Dissipative backward stochastic differential equations with locally
  Lipschitz nonlinearity
arXiv:0704.0509v1  [math.PR]  4 Apr 2007
Dissipative backward stochastic differential
equations with locally Lipschitz nonlinearity.
Fulvia Confortola
Dipartimento di Matematica, Politecnico di Milano
piazza Leonardo da Vinci 32, 20133 Milano, Italy
fulvia.confortola@polimi.it
November 4, 2018
Abstract
In this paper we study a class of backward stochastic differential
equations (BSDEs) of the form
dYt = −AYtdt−f0(t, Yt)dt−f1(t, Yt, Zt)dt+ZtdWt, 0 ≤ t ≤ T ; YT = ξ
in an infinite dimensional Hilbert space H , where the unbounded op-
erator A is sectorial and dissipative and the nonlinearity f0(t, y) is
dissipative and defined for y only taking values in a subspace of H . A
typical example is provided by the so-called polynomial nonlinearities.
Applications are given to stochastic partial differential equations and
spin systems.
Key words Backward stochastic differential equations, stochastic evo-
lution equations.
MSC classification. Primary: 60H15 Secondary: 35R60
1 Introduction
Let H,K be real separable Hilbert spaces with norms | · |H and | · |K . Let W
be a cylindrical Wiener process in K defined on a probability space (Ω,F ,P)
and let {Ft}t∈[0,T ] denote its natural augmented filtration. Let L
2(K,H) be
the Hilbert space of Hilbert-Schmidt operators from K to H.
http://arxiv.org/abs/0704.0509v1
We are interested in solving the following backward stochastic differential
equation
dYt = −AYtdt− f(t, Yt, Zt)dt+ ZtdWt, 0 ≤ t ≤ T, YT = ξ (1)
where ξ is a random variable with values in H, f(t, Yt, Zt) = f0(t, Yt) +
f1(t, Yt, Zt) and f0, f1 are given functions, and the operator A is an un-
bounded operator with domain D(A) contained in H. The unknowns are
the processes {Yt}t∈[0,T ] and {Zt}t∈[0,T ], which are required to be adapted
with respect to the filtration of the Wiener process and take values in H,
L2(K,H) respectively.
In finite dimensional framework such type of equations has been solved
by Pardoux and Peng [12] in the nonlinear case. They proved an existence
and uniqueness result for the solution of the equation (1) when A = 0, the
coefficient f(t, y, z) is Lipschitz continuous in both variables y and z, and
the data ξ and the process {f(t, 0, 0)}t∈[0,T ] are square integrable. Since this
first result, many papers were devoted to existence and uniqueness results
under weaker assumptions. In finite dimension, when A = 0, the Lipschitz
condition on the coefficient f with respect to the variable y is replaced by a
monotonicity assumption; moreover, more general growth conditions in the
variable y are formulated. Let us mention the contribution of Briand and
Carmona [1], for a study of polynomial growth in Lp with p > 2, and the
work of Pardoux [11] for an arbitrary growth. In [13] Pardoux and Rascanu
deal with a BSDE involving the subdifferential of a convex function; in
particular, one coefficient is not everywhere defined for y in Rk.
In other works the existence of the solution is proved when the data,
ξ and the process {f(t, 0, 0)}t∈[0,T ], are in L
p for p ∈ (1, 2). El Karoui,
Peng and Quenez [4] treat the case when f is Lipschitz continuous; in [2]
this result is generalized to the case of a monotone coefficient f (both for
equations on a fixed and on a random time interval) and is studied even the
case p = 1.
In the infinite-dimensional framework Hu and Peng [6], and Oksendal
and Zhang [10] give an existence and uniqueness result for the equation with
an operator A, infinitesimal generator of a strongly continous semigroup and
the coefficient f Lipschitz in y and z. Pardoux and Rascanu [14] replace the
operator A with the subdifferential of a convex function and assume that f is
dissipative, everywhere defined and continuous with respect to y, Lipschitz
with respect to z and with linear growth in y and z.
Special results deal with stochastic backward partial differential equa-
tions (BSPDEs): we recall in particular the works of Ma and Yong [8] and
[9]. Earlier, Peng [16] studied a backward stochastic partial differential
equation and regarded the classical Hamilton-Jacobi-Bellman equation of
optimal stochastic control as special case of this problem.
Our work extends these results in a special direction. We consider an
operator A which is the generator of an analytic contraction semigroup on
H and a coefficient f(t, y, z) of the form f0(t, y)+ f1(t, y, z). The coefficient
f1(t, y, z) is assumed to be bounded and Lipschitz with respect to y and z.
The term f0(t, y) is defined for y only taking values in a suitable subspaceHα
of H and it satisfies the following growth condition for some 1 < γ < 1/α,
S ≥ 0, P-a.s.
|f0(t, y)|H ≤ S(1 + ||y||
) ∀t ∈ [0, T ], ∀y ∈ Hα.
Following [6], we understand the equation (1) in the following integral
e(s−t)A[f0(s, Ys) + f1(s, Ys, Zs)]ds+
e(s−t)AZsdWs = e
(T−t)Aξ,
requiring, in particular, that Y takes values in Hα. This requires generally
that the final condition also takes values in the smaller spaceHα. We take as
Hα a real interpolation space which belongs to the class Jα between H and
the domain of an operator A (see Section 2). Moreover f0(t, ·) is assumed
to be locally Lipschitz from Hα into H and dissipative in H. We prove
(Theorem 5) that if ξ takes its values in the closure of D(A) in Hα and is
such that ||ξ||Hα is essentially bounded, then equation (2) has a unique mild
solution, i.e. there exists a unique pair of progressively measurable processes
Y : Ω×[0, T ] → Hα, Z : Ω×[0, T ] → L
2(K;H), satisfying P-a.s. equality (2)
for every t in [0, T ] and such that E supt∈[0,T ] ||Yt||
||Zt||
L2(K,H)
This result extends former results concerning the deterministic case to
the stochastic framework: see [7], where previous works of Fujita - Kato [5],
Pazy [15] and others are collected. In these papers similar assumptions are
made on the coefficients f0, f1 and on the operator A.
