E |ζ − ζ |
+
h ′
³Z
h 2pq 2
+E |ζ |
+

T

0

T

0

h
h
|mx,h
s (Us , Vs )

´2pq2 i

′
|Ax,h
s |ds

1
2q 2

−

′ ′
mxs ,h (Ush , Vsh )

h µZ
E

0

T

α

2

|Zsx+hei

−

+ δAs |ds

¶2pq2 i 1

′
Zsx+h ei |2 ds

q2

¶2pq2 i

(28)
1
2q 2

o
.

2

′

Condition (C2) implies that E[|ζ h − ζ h |2pq ] converges to zero as h, h′ → 0. Moreover, for some
′
2
open set O containing 0 we have suph′ ∈O\{0} (E|ζ h |2pq ) < ∞. Due to Condition (C1), we may
R T x,h′
2
also assume that suph′ ∈O\{0} E[( 0 |As |ds)2pq ] < ∞. Moreover,
Z
lim E(

h→0

T

0

|l(s, x + hei , Ysx , Zsx ) − l(s, x, Ysx , Zsx )|ds)β = 0

for all β ≥ 1, and therefore, with Theorem 5.1, the third summand on the right hand side of
(28) converges to zero as h, h′ → 0.
In order to prove convergence of the second summand let P ⊗ λ be the product measure of P
and the Lebesgue measure λ on [0, T ]. It follows from Theorem 5.1 that Z x+hei converges to Z x
in measure relative to P ⊗ λ. Moreover, for all t ∈ [0, T ], Ytx+hei converges to Ytx in probability.
Since the partial derivatives ly and lz are continuous and bounded, dominated convergence
´2pq2
³R
T
h , V h ) − m (U h , V h )|ds
= 0. Condition (C1) guarantees
(U
implies limh→0 EP 0 |mx,h
s
s
s
s
s
s
h
i
´2pq2
³R
′
T
= 0, and hence, limh,h′ →0 E supt∈[0,T ] |Uth − Uth |2p = 0.
limh→0 EP 0 |δAs |ds

Finally, Theorem 4.1 and an estimation similar to (28) yield
lim
E
′

h,h →0

µZ

0

T

¶p
′
|Vsh − Vsh |2 ds = 0.

Now let (hn ) be a sequence in R \ {0} converging to zero. Then, since R2p (R1 ) and L2p (Rd )
∂
Y x , and V hn to a process
are Banach spaces, the sequence U hn converges to a process ∂x
i t
∂
x
∂xi Zt with respect to the corresponding norms. By convergence term by term for the difference
1434

quotient version of the quadratic BSDE and its formal derivative, which follows from our a priori
∂
∂
estimates, we see that the pair ( ∂x
Y x , ∂x
Ztx ) is a solution of the BSDE
i t
i
Z Th
∂Zsx
∂xi l(s, x, Ysx , Zsx )
=
dWs +
∂xi
t
t
∂Y x
∂Z x
∂Z x i
+∂y l(s, x, Ysx , Zsx ) s + ∂z l(s, x, Ysx , Zsx ) s + 2αZsx s ds.
∂xi
∂xi
∂xi
i
h
∂
x |2p = 0 and
Y
Similarly to the first part one can show that limh→0 E supt∈[0,T ] |Uth − ∂x
i t
´p
³R
T
∂
= 0, and thus Rn → R2p (R1 ) × L2p (Rd ), x 7→ (Ytx , Ztx ) is
Zsx |2 ds
limh→0 E 0 |Vsh − ∂x
i
partially differentiable. The a priori estimates of Theorem 4.1 imply that the mapping x 7→
(∇Ytx , ∇Ztx ) is continuous and hence, (Ytx , Ztx ) is totally differentiable. Since differentiability
with respect to 2pth moments implies differentiability with respect to all inferior moments above
1, we have established the result.
∂Ytx
∂xi

∂
ξ(x) −
∂xi

Z

T

As a byproduct of the previous proof we obtain that for every x ∈ Rn there exists a solution
(∇Ytx , ∇Ztx ) of the BSDE (4).
We now proceed with the proof of Theorem 2.2, in which we claim pathwise continuous differentiability. To be consistent with the previous proof, we will again compare difference quotients
varying in h. To this end we need the following estimates.
Lemma 6.1. Suppose (C3) is satisfied and that l and the derivatives of l are all Lipschitz
continuous in (x, y, z). Then for all p > 1 there exists a constant C > 0, dependent only on p,
T , M and D, such that for all x, x′ ∈ Rn , h, h′ ∈ R and i ∈ {1, . . . , n},
"
¶p #
¯2p µZ T
¯
¯ x+hei
x′ +h′ ei ¯
x+hei
x′ +h′ ei 2
|Zs
− Zs
| ds
− Yt
E sup ¯Yt
¯ +
0

t∈[0,T ]

¡
¢p
≤ C |x − x′ |2 + |h − h′ |2 .

