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Probabilistic Representation of Weak Solutions of Partial
Differential Equations with Polynomial Growth
Coefficients
Qi Zhang a , Huaizhong Zhao
b
a
School of Mathematical Sciences, Fudan University, Shanghai, 200433, China.
Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK.
Emails: [email protected]; [email protected]
b
Summary. In this paper we develop a new weak convergence and compact embedding method to study
the existence and uniqueness of the L2ρ (Rd ; R1 ) ⊗ L2ρ (Rd ; Rd ) valued solution of backward stochastic
differential equations with p-growth coefficients. Then we establish the probabilistic representation of
the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.
Keywords: PDEs with polynomial growth coefficients, generalized Feynman-Kac formula,
probabilistic representation of weak solutions, backward stochastic differential equations, weak
convergence, compact embedding.
AMS 2000 subject classifications: 60H10, 60H30, 35K55.
1 Introduction
In this paper, we study the probabilistic representation of the weak solution of a class of
parabolic partial differential equations (PDEs) on Rd with p-growth coefficients

 ∂v
(t, x) = L v(t, x) + f x, v(t, x), (σ ∗ ∇v)(s, x) ,
0 ≤ t ≤ T,
∂t

v(0, x) = h(x),
(1.1)
by the solution of the corresponding backward stochastic differential equations (BSDEs) in
ρ-weighted L2 space. Here L is a second order differential operator
L =
d
d
X
1 X
∂2
∂
aij (x)
+
bi (x)
,
2 i,j=1
∂xi ∂xj
∂x
i
i=1
(1.2)
(aij (x)) is a symmetric matrix with a decomposition (aij (x)) = (σij (x))(σij (x))∗ , f : (x, y, z) 7→
f (x, y, z) is a function of polynomial growth in y and Lipschitz continuous in z. Many partial
differential equations arising in physics, engineering and biology have polynomial growth nonlinear terms e.g. KPP-Fisher equations, Allen-Cahn equations and Ginzburg-Landau equations.
The representation provides an important connection between stochastic flows generated by L
and the weak solutions of PDEs possibly with polynomial growth coefficients. In connection
with the classical solutions of the linear parabolic PDEs, the well-known Feynman-Kac formula
provides the probabilistic representation for them and originated many important developments
(Feynman [8], Kac [12]). An alternative probabilistic representation using only the values of a
2
Q. Zhang and H.Z. Zhao
finite (random) set of times to the linear heat equations was obtained recently by Dalang,
Mueller and Tribe [4]. This idea made it possible for them to obtain corresponding formula for
a wide class of linear PDEs such as some wave equations with potentials. The Feynman-Kac
formula has played important roles in problems such as the large deviation theory of Donsker
and Varadhan [6], Wentzell and Freidlin [23], small time asymptotics of heat kernel and its
logarithmic derivatives, in particular on Riemannian manifolds (Elworthy [10], Malliavin and
Stroock [16]). The Feynman-Kac formula has been extended and used to quasi-linear parabolic
type partial differential equations, especially, in the study of the generalized KPP equations
using the large deviation theory method by Freidlin [9], using the semi-classical probabilistic
method by Elworthy, Truman and Zhao [7]. The study of the quasi-linear parabolic type PDE
is based on an equation of the Feynman-Kac type integration of stochastic functionals. The
approach of the backward stochastic differential equations, pioneered by Pardoux and Peng
[19], [20] originally, provided an alternative approach to the classical solution of the parabolic
type PDEs, when the coefficients of the PDE are sufficiently regular and Lipschitz continuous.