The plan of the paper is as follows. In Section 2 some notations and
definitions are fixed. In Section 3 existence and uniqueness of the solution
of a simplified equation are proved, where f1 is a bounded progressively
measurable process which does not depend on y and z. In Section 4, applying
the previous result, a fixed point argument is used in order to prove our
main result on existence and uniqueness of a mild solution of (2). Section 5
is devoted to applications.
2 Notations and setting
The letters K and H will always denote two real separable Hilbert spaces.
Scalar product is denoted by 〈·, ·〉; L2(K;H) is the separable Hilbert space of
Hilbert-Schmidt operators from K to H endowed with the Hilbert-Schmidt
norm. W = {Wt}t∈[0,T ] is a cylindrical Wiener process with values in K,
defined on a complete probability space (Ω,F ,P). {Ft}t∈[0,T ] is the natural
filtration of W , augmented with the family of P-null sets of F .
Next we define several classes of stochastic processes with values in a
Banach space X.
• L2(Ω× [0, T ];X) denotes the space of measurable X-valued processes
Y such that
|Yτ |
is finite, identified up to modification.
• L2(Ω;C([0, T ];X)) denotes the space of continuousX-valued processes
Y such that
E sup
τ∈[0,T ]
|Yτ |
is finite, identified up to indistinguisha-
bility.
• Cα([0, T ];X) denotes the space of α-Hölderian functions on [0, T ] with
values in X such that [f ]α = sup
0≤x<y≤T
|f(x)− f(y)|
(y − x)α
Now we need to recall several preliminaries on semigroup and interpo-
lation spaces. We refer the reader to [7] for the proofs and other related
results.
A linear operator A in a Banach space X, with domain D(A) ⊂ X, is
called sectorial if there are constants ω ∈ R, θ ∈ (π/2, π), M > 0 such that
(i) ρ(A) ⊇ Sθ,ω = {λ ∈ C : λ 6= ω, |arg(λ− ω)| < θ},
(ii) ||(λI −A)−1||L(X) ≤
|λ−ω|
∀λ ∈ Sθ,ω
where ρ(A) is the resolvent set of A. For every t > 0, (3) allows us to define
a linear bounded operator etA in X, by means of the Dunford integral
etA =
ω+γr,η
etλ(λI −A)−1dλ, t > 0, (4)
where, r > 0, η ∈ (π/2, π) and γr,η is the curve {λ ∈ C : |argλ| = η, |λ| ≥
r} ∪ {λ ∈ C : |argλ| ≤ η, |λ| = r}, oriented counterclockwise. We also set
e0Ax = x,∀x ∈ X. Since the function λ 7→ etλR(λ,A) is holomorphic in
Sθ,ω, the definition of e
tA is independent of the choice of r and η. If A is
sectorial, the function [0,+∞) → L(X), t 7→ etA, with etA defined by (4)
is called analytic semigroup generated by A in X. We note that for every
x ∈ X the function t 7→ etAx is analytic (and hence continuous) for t > 0.
etA is a strongly continuous semigroup if and only if D(A) is dense in X; in
particular this holds if X is a reflexive space.
We need to introduce suitable classes of subspaces of X.
Definition 1. Let (α, p) be two numbers such that 0 < α < 1, 1 ≤ p ≤ ∞
or (α, p) = (1,∞). Then we denote with DA(α, p) the space
DA(α, p) = {x ∈ X : t 7→ v(t) = ||t
1−α−1/pAetAx|| ∈ Lp(0, 1)}
where ||x||DA(α,p) = ||x||X + [x]α = ||x||X + ||v||Lp(0,1).
(We set as usual 1/∞ = 0).
We recall here some estimates for the function t 7→ etA when t → 0,
which we will use in the sequel. For convenience, in the next proposition we
set DA(0, p) = X, p ∈ [1,∞].
Proposition 1. Let (α, p), (β, p) ∈ (0, 1)× [1,+∞]∪{(1,∞)}, α ≤ β. Then
there exists C = C(p;α, β) such that
||t−α+βetA||L(DA(α,p),DA(β,p)) ≤ C, 0 < t ≤ 1.
Definition 2. Let 0 ≤ α ≤ 1 and let D,X be Banach spaces, D ⊂ X. A Ba-
nach space Y such that D ⊂ Y ⊂ X is said to belong to the class Jα between
X and D if there is a constant C such that ||x||Y ≤ C||x||
X ||x||
D, ∀x ∈
D. In this case we write Y ∈ Jα(X,D).
Now we give the definition of solution to the BSDE:
e(s−t)A[f0(s, Ys) + f1(s, Ys, Zs)]ds+
e(s−t)AZsdWs = e
(T−t)Aξ,
Definition 3. A pair of progressively measurable processes (Y,Z) is called
mild solution of (5) if it belongs to the space L2(Ω;C([0, T ];Hα))×L
[0, T ];L2(K,H)) and P-a.s.solves the integral equation (5) on the interval
[0, T ].
We finally state a lemma needed in the sequel. It is a generalization of
the well known Gronwall’s lemma. Its proof is given in the Appendix.
Lemma 1. Assume a, b, α, β are nonnegative constants, with α < 1, β > 0
and 0 < T < ∞. For any nonnegative process U ∈ L1(Ω× [0, T ]), satisfying
P-a.s. Ut ≤ a(T − t)
(s− t)β−1EFtUsds for almost every t ∈ [0, T ], it
holds P-a.s. Ut ≤ aM(T − t)
−α, for almost every t ∈ [0, T ]. M is a constant
depending only on b, α, β, T .