Proof. This follows from Theorem 5.1, where we put l1 (s, y, z) = l(s, x + hei , y, z), l2 (s, y, z) =
l(s, x′ + h′ ei , y, z).
The preceding Lemma immediately implies a first pathwise smoothness result in x for the process
Y x . In fact, Kolmogorov’s continuity criterion applies and yields a modification of Y x which is
continuous in x. More precisely:
Corollary 6.2. There exists a process Ŷ x such that for all (t, ω) ∈ [0, T ] × Ω, the function
x 7→ Ŷtx (ω) is continuous, and for all (t, x) we have Ŷtx = Ytx almost surely.
Let ei be a unit vector in Rn . For all x ∈ Rn and h 6= 0, let Utx,h = h1 (Ytx+hei − Ytx ), Vtx,h =
¡
¢
x+hei
∂
∂
1
−Ztx ) and ζ x,h = h1 ξ(x+hei )−ξ(x) . If h = 0, then define Utx,0 = ∂x
Y x , Vtx,0 = ∂x
Zx
h (Zt
i
i
∂
and ζ x,0 = ∂x
ξ(x). The proof of Theorem 2.2 will be based on the following result on the usual
i
difference of difference quotients. Knowing a ”good candidate“ for the derivative from Theorem
2.1 we allow h = 0 this time, by replacing the difference quotient with this candidate.

1435

Lemma 6.3. Let p > 1 and O ⊂ Rn+1 be an open set contained in a ball of radius κ. Suppose
that Condition (C3) holds and that l and the derivatives of l in (x, y, z) are Lipschitz continuous
in (x, y, z) with Lipschitz constant L > 0. Then there exists a constant C, depending on κ, L,
p, T , M , D, such that for all (x, h) and (x′ , h′ ) ∈ O,
h
i
′ ′
E sup |Utx,h − Utx ,h |2p ≤ C(|x − x′ |2 + |h − h′ |2 )p .
(29)
t∈[0,T ]

Proof. Throughout the proof, C1 , C2 , . . . are constants depending on κ, L, p, T , M , D.
Since O is bounded, (C3) implies that for every r > 1 there exists a constant C1 such that for all
(x, h) ∈ O we have E(supt∈[0,T ] |ζtx,h |2r ) < C1 . Now let sx,h , mx,h , Ax,h , Gx,h , I x,h and U x,h be
∂l
∂l
defined as in the proof of Theorem 2.2, and denote Ax,0 = ∂x
(x, Y x , Z x ), Gx,0 = ∂y
(x, Y x , Z x ),
etc. Then the estimate (29) will be deduced from the inequality
i
h
′ ′
(30)
E sup |Utx,h − Utx ,h |2p
t∈[0,T ]

n h
i 12
′ ′
2
≤ C2 E |ζ x,h − ζ x ,h |2pq q
´2pq2 i 12
h³ Z T
′ ′
q
x,h
x,h
x,h
x′ ,h′
|msx ,h (Usx,h , Vsx,h ) − mx,h
(U
,
V
)|
+
|A
−
A
|ds
+ E
s
s
s
s
s
0

Z
h ′ ′
¡
x ,h 2pq 2
+ E |ζ
|
+

T

0

¢2pq2
′ ′
|Asx ,h |ds

i

1
2q 2

h µZ
E

T

α
0

2

′
′
|Zsx +h ei

−

Zsx+hei |2 ds

¶2pq2 i

1
2q 2

o

which follows from Theorem 4.1. We first analyze the order of the convergence of
h µZ
B1 (x, x , h, h ) = E
′

′

0

T

′ ′
|mxs ,h (Usx,h , Vsx,h )

−

¶2pq2 i 1

q2

x,h
x,h
mx,h
s (Us , Vs )|ds

as h, h′ → 0.

To this end notice that
B1 (x, x′ , h, h′ ) ≤ C3

n

à µZ
E

0

T

x′ ,h′

|Gt

à µZ
+ E

0

1436

T

x,h
− Gx,h
t ||Ut |dt

x′ ,h′

|It

¶2pq2 ! q12

− Itx,h ||Vtx,h |dt

¶2pq2 ! q12 o

.

Then
Z

T

′

′

x,h
|Gxt ,h − Gx,h
t ||Ut |dt
0
Z T
′ ′
x,h
|Gxt ,h − Gx,h
≤ sup |Ut |
t |dt
0

t∈[0,T ]

≤ sup |Utx,h |

Z

≤ sup |Utx,h |

Z

t∈[0,T ]

t∈[0,T ]

T

0
T

0

µZ
Z

1

0

1

¶
|∂y l(sx′ ,h′ (θ)) − ∂y l(sx,h (θ))|dθ dt

L|sx′ ,h′ (θ) − sx,h (θ)|dθdt

0

³
′
′
′
≤ C4 sup |Utx,h | |x′ − x| + |h′ − h| + sup |Ytx − Ytx | + sup |Ytx +h ei − Ytx+hei |
t∈[0,T ]

+

T

Z

0

t∈[0,T ]

′

t∈[0,T ]

´

′

(|Ztx − Ztx | + |Ztx +h ei − Ztx+hei |)dt ,
′

and, by applying Hölder’s inequality we obtain with Lemma 6.1
à µZ
¶2pq2 ! q12
T
¡
¢p
′ ′
x,h
|Gxt ,h − Gx,h
E
≤ C5 |h − h′ |2 + |x − x′ |2 .
t ||Ut |dt
0

µ h
i2pq2 ¶ q12
¡
¢p
R T x′ ,h′
x,h
x,h
≤ C6 |h − h′ |2 + |x − x′ |2 , and so we con− It ||Vt |dt
Similarly, E 0 |It
¡
¢p
clude B1 (x, x′ , h, h′ ) ≤ C7 |h − h′ |2 + |x − x′ |2 .