This was extended to the viscosity solution of a large class of partial differential equations and
BSDEs. They include the linear growth case considered by Lepeltier and San Martin [15], the
quadratic coefficients (in z) considered by Kobylanski [13], Briand and Hu [3], and the polynomial growth coefficients in Pardoux [18]. The solution of the BSDEs in above cases gives the
probabilistic representation of the classical or viscosity solution of the PDEs as a generalization
to the Feynman-Kac formula. Applications of BSDEs have been found in some problems such
as a model in mathematics of finance (El Karoui, Peng and Quenez [11]), as an efficient method
for constructing Γ -martingales on Riemannian manifolds (Darling [5]), and as an intrinsic tool
to construct the pathwise stationary solution for stochastic PDEs (Zhang and Zhao [24], [25]).
The Feynman-Kac approach to a Sobolev or L2 space valued weak solution of PDEs has
been concentrated mainly on linear problems. Many important progress has been made e.g.
in quantum field theory (see [22]). The probabilistic approach to the weak solution of quasilinear PDEs stayed behind. Regularity of the solutions, even in the sense of weak derivative,
was not given in Freidlin’s probabilistic approach of generalized solution formally represented
by the Feynman-Kac formula ([9]). The BSDEs start to show some usefulness in this aspect,
when the coefficients are of Lipschitz continuous in the space L2ρ (Rd ; R1 ) ⊗ L2ρ (Rd ; Rd ) or of
linear growth, and monotone, from the work of Barles and Lesigne [2], Bally and Matoussi [1],
Zhang and Zhao [24], [25]. The objective of this paper is to move away from the assumption
of the linear growth of f and from considering the classical or viscosity solution of PDEs to
establish the probabilistic representation for the weak solution of such polynomial growth PDEs.
Although the connection of BSDEs with the viscosity solution for the cases of quadratic and
polynomial growth has been obtained in [13], [18] respectively, the existing methods in the study
of BSDEs for finding the solution of the BSDEs in L2ρ (Rd ; R1 ) ⊗ L2ρ (Rd ; Rd ) are not adequate
to solve the problem of the weak solution of BSDEs with p-growth coefficients. The fixed
point method in M 2 ([t, T ]; L2ρ (Rd ; R1 )) ⊗ M 2 ([t, T ]; L2ρ (Rd ; Rd )), which is equivalent to finding
a strongly convergent sequence in the same space, seems difficult to work for the problem with
p-growth coefficients. It is also inadequate to use a combination of the weak convergence in
finite dimensional space developed by Pardoux [18] and the weak solution method developed by
Bally and Matoussi [1], Zhang and Zhao [24], [25] to solve this problem. We need to introduce
Weak Solutions of PDEs with p-Growth Coefficients
3
some new ideas to the study of BSDEs. The progress of this problem was made when we
realized that, in addition to the method of Zhang and Zhao ([24], [25]), as well as the standard
approach using Alaoglu lemma to find a weakly convergent sequence (Y n , Z n ), we can use the
equivalence of norm principle and Rellich-Kondrachov Compactness Theorem to get a strongly
convergent sequence Y n . Our recent result on the S 2 ([t, T ], L2ρ (Rd ; R1 ))⊗M 2 ([t, T ]; L2ρ (Rd ; Rd ))
valued solution of BDSDEs with nonlipschitz linear growth coefficients made it possible for us
d
1
2
2
d
d
to study the BSDEs in S 2p ([t, T ], L2p
ρ (R ; R )) ⊗ M ([t, T ]; Lρ (R ; R )) with polynomial growth
coefficients, even without assuming f being locally Lipschitz continuous in y. Of course, we need
to assume the monotonicity condition of f in y. Moreover, it is also an essential step to prove
the strong convergence of Z n in M 2 ([t, T ], L2ρ (Rd ; Rd )) from the result of the strong convergence
of Y n and Itˆ
o’s formula. The weak convergence and compact embedding method has been used
in the study of PDEs. However, as far as we know, to use this kind of argument to the study
of BSDE, this paper is the first time in literature. The equivalence of norm principle and very
careful probabilistic (measure theoretical) and analytic arguments including localization made
it work in the probabilistic context. However, the probabilistic case is a lot more complicated
than the deterministic PDEs case as we need to work on the space Ω ⊗ [0, T ] ⊗ Rd and solve
the equation with probabilistic one, instead only work on [0, T ] ⊗ Rd in the deterministic PDEs
case. The probabilistic representation can be regarded as a generalized Feynman-Kac formula
to the weak solution of the PDEs with p-growth coefficients and is new in literature. We believe
our method will be useful to other types of BSDEs and PDEs as well.