3 A simplified equation
As a preparation for the study of (2), in this section we consider the following
simplified version of that equation:
e(s−t)A[f0(s, Ys)ds+ f1(s)]ds +
e(s−t)AZsdWs = e
(T−t)Aξ, (6)
for all t ∈ [0, T ].
We suppose that the following assumptions hold.
Hypothesis 2.
1. A : D(A) ⊂ H → H is a sectorial operator. We also assume that A is
dissipative, i.e. it satisfies < Ay, y >≤ 0,∀y ∈ D(A);
2. for some 0 < α < 1 there exists a Banach space Hα continuously embed-
ded in H and such that
(i) DA(α, 1) ⊂ Hα ⊂ DA(α,∞);
(ii) the part of A in Hα is sectorial in Hα.
3. the final condition ξ is an FT -measurable random variable defined on
Ω with values in the closure of D(A) with respect to Hα-norm. We
denote this set D(A)
. Moreover ξ belongs to L∞(Ω,FT ,P;Hα);
4. f0 : Ω× [0, T ]×Hα → H satisfies:
i) {f0(t, y)}t∈[0,T ] is progressively measurable ∀y ∈ Hα;
ii) there exist constants S > 0, 1 < γ < 1/α such that P-a.s.
|f0(t, y)|H ≤ S(1 + ||y||
) t ∈ [0, T ], y ∈ Hα;
iii) for every R > 0 there is LR > 0 such that P-a.s.
|f0(t, y1)− f0(t, y2)|H ≤ LR||y1 − y2||Hα
for t ∈ [0, T ] and yi ∈ Hα with ||yi||Hα ≤ R;
iv) there exists a number µ ∈ R such that P-a.s., ∀t ∈ [0, T ], y1, y2 ∈
< f0(t, y1)− f0(t, y2), y1 − y2 >H≤ µ|y1 − y2|
H ; (7)
5. f1 : Ω × [0, T ] → H is progressively measurable and for some constant
C > 0 it satisfies P-a.s. |f1(t)|H ≤ C, for t ∈ [0, T ].
Remark 1. We note that the pair (Y,Z) solves the BSDE (6) with final con-
dition ξ and drift f = f0+f1 if and only if the pair (Ȳ , Z̄) := (e
λtYt, e
λtZt) is
a solution of the same equation with final condition eλT ξ and drift f ′(t, y) :=
0(t, y) + f
1(t) where f
0(t, y) = e
λt(f0(t, e
−λty)− λy), f
1(t) = e
λtf1(t). If we
choose µ = λ, then f
0 satisfies the same assumption as f0, but with (7) re-
placed by < f0(t, y1)− f0(t, y2), y1 − y2 >H≤ 0. If this last condition holds,
then f0 is called dissipative. Hence, without loss of generality, we shall
assume until the end that f0 is dissipative, or equivalently that µ = 0 in (7).
3.1 A priori estimates
We prove a basic estimate for the solution in the norm of H.
Proposition 2. Suppose that Hypothesis 2 holds; if (Y,Z) is a mild solution
of (6) on the interval [a, T ], 0 ≤ a ≤ T , then there exists a constant C1,
which depends only on ||ξ||L∞(Ω;H) and on the constants S of 4.ii) and C of
5. such that P-a.s. supa≤t≤T ||Yt||H ≤ C1. In particular the constant C1 is
independent of a.
Proof. Let the pair (Y,Z) ∈ L2(Ω, C([a, T ];Hα)× L
2(Ω × [a, T ];L2(K;H))
satisfy (6). Let us introduce the operators Jn = n(nI − A)
−1, n > 0. We
note that the operators AJn are the Yosida approximations of A and they
are bounded. Moreover |Jnx − x| → 0 as n → ∞, for every x ∈ H. We
set Y nt = JnYt, Z
t = JnZt. It is readily verified that Y
n admits the Itô
differential
dY nt = −AY
t dt− Jnf(t, Yt)dt− Jnf1(t)dt+ Z
t dWt, and Y
T = Jnξ.
Applying the Ito formula to |Y nt |
H , using the dissipativity of A, we obtain
|Y nt |
||Zns ||
L2(K;H)ds ≤ |Jnξ|
H + 2
< Jnf0(s, Ys), Y
s >H ds+
< Jnf1(s), Y
s >H ds− 2
< Y ns , Z
s dWs >H .
We note that
< Jnf0(s, Ys) + Jnf1(s), Y
s >H ds →
< f0(s, Ys) +
f1(s), Ys >H ds by dominated convergence, as n → ∞. Moreover by
the dominated convergence theorem we have
||(Zns )
∗Y ns − Z
sYs||
Kds →
0 P-a.s. and it follows that
< Y ns , Z
s dWs >H→
< Ys, ZsdWs >H
in probability. If we let n → ∞ in (8) we obtain
||Zs||
L2(K;H)ds ≤ |ξ|
H + 2
< f0(s, Ys) + f1(s), Ys >H ds
< Ys, ZsdWs >H .
Recalling (7), that we assume to hold with µ = 0, it follows that
||Zs||
L2(K,H) ≤
≤ |ξ|2H + 2
< f0(s, 0), Ys >H +2
< f1(s), Ys >H ds+
< Ys, ZsdWs >H
≤ |ξ|2H +
|f(s, 0)|2Hds+
|f1(s)|
Hds + 2
< Ys, ZsdWs >H .