By using similar arguments we get
E

·³ Z

0

T

|Ax,h
t

−

′ ′
Axt ,h |dt

´2pq2 ¸ q12

≤ E

"µZ

T

0

≤ C8 E

"µZ

Z

1

|∂x l(sx′ ,h′ (θ)) − ∂x l(sx,h (θ))|dθdt

0

T
0

Z

1

|sx′ ,h′ (θ) − sx,h (θ)|dθdt
0

¡
¢p
≤ C9 |h − h′ |2 + |x − x′ |2 .

¶2pq2 # q12

¶2pq2 # q12

Theorem 5.1 and the Lipschitz continuity of l imply
¶2pq2 i 1
h µZ T
2q 2
x′ +h′ ei 2
x+hei
− Zs
| ds
|Zs
E
ht
4
≤ C10 E |(ξ(x + hei ) − ξ(x′ + h′ ei )|4pq
´4pq4 i 14
³Z T
2q
x+hei
x+hei
′
′
x+hei
x+hei
|l(s, x + hei , Y
,Z
) − l(s, x + h ei , Y
,Z
)|ds
+
0

≤ C11 (|x − x′ |2 + |h − h′ |2 )p .
′

′

2

1

Finally, (C3) yields (E|ζ x,h − ζ x ,h |2pq ) q2 ≤ C12 (|x − x′ |2 + |h − h′ |2 )p , and hence
h
i
′ ′
E sup |Utx,h − Utx ,h |2p ≤ C13 (|x − x′ |2 + |h − h′ |2 )p .
t∈[0,T ]

1437

Proof of Theorem 2.2. To simplify notation we may assume that (29) is satisfied for O = Rn+1 .
Assume that Ytx is continuous in x (see Corollary 6.2). Lemma 6.3 and Kolmogorov’s continuity
∂
Y x = Ûtx,0
criterion imply that Utx,h has a modification Ûtx,h continuous in (x, h). Define ∂x
i t
and note that we obtain thus a continuous version of the solution of the BSDE (4). For all
(x, h) ∈ Qn+1Slet N (x, h) be a null set such that for all ω ∈
/ N (x, h) we have Ûtx,h (ω) = Utx,h (ω).
/ N the following implication
Then, N = (x,h)∈Qn+1 N (x, h) is a null set such that for all ω ∈
holds: If qk ∈ Qn and rk ∈ Q \ {0} are sequences with limk→∞ qk = x ∈ Rn and limk→∞ rk = 0,
then
1
∂ x
lim
Y .
(Ytqk +rk ei − Ytqk ) =
k→∞ rk
∂xi t
As a consequence of this and the subsequent Lemma 6.4, Ytx (ω) is continuously partially differ/ N . Since we can choose such a null set for any i ∈ {1, . . . , n}, total
entiable relative to xi if ω ∈
differentiability follows and the proof is complete.
Lemma 6.4. Let f : Rn → R be a continuous function and g : Rn → Rn a continuous vector
field. Suppose that for all sequences qk ∈ Qn with qk → x ∈ Rn and rk ∈ Q \ {0} with rk → 0 we
have
1
lim
(f (qk + rk ei ) − f (qk )) = gi (x),
k→∞ rk
where 1 ≤ i ≤ n. Then f is differentiable and ∇f = g.
Proof. To simplify notation assume that n = 1. Let xk ∈ R with xk → x ∈ R and hk ∈ R \ {0}
with hk → 0. Since f is continuous we may choose qk ∈ Q and rk ∈ Q \ {0} such that
|f (qk ) − f (xk )| ≤ |h2kk | , |f (qk + rk ) − f (xk + hk )| ≤ |h2kk | and | r1k − h1k | ≤ 21k . Then
|

1
(f (xk + hk ) − f (xk )) − g(x)|
hk
¯1 ¡
¢ ¡
¢ ¯¯
¯
≤ ¯ [ f (xk + hk ) − f (xk ) − f (qk + rk ) − f (qk ) ]¯
hk
¢
1
1
1¡
+|( − )(f (qk + rk ) − f (qk ))| + |
f (qk + rk ) − f (qk ) − g(x)|
hk
rk
rk
1
1
1
≤ 2 k + k |f (qk + rk ) − f (qk )| + | (f (qk + rk ) − f (qk )) − g(x)|
2
2
rk
→ 0,
(k → ∞),

and hence f is partially differentiable. Since the partial derivatives gi are continuous, f is also
to