After this paper was completed, we were informed the paper Matoussi and Xu [17]. But
we would like to point out what we have proved as well as our methods are different. Notice the convergence (Yst,x,n , Zst,x,n ) is only a weak convergence along a subsequence according
to the Alaoglu lemma. If one considers weak convergence in M 2 ([t, T ], R1 ) ⊗ M 2 ([t, T ]; Rd )),
which worked well in Pardoux [18] for the case of viscosity solutions of the PDEs, then each
weak convergence is for a fixed x, and the choice of subsequence may depend on x. However, this will cause serious problems when one considers weak solutions. Our approach to
avoid this essential difficulty is to find a subsequence of the weak convergence in the space
M 2 ([t, T ], L2ρ (Rd ; R1 )) ⊗ M 2 ([t, T ]; L2ρ (Rd ; Rd )). The whole point and major difficulty of this
approach are to pass the limit term by term in the approximating equation to the desired limit.
This is achieved in our paper by obtaining a strong convergent subsequence of (Yst,x,n , Zst,x,n )
in M 2 ([t, T ], L2ρ (Rd ; R1 )) ⊗ M 2 ([t, T ]; L2ρ (Rd ; Rd )) using the Rellich-Kondrachov compactness
theorem and generalized equivalence of norm principle as we have already mentioned.
2 The main results
In this paper, we study the weak solutions of a class of parabolic PDEs with p-growth coefficients, their corresponding backward stochastic differential equations (BSDEs) in a Hilbert
space (ρ-weighted L2 space) and the probabilistic representation of the weak solutions of (1.1)
by using the solutions of BSDEs. We start from the following SDE:
Z s
Z s
t,x
t,x
Xs = x +
b(Xr )dr +
σ(Xrt,x )dWr , s ≥ t,
(2.1)
t
t
4
Q. Zhang and H.Z. Zhao
where W is a Rd Brownian motion on a probability space (Ω, F , P ), and b : Rd → Rd , σ : Rd →
Rd×d are measurable. We consider a slightly more general BSDEs by allowing f depending on
time explicitly:
Yst,x = h(XTt,x ) +
T
Z
f (r, Xrt,x , Yrt,x , Zrt,x )dr −
s
T
Z
hZrt,x , dWr i,
(2.2)
s
where f : [0, T ] × Rd × R1 × Rd → R1 and h : Rd → R1 are measurable. More conditions on b, σ,
f are needed and will be specified later. The Hilbert space L2ρ (Rd ; Rk ) is the space containing
R
all Borel measurable functions l: Rd → Rk such that Rd < l(x), l(x) > ρ−1 (x)dx < ∞, with
the inner product
Z
hu1 , u2 i =
< u1 (x), u2 (x) > ρ−1 (x)dx,
Rd
where ρ(x) = (1 + |x|)q , q > d, is a weight function. The Banach space L2p
(Rd ; R1 ) is the
R 2p ρ −1
d
1
space containing all Borel measurable functions l: R → R such that Rd l (x)ρ (x)dx < ∞
R
1
with the norm ||l||L2p
= ( Rd l2p (x)ρ−1 (x)dx) 2p . It is easy to see that ρ(x) : Rd −→ R1 is
d
ρ (R )
R
a continuous positive function satisfying Rd ρ−1 (x)dx < ∞. Note that we can consider more
general ρ which satisfies the above condition and conditions in [1] and all the results of this
paper still hold. For k ≥ 0, we denote by Cbk the set of C k -functions whose partial derivatives
of order less than or equal to k are bounded and by Hρk the ρ-weighted Sobolev space (See e.g.