Now, since sup0≤t≤T |f(t, 0)|
H ≤ S
2 and since the stochastic integral
< Ys, ZsdWs >H , t ∈ [a, T ] is a martingale, if we take the conditional
expectation given Ft we have
H ≤ E
Ft |ξ|2H + 2E
|f(s, 0)|2Hds+ E
|f1(s)|
≤ |ξ|2L∞(Ω,H) + (S
2 + C2)T + 2
Ft |Ys|
Since Y belongs to L2(Ω;C([a, T ];Hα)) and, consequently, ||Y ||
L1(Ω× [0, T ]), we can apply Lemma 1 to |Y |2H and conclude that
H ≤ (|ξ|
L∞(Ω,H)
+ [S2 + C2]T )(1 + 2Te2T ).
Now we will show that the result of Proposition 2, together with the
growth condition satisfied by f0, yields an a priori estimate on the solution
in the Hα-norm.
Let 0 < α < 1 and let γ > 1 be given by 4.ii). We fix θ = αγ and consider
the Banach space DA(θ,∞) introduced in Definition 1. It is easy to check
(see [7]) that, if we take θ ∈ (0, 1), θ > α, then Hα contains DA(θ,∞) and
belongs to the class Jα/θ between DA(θ,∞) and H, hence the following
inequality is satisfied:
|x|Hα ≤ c|x|
DA(θ,∞)
H , x ∈ DA(θ,∞). (9)
Proposition 3. Suppose that Hypothesis 2 is satisfied. Let (Y,Z) be a mild
solution of (6) in [a, T ], a ≥ 0 and assume that there exists two constants
R > 0 and K > 0, possibly depending on a, such that, P-a.s.,
t∈[a,T ]
||Yt||Hα ≤ R, sup
t∈[a,T ]
|Yt|H ≤ K. (10)
Then the following inequality holds P-a.s.:
|Yt|L∞(Ω,DA(θ,∞)) ≤ C2
(T − t)θ−α
, a ≤ t < T (11)
with C2 depending on the operator A, ||ξ||L∞(Ω,Hα), θ, α, K, C of 5. and S
of 4.ii) of Hypothesis 2.
Proof. Taking the conditional expectation given Ft in equation (6) we find
Yt = E
e(T−t)Aξ +
e(s−t)A[f0(s, Ys) + f1(s)]ds
, a ≤ t ≤ T.
Consequently, we have
||Yt||DA(θ,∞) ≤ E
Ft ||e(T−t)Aξ||DA(θ,∞)
||e(s−t)A[f0(s, Ys) + f1(s)]||DA(θ,∞)ds, a ≤ t ≤ T .
Since Hα ⊂ DA(α,∞), we have
Ft ||e(T−t)Aξ||DA(θ,∞) ≤
≤ EFt ||e(T−t)A||L(DA(α,∞),DA(θ,∞))||ξ||L∞(Ω,DA(α,∞))
(T − t)θ−α
||ξ||L∞(Ω,Hα),
with C0 = C0(α, θ,∞), where in the last inequality we use Proposition 1.
Moreover
||e(s−t)A[f0(s, Ys) + f1(s)]||DA(θ,∞)ds ≤
≤ EFt
||e(s−t)A||L(H,DA(θ,∞))|f0(s, Ys) + f1(s)|Hds ≤
≤ EFt
(s− t)θ
[|f0(s, Ys)|H + |f1(s)|H ]ds
≤ EFt
(s− t)θ
[S(1 + ||Ys||
) + C]ds.
In the inequality we used Hypotheses 4.ii) and 5. and Proposition 1. Re-
calling (9), we conclude that the last term is dominated by
(s − t)θ
S(1 + c|Ys|
γ(1−α)/θ
H ||Ys||
DA(θ,∞)
) + C
= EFt
(s − t)θ
S(1 + c|Ys|
γ(1−α)/θ
H ||Ys||DA(θ,∞)) + C
by choosing θ = αγ. By the second inequality in (10) this can be estimated
(s− t)θ
S(1 + cKγ(1−α)/θEFt ||Ys||DA(θ,∞) + C)ds
(s− t)θ
(C + S)ds +
(s− t)θ
ScKγ(1−α)/θEFt ||Ys||DA(θ,∞)ds.
Hence by (13) and (14) it follows
||Yt||DA(θ,∞) ≤
(T − t)θ−α
||ξ||L∞(Ω,Hα) +
(s− t)θ
(C + S)ds
(s− t)θ
ScKγ(1−α)/θEFt ||Ys||DA(θ,∞)ds,
and (11) follows from Lemma 1. In order to justify the application of Lemma
1, we need to prove that ||Y ||DA(θ,∞) belongs to L
1(Ω × [a, T ]). This also
follows from(13) and (14) since, for some constant K1,
||Yt||DA(θ,∞) ≤
(T − t)θ−α
||ξ||L∞(Ω,Hα) + E
Ft [ sup
s∈[a,T ]
(1 + ||Ys||
(s − t)θ
(T − t)θ−α
||ξ||L∞(Ω,Hα) + (1 +R
(s − t)θ
3.2 Local existence and uniqueness
We prove that, under Hypothesis 2, there exists a unique solution of (6) on
an interval [T − δ, T ] with δ sufficiently small.
To treat the ordinary integral in the left hand side of (6), we need the
following result, whose proof can be found in [7], Proposition 4.2.1 and
Lemma 7.1.1.
Lemma 3. Let φ ∈ L∞((a, T );H), 0 < a < T and set
v(t) =
e(s−t)Aφ(s)ds, a ≤ t ≤ T.
If 0 < α < 1, then v ∈ C1−α([a, T ];DA(α, 1)) and there is G0 > 0, not
depending on a, such that
||v||C1−α([a,T ];DA(α,1)) ≤ G0||φ||L∞((a,T );H).
Since DA(α, 1) ⊂ Hα, we also have v ∈ C
1−α([a, T ];Hα) and there is G > 0,
not depending on a, such that
||v||C1−α([a,T ];Hα) ≤ G||φ||L∞((a,T );H).
Theorem 4. Let us assume that Hypothesis 2 holds, except possibly 4.iv).
Then there exists δ > 0 such that the equation (6) has a unique local mild
solution (Y,Z) ∈ L2(Ω;C([T − δ, T ];Hα))× L
2(Ω× [T − δ, T ];L2(K;H)).
Remark 2. The dissipativity condition 4.iv) only plays a role in obtaining
the a priori estimate in H (Proposition 2) and consequently global existence,
as we will see later.
Proof. Let Mα := sup0≤t≤T ||e
tA||L(Hα). We fix a positive number R such
that R ≥ 2Mα||ξ||L∞(Ω;Hα). This implies that sup0≤t≤T ||e
tAξ||Hα ≤ R/2
P-a.s. Moreover, let LR be such that
|f0(t, y1)− f0(t, y2)|H ≤ LR||y1 − y2||Hα 0 ≤ t ≤ T, ||yi||Hα ≤ R
We recall that the space L2(Ω;C([T − δ, T ];Hα)) is a Banach space en-
dowed with the norm Y →
E supt∈[T−δ,T ] ||Yt||
. We define
K = {Y ∈ L2(Ω;C([T − δ, T ],Hα)) : sup
t∈[T−δ,T ]
||Yt||Hα ≤ R, a.s.}.
It easy to check that K is a closed subset of L2(Ω;C([T − δ, T ],Hα)),
hence a complete metric space (with the inherited metrics). We look for a
local mild solution (Y,Z) in the space K. We define a nonlinear operator
Γ : K → K as follows: given U ∈ K, Y = Γ(U) is the first component of the
mild solution (Y,Z) of the equation
e(s−t)A[f0(s, Us)ds+ f1(s)]ds+
e(s−t)AZsdWs = e
(T−t)Aξ (15)
for t ∈ [T − δ, T ]. Since U ∈ K we have P-a.s.
|f0(t, Ut) + f1(t)|H ≤ S(1 + ||Ut||
) + C ≤ S(1 +Rγ) + C, (16)
for all t in [T − δ, T ]. Hence f0(·, U·)+ f1(·) belongs to L
2(Ω× [T − δ, T ];H)
and, by a result of Hu and Peng [6], there exists a unique pair (Y,Z) ∈
L2(Ω× [T − δ, T ];H)× L2(Ω× [T − δ, T ];L2(K;H)) satisfying (15). More-
over, by taking the conditional expectation given Ft, Y has the following
representation
Yt = E
e(T−t)Aξ +
e(s−t)A[f0(s, Us) + f1(s)]ds
We will show that Γ is a contraction for the norm of L2(Ω, C([T − δ, T ];Hα)
and maps K into itself, if δ is sufficiently small; clearly, its unique fixed point
is the required solution of the BSDE.
We first check the contraction property. Let U1, U2 ∈ K. Then
Γ(U1)t − Γ(U
2)t = Y
t − Y
t = E
e(s−t)A(f0(s, U
s )− f0(s, U
Let v(t) =
e(s−t)A
f0(s, U
s )− f0(s, U
ds. Then, noting that v(T ) = 0
and recalling Lemma 3, for t ∈ [T − δ, T ]
||Y 1t − Y
t ||Hα =
= ||EFtv(t)||Hα ≤ E
Ft ||v(t)||Hα
≤ δ1−αEFt ||v||C(1−α)([T−δ,T ],Hα)
≤ Gδ(1−α)EFt ||f0(·, U
· )− f0(·, U
· )||L∞([T−δ,T ],H)
≤ Gδ(1−α)LRE
Ft sup
t∈[T−δ,T ]
||U1t − U
t ||Hα =: Mt,
where {Mt, t ∈ [T − δ, T ]} is a martingale. Hence, by Doob’s inequality
E sup
t∈[T−δ,T ]
||Y 1t − Y
≤ E sup
t∈[T−δ,T ]
2 ≤ 2E|MT |
= 2G2L2Rδ
2(1−α)
E sup
t∈[T−δ,T ]
||U1t − U
If δ ≤ δ0 = 2GLR
(1−α) , then Γ is a contraction with constant 1/2.
Next we check that Γ mapsK into itself. For each U ∈ K and t ∈ [T−δ, T ]
with δ ≤ δ0 we have
t∈[T−δ,T ]
||Γ(U)t||Hα = sup
t∈[T−δ,T ]
||Yt||Hα ≤ sup
t∈[T−δ,T ]
Ft ||e(T−t)Aξ||Hα+
+ sup
t∈[T−δ,T ]
Ft ||
e(s−t)A[f0(s, Us) + f1(s)]ds||Hα
≤ R/2 + sup
t∈[T−δ,T ]
||e(s−t)A[f0(s, Us) + f1(s)]||Hαds
≤ R/2 + sup
t∈[T−δ,T ]
||e(s−t)A[f0(s, Us) + f1(s)]||DA(α,1)ds,
where in the last inequality we have used the fact that DA(α, 1) ⊂ Hα. Now,
by Proposition 1, and from 4.ii) and 5., it follows that
||e(s−t)A[f0(s, Us) + f1(s)]||DA(α,1) ≤
≤ ||e(s−t)A||L(H,DA(α,1))|f0(s, Us) + f1(s)|H
(s− t)α
[S(1 + ||Us||
) + C].