[1]). Now we assume the following conditions for the coefficients in SDE (2.1) and BSDE (2.2):
R
(H.1). For a given p ≥ 1, Rd |h(x)|2p ρ−1 (x)dx < ∞.
RT R
(H.2). There exists a constant C ≥ 0 and a function f0 with 0 Rd |f0 (s, x)|2p ρ−1 (x)dxds < ∞
s.t. |f (s, x, y, z)| ≤ C(|f0 (s, x)| + |y|p + |z|), where p is the same as in (H.1).
(H.3). There exists a constant µ ∈ R1 s.t. for any s ∈ [0, T ], y1 , y2 ∈ R1 , x, z ∈ Rd ,
2
(y1 − y2 ) f (s, x, y1 , z) − f (s, x, y2 , z) ≤ µ|y1 − y2 | .
(H.4). The function (y, z) → f (s, x, y, z) is continuous and z → f (s, x, y, z) is globally Lipschitz
continuous with Lipschitz constant L ≥ 0, i.e. for any s ∈ [0, T ], y ∈ R1 , x, z1 , z2 ∈ Rd ,
|f (s, x, y, z1 ) − f (s, x, y, z2 )| ≤ L|z1 − z2 |.
(H.5). The diffusion coefficients b ∈ Cb2 (Rd ; Rd ), σ ∈ Cb3 (Rd ; Rd × Rd ) and σ satisfies the
uniform ellipticity condition, i.e. there exists a constant D > 0 s.t. ξ ∗ (σσ ∗ )(x)ξ ≥ Dξ ∗ ξ for
any ξ ∈ Rd .
It is easy to see that for a.e. x ∈ Rd , (Yst,x , Zst,x ) solves BSDE (2.2) if and only if (Y˜st,x , Z˜st,x ) =
(e Yst,x , eµs Zst,x ) solves the following BSDE:
µs
Y˜st,x = eµT h(XTt,x ) +
Z
s
T
f˜(r, Xrt,x , Y˜rt,x , Z˜rt,x )dr −
Z
T
hZ˜rt,x , dWr i,
(2.3)
s
where f˜(r, x, y, z) = eµr f (r, x, e−µr y, e−µr z) − µy. We can verify that f˜ satisfies Conditions
(H.2), (H.3) and (H.4). But, by Condition (H.3), for y1 , y2 ∈ R1 , and x, z ∈ R1 ,
Weak Solutions of PDEs with p-Growth Coefficients
(y1 − y2 ) f˜(s, x, y1 , z) − f˜(s, x, y2 , z)
5
= e2µs (e−µs y1 − e−µs y2 ) f (s, x, e−µs y1 , e−µs z) − f (s, x, e−µs y2 , e−µs z) − µ(y1 − y2 )(y1 − y2 )
≤ µe2µs |e−µs y1 − e−µs y2 |2 − µ|y1 − y2 |2 = 0.
Now we give the definition for the solution of BSDE (2.2) in the ρ-weighted L2 space. First
define the space for the solution (Y·t,· , Z·t,· ). We denote by N the class of P -null sets of F and
W
let Ft , FtW N , for 0 ≤ t ≤ T . We recall some definitions.
Definition 2.1 (Definitions 2.2 in [24]) Let S be a Banach space with norm k · kS and Borel
σ-field S and q ≥ 2 be a real number. We denote by M q ([t, T ]; S) the set of B([t, T ]) ⊗ F /S
measurable random processes {φ(s)}t≤s≤T with values in S satisfying
(i) φ(s) : Ω → S is Fs measurable for t ≤ s ≤ T ;
RT
(ii) E[ t kφ(s)kqS ds] < ∞.
Also we denote by S q ([t, T ]; S) the set of B([t, T ]) ⊗ F /S measurable random processes
{ψ(s)}t≤s≤T with values in S satisfying
(i) ψ(s) : Ω → S is Fs measurable for t ≤ s ≤ T and ψ(·, ω) is continuous P -a.s.;
(ii) E[supt≤s≤T kψ(s)kqS ] < ∞.
Definition 2.2 (Definitions 3.1 in [24]) A pair of processes (Yst,x , Zst,x ) is called a solution
N 2
d
1
of BSDE (2.2) if (Y·t,· , Z·t,· ) ∈ S 2p ([t, T ]; L2p
M ([t, T ]; L2ρ (Rd ; Rd )) and (Yst,x , Zst,x )
ρ (R ; R ))
satisfies (2.2) for a.e. x, with probability one.
d
d
d
1
2
2
˜ t,· ˜ t,·
Since (Y·t,· , Z·t,· ) ∈ S 2p ([t, T ]; L2p
ρ (R ; R ))⊗M ([t, T ]; Lρ (R ; R )) if and only if (Y· , Z· ) ∈
t,x
d
d
t,x
d
1
2
2
S 2p ([t, T ]; L2p
ρ (R ; R ))⊗M ([t, T ]; Lρ (R ; R )), so we claim (Ys , Zs ) is the solution of BSDE
(2.2) in the ρ-weighted L2 space if and only if (Y˜st,x , Z˜st,x ) is the solution of BSDE (2.3) in ρweighted L2 space. Therefore we can replace, without losing any generality, Condition (A.4)
by
∗
(H.3) . For any s ∈ [0, T ], y1 , y2 ∈ R1 , x, z ∈ Rd ,
(y1 − y2 ) f (s, x, y1 , z) − f (s, x, y2 , z) ≤ 0.
The main purpose of this paper is to prove the following two theorems. The first one is
about the existence and uniqueness of solutions to BSDE (2.2):
Theorem 2.3 Under Conditions (H.1), (H.2), (H.3)∗ , (H.4) and (H.5), BSDE (2.2) has a
d
1
2
2
d
d
unique solution (Y·t,· , Z·t,· ) ∈ S 2p ([t, T ]; L2p
ρ (R ; R )) ⊗ M ([t, T ]; Lρ (R ; R )).
BSDE (2.2) corresponds to the following PDE with p-growth coefficients:

 ∂u
(t, x) = −L u(t, x) − f t, x, u(t, x), (σ ∗ ∇u)(s, x) , 0 ≤ t ≤ T,
∂t

u(T, x) = h(x).
(2.4)
The other main theorem is the probabilistic representation of PDE (2.4) in the ρ-weighted
L space through its corresponding BSDE:
2
6
Q. Zhang and H.Z. Zhao
Theorem 2.4 Define u(t, x) = Ytt,x , where (Yst,x , Zst,x ) is the solution of BSDE (2.2) under
Conditions (H.1), (H.2), (H.3)∗ , (H.4) and (H.5), then u(t, x) is the unique weak solution of
PDE (2.4). Moreover,
u(s, Xst,x ) = Yst,x , (σ ∗ ∇u)(s, Xst,x ) = Zst,x for a.a. s ∈ [t, T ], x ∈ Rd a.s.
Noticing f is of p-growth on y, we recall the definition for the weak solution of PDE (2.4):
Definition 2.5 Function u is called the weak solution of PDE (2.4) if (u, σ ∗ ∇u) ∈ L2p ([0, T ]; L2p
ρ
N
(Rd ; R1 )) L2 ([0, T ]; L2ρ (Rd ; Rd )) and for an arbitrary ϕ ∈ Cc∞ (Rd ; R1 ),
Z
Z
1
u(t, x)ϕ(x)dx −
u(T, x)ϕ(x)dx −
2
Rd
Rd
Z TZ
˜ (x)dxds
u(s, x)div (b − A)ϕ
−
T
Z
=
t
Z
∗
(σ ∗ ∇u)(s, x) (σ ∗ ∇ϕ)(x)dxds
Rd
f s, x, u(s, x), (σ ∗ ∇u)(s, x) ϕ(x)dxds.