Then, since U ∈ K, we arrive at
t∈[T−δ,T ]
||Γ(U)t||Hα ≤
≤ R/2 + sup
t∈[T−δ,T ]
(s− t)α
[S(1 + ||Us||
) + C]ds
≤ R/2 + sup
t∈[T−δ,T ]
(s− t)α
[S(1 +Rγ) + C]ds
≤ R/2 + CαS
[(1 +Rγ) + C]
δ1−α,
where Cα depends on A, α. Hence, if δ ≤ δ0 is such that CαS
[(1+Rγ )+C]
is less or equal to R/2, then sup
t∈[T−δ,T ]
||Γ(U)t||Hα ≤ R. Due to Lemma 3, P-
a.s. the function t 7→ Yt−E
Fte(T−t)Aξ belongs to C[T−δ, T ];Hα); moreover,
the map t 7→ EFte(T−t)Aξ belongs to C[T − δ, T ];Hα), since ξ is a random
variable taking values in D(A)
. Therefore, P-a.s. Y· ∈ C([T − δ, T ];Hα)
and Γ maps K into itself and has a unique fixed point in K.
Remark 3. By Lemma 3, using properties of analytic semigroups, it can
be proved that for every fixed ω the range of the map Γ is contained in
C1−β([T − δ, T − ǫ];DA(β, 1)) for every ǫ ∈ (0, δ), β ∈ [0, 1].
3.3 Global existence
Now we are able to prove a global existence theorem for the solution of the
equation (6), using all the results presented above.
Theorem 5. If Hypothesis 2 is satisfied, the equation (6) has a unique mild
solution (Y,Z) ∈ L2(Ω;C([0, T ],Hα))× L
2(Ω× [0, T ]);L2(K;H)).
Proof. By Theorem 4 equation (6) has a unique mild solution (Y 1, Z1) ∈
L2(Ω;C([T − δ1, T ],Hα)) × L
2(Ω × [T − δ1, T ]);L
2(K;H)) on the interval
[T − δ1, T ], for some δ1 > 0. By Proposition 2 we know that there exists a
constant C1 such that P-a.s.
|YT−δ1 |H ≤ C1. (17)
We recall that the constant C1 depends only on |ξ|L∞(Ω;H) and on the con-
stants S of 4.ii) and C of 5. and is independent of δ1. Moreover, by Propo-
sition 3, there exists a constant C2 such that P-a.s.
||YT−δ1 ||L∞(Ω,DA(θ,∞)) ≤ C2
δθ−α1
, (18)
with C2 depending on the operator A, ||ξ||L∞(Ω,Hα), θ, α, C1. This implies
that YT−δ1 belongs to L
∞(Ω;Hα) and it can be taken as final value for the
problem
∫ T−δ1
e(s−t)A[f0(s, Ys)ds+ f1(s)]ds +
∫ T−δ1
e(s−t)AZsdWs =
= e(T−δ1−t)AYT−δ1
on an interval [T − δ1 − δ2, T − δ1], for some δ2 > 0. As in the proof of
Theorem 4, we fix a positive number R2 such that
R2 = 2Mα
≥ 2Mα||YT−δ1 ||L∞(Ω,DA(θ,∞)).
By Theorem 4 there exists a pair of progressively measurable processes
(Y 2, Z2) in L2(Ω;C([T − δ1 − δ2, T − δ1];Hα)) × L
2(Ω × [T − δ1 − δ2, T −
δ1];L
2(K,H)) which solves (19) on the interval [T − δ1 − δ2, T − δ1] where
δ2 depends on the operator A, α, R2. We note that the continuity in T − δ1
of Y 2 follows from the fact that YT−δ1 takes values in DA(α, 1) (see Remark
3), so that YT−δ1 takes values in D(A)
. Now, the process Yt defined by
Y 1t on the interval [T − δ1, T ] and by Y
t on [T − δ1 − δ2, T − δ1] belongs
to L2(Ω;C([T − δ1 − δ2, T ];Hα)) and it easy to see that it satisfies (6) in
the whole interval [T − δ1 − δ2, T ]. Consequently, by Proposition 2, P-a.s.,
|YT−δ1−δ2 |H ≤ C1 with C1 the constant in (17), and by (18)
||YT−δ1−δ2 ||L∞(Ω,DA(θ,∞)) ≤
(δ1 + δ2)θ−α
, (20)
where C2 is the same constant as in (18). Again, YT−δ1−δ2 can be taken as
initial value for problem
∫ T−δ1−δ2
e(s−t)A[f0(s, Ys)ds+ f1(s)]ds +
∫ T−δ1−δ2
e(s−t)AZsdWs =
e(T−δ1−δ2−t)AYT−δ1−δ2
on the interval [T − δ1 − δ2 − δ3, T − δ1 − δ2], where δ3 will be fixed later.
In this case, by (20), we can choose
R3 = R2 = 2Mα
≥ 2Mα||YT−δ1−δ2 ||L∞(Ω,DA(θ,∞))
and prove that there exists a unique mild solution (Y 3, Z3) of (21) on the
interval [T−δ1−δ2−δ3, T−δ1−δ2], with δ3 = δ2 . So we extend the solution
to [T − δ1 − 2δ2, T ]. Proceeding this way we prove the global existence to
(6) on [0, T ].
4 The general case
We can now study the equation:
e(s−t)A[f0(s, Ys) + f1(s, Ys, Zs)]ds +
e(s−t)AZsdWs = e
(T−t)Aξ
We require that the function f1 satisfy the following assumptions:
Hypothesis 6.
1. there exists K ≥ 0 such that P-a.s.
|f1(t, y, z)− f1(t, y
)|H ≤ K|y − y
|H +K||z − z
||L2(K;H),
for every t ∈ [0, T ], y, y
∈ H, z, z
∈ L2(K;H),
2. there exists C ≥ 0 such that P-a.s. |f1(t, y, z)|H ≤ C, for every t ∈
[0, T ], y ∈ H, z ∈ L2(K;H).
Theorem 7. If Hypotheses 2 and 6 hold, then equation (22) has a unique
solution in L2(Ω;C([0, T ];Hα))× L
2(Ω× [0, T ];L2(K;H)).