(2.5)
Rd
t
Here A˜j ,
T
Rd
t
Z
Z
1
2
Pd
i=1
∂aij (x)
∂xi ,
and A˜ = (A˜1 , A˜2 , · · ·, A˜d )∗ .
We give the proofs of these two theorems in the latter sections.
In Sections 3-5, by making use of truncated BSDEs, we first deal with BSDE (2.2). To prove
BSDE (2.2) has a unique solution, we use the Alaoglu lemma to derive a weak convergence
sequence in Section 3 and further use the equivalence of norm principle and Rellich-Kondrachov
Compactness Theorem to get a strong convergence sequence in Section 4. Then we complete
the proofs of Theorem 2.3 in Section 5 and consider the corresponding PDE (2.4) to obtain
Theorem 2.4 in Section 6 which gives the probabilistic representation to the weak solution of
PDE (2.4).
Remark 2.6 Let u be the weak solution of PDE (2.4) with coefficient f x, u, (σ ∗ ∇u) which
is independent of t, we can see easily that v(t) , u(T − t) is the unique weak solution of PDE
(1.1).
3 The weak convergence
∗
Assume f satisfies Conditions (H.2), (H.3) and (H.4). We first use a standard cut-off technique
to study a sequence of BSDEs with nonlinear function fn satisfying the linear growth condition
d
1
2
2
d
d
on y. The S 2p ([t, T ]; L2p
ρ (R ; R ))⊗M ([t, T ]; Lρ (R ; R )) valued solution for this kind equation
was studied in [25]. For this, we define for each n ∈ N
fn (s, x, y, z) = f s, x, Πn (y), z ,
where Πn (y) =
inf(n,|y|)
y.
|y|
Then fn : [0, T ] × Rd × R1 × Rd → R1 satisfies
(H.2)0 . For any s ∈ [0, T ], y ∈ R1 , x, z ∈ Rd and the constant C given in (H.2),
|fn (s, x, y, z)| ≤ C(|f0 (s, x)| + |n|p + |z|).
(3.1)
Weak Solutions of PDEs with p-Growth Coefficients
7
(H.3)0 . For any s ∈ [0, T ], y1 , y2 ∈ R1 , x ∈ Rd ,
(y1 − y2 ) fn (s, x, y1 , z) − fn (s, x, y2 , z) ≤ 0.
(H.4)0 . The function (y, z) → fn (s, x, y, z) is continuous, and for any s ∈ [0, T ], y ∈ R1 ,
x, z1 , z2 ∈ Rd and the constant L given in (H.4),
|fn (s, x, y, z1 ) − fn (s, x, y, z2 )| ≤ L|z1 − z2 |.
To see (H.3)0 , if Πn (y1 ) = Πn (y2 ), it is obvious; if Πn (y1 ) 6= Πn (y2 ), then
(y1 − y2 ) fn (s, x, y1 , z) − fn (s, x, y2 , z)
y1 − y2
= (Πn (y1 ) − Πn (y2 )) f (s, x, Πn (y1 ), z) − f (s, x, Πn (y2 ), z)
≤ 0.
Πn (y1 ) − Πn (y2 )
We then study the following BSDE with the global Lipschitz coefficient fn :
Z T
Z T
Yst,x,n = h(XTt,x ) +
fn (r, Xrt,x , Yrt,x,n , Zrt,x,n )dr −
hZrt,x,n , dWr i.