Proof. LetM be the space of progressive processes (Y,Z) in the space L2(Ω×
[0, T ];H) × L2(Ω× [0, T ];L2(K;H)) endowed with the norm
|||(Y,Z)|||2β = E
eβs(|Ys|
H + ||Zs||
L2(K;H))ds,
where β will be fixed later. We define Φ : M → M as follows: given (U, V ) ∈
M, (Y,Z) = Φ(U, V ) is the unique solution on the interval [0, T ] of the
equation
e(s−t)A[f0(s, Ys)ds+f1(s, Us, Vs)]ds+
e(s−t)AZsdWs = e
(T−t)Aξ.
By Theorem 5 the above equation has a unique mild solution (Y,Z) which
belongs to L2(Ω;C([0, T ];Hα))×L
2(Ω×[0, T ];L2(K;H)). Therefore Φ(M) ⊂
M. We will show that Φ is a contraction for a suitable choice of β; clearly,
its unique fixed point is the required solution of (22). We take another
pair (U
) ∈ M and apply Proposition 3.1 in [3] to the difference of two
equations. We obtain
β|Y 1t − Y
H + ‖Z
s − Z
L2(K;H)ds
eβs < f0(s, Y
s ) + f1(s, U
s , V
−f0(s, Y
s )− f1(s, U
s , V
s ), Y
s − Y
s >H ds
eβsK(|U1s − U
s |H + ||V
s − V
s ||L2(K;H))|Y
s − Y
s |Hds
eβs(|U1s − U
H + ||V
s − V
L2(K;H))/2 + 4K
2|Y 1s − Y
where we have used 4.iv) of Hypothesis 2 and 1. of Hypothesis 6. Choosing
β = 4K2 + 1, we obtain the required contraction property.
5 Applications
In this section we present some backward stochastic partial differential prob-
lems which can be solved with our techniques.
5.1 The reaction-diffusion equation
Let D be an open and bounded subset of Rn with a smooth boundary
∂D. We choose K = L2(D). This choice implies that dWt/dt is the
so-called ”space-time white noise”. Moreover, since Hilbert-Schmidt op-
erators on L2(D) are represented by square integrable kernels, the space
L2(L2(D), L2(D)) can be identified with L2(D ×D). We are given a com-
plete probability space (Ω,F ,P) with a filtration (Ft)t∈[0,T ] generated by
W and augmented in the usual way. Let us consider a non symmetric
bilinear, coercive continuous form a : H10 (D) × H
0 (D) → R defined by
a(u, v) := −
i,j aij(x)Diu(x)Djv(x)dx, where the coefficients aij are
Lipschitz continuous and there exists α > 0 such that
i,j=1 aij(x)ξiξj ≥
α|ξ|2 for every x ∈ D, ξ ∈ Rn. Let A be the operator associated with the bi-
linear form a such that < Au, v >L2(D)= a(u, v), v ∈ H
0 (D) and u ∈ D(A).
It is known that, in this case, D(A) = H2(D) ∩H10 (D), where H
2(D) and
H10 (D) are the usual Sobolev spaces.
We consider for t ∈ [0, T ] and x ∈ D the backward stochastic problem
written formally
∂Y (t, x)
= AY (t, x) + r(Y (t, x)) + g(t, Y (t, x), Z(t, x), x)+
+ Z(t, x)
∂W (t, x)
on Ω× [0, T ]× D̄
Y (T, x) = ξ(x) on Ω× D̄
Y (t, x) = 0 on Ω× [0, T ]× ∂D
We suppose the following.
Hypothesis 8.
1. r : R → R is a continuous, increasing and locally Lipschitz function;
2. r satisfies the following growth condition: |r(x)| ≤ S(1 + |x|γ) ∀x ∈ R
for some γ > 1;
3. g is a measurable real function defined on [0, T ] × R × L2(D × D) ×D
and there exists a constant K > 0 such that
|g(t, y1, z1, x)− g(t, y2, z2, x)| ≤ K(|y1 − y2|+ ||z1 − z2||L2(D×D))
for all t ∈ [0, T ], y1, y2 ∈ R, z1, z2 ∈ L
2(D), x ∈ D;
4. there exists a real function h in L2(D×D) such that P-a.s. |g(t, y, z, x)| ≤
K1h(x) for all t ∈ [0, T ], y ∈ R, z ∈ L
2(D), x ∈ D;
5. ξ belongs to L∞(Ω;H2(D) ∩H10 (D)).
We define the operator A by (Ay)(x) = Ay(x) with domain D(A) =
H2(D) ∩ H10 (D). We set f0(t, y)(x) = −r(y(t, x)) for t ∈ [0, T ], x ∈ D
and y in a suitable subspace of H which will be determined below. For
t ∈ [0, T ], x ∈ D, y ∈ L2(D), z ∈ L2(D × D) we define f1 as the operator
f1(t, y, z)(x) = −g(t, y(t, x), z(t, x), x). Then problem (23) can be written
in abstract way as
dYt = −AYtdt− f0(t, Yt)dt− f1(t, Yt, Zt)dt+ ZtdWt, YT = ξ.