s
(3.2)
s
Notice that under the conditions of Theorem 2.3, the coefficients h and fn satisfy Conditions
(H.1), (H.2)0 and (H.4)0 . Hence by Theorems 2.2 and 2.3 in [25], we have the following proposition:
Proposition 3.1 ([25]) Under the conditions of Theorem 2.3, for fn defined in (3.1), BSDE
(3.2) has a unique solution (Yst,x,n , Zst,x,n ) ∈ S 2 ([t, T ]; L2ρ (Rd ; R1 )) ⊗ M 2 ([t, T ]; L2ρ (Rd ; Rd )). If
we define Ytt,x,n = un (t, x), then un (t, x) is the unique weak solution of the following PDE

 ∂un
(t, x) = −L un (t, x) − fn t, x, un (t, x), (σ ∗ ∇u)(t, x) ,
0 ≤ t ≤ T,
∂t

un (T, x) = h(x).
(3.3)
Moreover,
un (s, Xst,x ) = Yst,x,n , (σ ∗ ∇un )(s, Xst,x ) = Zst,x,n for a.a. s ∈ [t, T ], x ∈ Rd a.s.
The key is to pass the limits in (3.2) and (3.3) in some desired sense. For this we need some
estimates that go beyond those in [24] and [25]. Before we derive some useful estimations to
the solution of BSDEs (3.2), we give the generalized equivalence of norm principle which is an
extension of equivalence of norm principle given in [14], [2], [1] to the cases when ϕ and Ψ are
random.
Lemma 3.2 (generalized equivalence of norm principle [24]) Let ρ be the weight function defined at the beginning of Section 1 and X be a diffusion process defined in (2.1). If s ∈ [t, T ],
ϕ : Ω × Rd → R1 is independent of the σ-field σ{Wr − Wt , t ≤ r ≤ s} and ϕρ−1 ∈ L1 (Ω ⊗ Rd ),
then there exist two constants c > 0 and C > 0 such that
Z
Z
Z
cE[
|ϕ(x)|ρ−1 (x)dx] ≤ E[
|ϕ(Xst,x )|ρ−1 (x)dx] ≤ CE[
|ϕ(x)|ρ−1 (x)dx].
Rd
Rd
Rd
W
Moreover if Ψ : Ω × [t, T ] × Rd → R1 , Ψ (s, ·) is independent of Ft,s
and Ψ ρ−1 ∈ L1 (Ω ⊗ [t, T ] ⊗
Rd ), then
8
Q. Zhang and H.Z. Zhao
T
Z
cE[
Z
Z
|Ψ (s, x)|ρ−1 (x)dxds] ≤ E[
Rd
t
Z
≤ CE[
t
T
Z
T
Z
Rd
t
|Ψ (s, Xst,x )|ρ−1 (x)dxds]
|Ψ (s, x)|ρ−1 (x)dxds].
Rd
First we deduce a useful estimate.
Lemma 3.3 Under Conditions (H.1), (H.2), (H.3)∗ , (H.4) and (H.5), if (Y·t,·,n , Z·t,·,n ) is the
solution of BSDE (3.2), then we have
T
Z
E[
Z
n
t
T
Z
|Yst,x,n |2p ρ−1 (x)dxds] + sup E[
sup
n
Rd
Z
Rd
t
|Yst,x,n |
2p−2
|Zst,x,n |2 ρ−1 (x)dxds] < ∞.
Proof. For M , N > 0 and m ≥ 2, define
ψM (y) = y 2 I{−M ≤y<M } + M (2y − M )I{y≥M } − M (2y + M )I{y<−M }
and
m
ϕN,m (y) = y 2 I{0≤y<N } + N
m−2
2
(
m−2
m
y−
N )I{y≥N } .