Under the conditions in Hypothesis 8, the assumptions in Hypotheses
2, 6 are satisfied. The operator A is a closed operator in L2(D) and it
is the infinitesimal generator of an analytic semigroup in L2(D) satisfying
‖etA‖L(H) ≤ 1 (see [17], Chapter 3). In particular, by Lumer-Philips theo-
rem, A is dissipative. The non linear function f0(t, ·) : L
2γ(D) → L2(D),
y 7→ −r(y) is locally Lipschitz. We look for a space of class Jα between H
and D(A) where f0 is well defined and locally Lipschitz. It is well known
(see [18]) that the fractional order Sobolev space W β,2(D) is of class Jβ/2
between L2(D) and H2(D) for every β ∈ (0, 2). Hence the space Hα defined
by Hα = W
β,2(D) if β < 1, by W β,2(D) ∩H10 (D) if β ≥ 1 is of class Jβ/2
between H and D(A). Moreover the restriction of A on Hα is a sectorial
operator ([18]). By the Sobolev embedding theorem, W β,2 is contained in
Lq(D) for all q if β ≥ n
, and in L2n/(n−2β)(D) if β < n
. If we choose
β ∈ (0, 2) we have W β,2(D) ⊂ L2γ(D) for n < 4
. It is clear that f0
is locally Lipschitz with respect to y from Hα into H. It is easy to verify
that f0 satisfies 4.ii) of Hypothesis 2 with γ = 2n + 1 and that it is dis-
sipative with constant µ = 0. The function f1 is Lipschitz uniformly with
respect to y and z and it is bounded. The final condition ξ takes values in
and belongs to L∞(Ω;Hα). Hence we can apply the global exis-
tence theorem and state that the above problem has a unique mild solution
(Y,Z) ∈ L2(Ω;C([0, T ];Hα))× L
2(Ω× [0, T ];L2(K,H)).
5.2 A spin system
Let Z be the one-dimensional lattice of integers. Its elements will be inter-
preted as atoms. A configuration is a real function y defined on Z. The
value y(n) of the configuration y at the point n can be viewed as the state
of the atom n.
We consider an infinite system of equations
dY nt = −anY
t dt+
|n−j|≤1
V (Y nt − Y
t )dt+ Z
t n ∈ Z, 0 ≤ t ≤ T
Yn(T ) = ξn n ∈ Z,
where Y n and Zn are real processes, and V : R → R.
Let l2(Z) be the usual Hilbert space of square summable sequences. To
study system (24) we apply results of previous sections. To fit our assump-
tion in Hypotheses 2 and 6, we suppose the following
Hypothesis 9.
1. W n, n ∈ Z are independent standard real Wiener processes;
2. a = {an}n∈Z is a sequence of nonnegative real numbers;
3. ξ = {ξn}n∈Z is a random variable belonging to L
∞(Ω, l2(Z));
4. the function V : R → R is defined by V (x) = x2k+1 k ∈ N.
We will study system (24) regarded as a backward stochastic evolution
equation for t ∈ [0, T ]
dYt = (AYt + f0(t, Yt))dt+ ZtdWt, YT = ξ (25)
on a properly chosen Hilbert space H of functions on Z.
To reformulate problem (24) in the abstract form (25), we set K = H =
l2(Z). We set Wt = {W
t }n∈Z, t ∈ [0, T ]. By 1. of Hypothesis 9, W is a
cylindrical Wiener process inH defined on (Ω,F , P ). We define the operator
A(y) = (anyn)n, D(A) = {y ∈ l
2(Z) such that
n∈Z a
n < ∞}.
It is easy to prove that A is a self-adjoint operator in l2(Z), hence the
infinitesimal generator of a sectorial semigroup. The coefficient f0 is given
by (f0(t, y))n = (V (yn+1−yn)+V (yn−1−yn)), t ∈ [0, T ], y ∈ D(f0) where
D(f0) = {y ∈ l
2(Z) such that
n∈Z |xn+1 − xn|
2(2k+1) < +∞}. Under Hy-
pothesis 9, A, f0, ξ satisfy Hypotheses 2 and 6. We observe that in this case
the domain of f0 is the whole space H: if y ∈ l
2(Z) then
|yn+1 − yn|
2(2k+1)}
2(2k+1) ≤ {
|yn+1 − yn|
2 ≤ 2||y||l2(Z).
Consequently, we can take Hα with α = 0, i.e. H0 = H. The function f0 is
dissipative. Namely
< f0(t, y)− f0(t, y
′), y − y′ >l2(Z) =
{[(yn+1 − yn)
(2k+1) + (yn−1 − yn)
(2k+1)]+
+[(y′n+1 − y
(2k+1) + (y′n−1 − y
(2k+1)])[yn − y
[(yn+1 − yn)
(2k+1) − (y′n+1 − y
(2k+1)][(yn+1 − yn)− (y
n+1 − y
and the last term is negative. Moreover, f0 satisfies 4.ii) of Hypothesis 2
with γ = 2k + 1. The map f0 is also locally Lipschitz from H in to H.
Then by Theorem 7, problem (25) has a unique mild solution (Y,Z) which
belongs to L2(Ω, C([0, T ];H)) × L2(Ω× [0, T ];L2(K,H)).
6 Appendix
This section is devoted to the proof of Lemma 1. Assume first that β =
1. Using recursively the inequality Ut ≤ a(T − t)
−α + b
FtUsds we can
easily prove that
≤ a(T − t)−α +
(r − t)k−1
(k − 1)!
(T − r)α
+bEFt
(b(r − t))n−1
(n− 1)!
Urdr.
The last term in the above inequality tends to zero as n tends to infinity
for each t in the interval [0, T ]. Thus
Ut ≤ a(T − t)
−α + a
bk(T − t)k−1
(k − 1)!
(T − r)α
≤ a(T − t)−α + abeb(T−t)
(T − r)α
≤ a(T − t)−α + abeb(T−t)
(T − t)1−α ≤ a(T − t)−αM
where M = 1 + bebT 1
In the case β 6= 1 a similar proof can be given, based on recursive use of
the inequality Ut ≤ a(T − t)
−α + b
(s − t)β−1EFtUsds.
Acknowledgments:
I wish to thank Giuseppe Da Prato for hospitality at the Scuola Normale
Superiore in Pisa, suggestions and helpful discussions. I would like to express
my gratitude to Marco Fuhrman: I am indebted to him for his precious help
and encouragement. Special thanks are due to Alessandra Lunardi, who
gave me valuable advice and support.
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