2
2
Applying Itˆ
o’s formula to eKr ϕN,m ψM (Yrt,x,n ) for a.e. x ∈ Rd , we have
eKs ϕN,m ψM (Yst,x,n ) + K
T
Z
eKr ϕN,m ψM (Yrt,x,n ) dr
s
+
1
2
Z
s
T
+
s
KT
=e
T
Z
00
0
eKr ϕN,m ψM (Yrt,x,n ) |ψM (Yrt,x,n )|2 |Zrt,x,n |2 dr
0
eKr ϕN,m ψM (Yrt,x,n ) I{−M ≤Yrt,x,n <M } |Zrt,x,n |2 dr
ϕN,m ψM (h(XTt,x ))
Z
−
T
0
Z
T
0
0
eKr ϕN,m ψM (Yrt,x,n ) ψM (Yrt,x,n )fn (r, Xrt,x , Yrt,x,n , Zrt,x,n )dr
+
s
0
heKr ϕN,m ψM (Yrt,x,n ) ψM (Yrt,x,n )Zrt,x,n , dWr i.
(3.4)
s
N
From [24], we note first (Y·t,·,n , Z·t,·,n ) ∈ S 2 ([0, T ]; L2ρ (Rd ; R1 )) M 2 ([0, T ]; L2ρ (Rd ; Rd )). Also
0
0
it is obvious that ϕN,m ψM (Yrt,x,n ) ψM (Yrt,x,n ) is bounded, hence we can use the stochastic
Fubini theorem and take the conditional expectation w.r.t. Fs . Note that the stochastic integral
0
ψM (y)
y
has zero conditional expectation. So if we define
Z
Rd
eKs ϕN,m ψM (Yst,x,n ) ρ−1 (x)dx + E[K
Z
s
Z
T
Z
T
Z
Rd
= 2 when y = 0, we have
eKr ϕN,m ψM (Yrt,x,n ) ρ−1 (x)dxdr|Fs ]
00
0
1
+ E[
eKr ϕN,m ψM (Yrt,x,n ) |ψM (Yrt,x,n )|2 |Zrt,x,n |2 ρ−1 (x)dxdr|Fs ]
2
d
s
R
Z TZ
0
+E[
eKr ϕN,m ψM (Yrt,x,n ) I{−M ≤Yrt,x,n <M } |Zrt,x,n |2 ρ−1 (x)dxdr|Fs ]
d
Z s R
= E[
eKT ϕN,m ψM (h(XTt,x )) ρ−1 (x)dx|Fs ]
Rd
Weak Solutions of PDEs with p-Growth Coefficients
9
Z TZ
0
0
+E[
eKr ϕN,m ψM (Yrt,x,n ) ψM (Yrt,x,n )fn (r, Xrt,x , Yrt,x,n , Zrt,x,n )ρ−1 (x)dxdr|Fs ]
d
Z s R
= E[
eKT ϕN,m ψM (h(XTt,x )) ρ−1 (x)dx|Fs ]
Rd
Z
+E[
T
s
Z
+E[
T
s
0
ψ (Yrt,x,n ) t,x,n
0
Yr
eKr ϕN,m ψM (Yrt,x,n ) M t,x,n
Yr
Rd
× fn (r, Xrt,x , Yrt,x,n , Zrt,x,n ) − fn (r, Xrt,x , 0, Zrt,x,n ) ρ−1 (x)dxdr|Fs ]
Z
0
0
eKr ϕN,m ψM (Yrt,x,n ) ψM (Yrt,x,n )
d
R
× fn (r, Xrt,x , 0, Zrt,x,n ) − fn (r, Xrt,x , 0, 0) ρ−1 (x)dxdr|Fs ]
Z
0
0
eKr ϕN,m ψM (Yrt,x,n ) ψM (Yrt,x,n )fn (r, Xrt,x , 0, 0)ρ−1 (x)dxdr|Fs ]
Z
Z T
+E[
d
Z s R
≤ E[
eKT ϕN,m ψM (h(XTt,x )) ρ−1 (x)dx|Fs ]
Rd
Z
+LE[
Z
+E[
T
Z
s
Rd
T Z
Rd
s
0
0
eKr |ϕN,m ψM (Yrt,x,n ) ||ψM (Yrt,x,n )||Zrt,x,n |ρ−1 (x)dxdr|Fs ]
0
0
eKr |ϕN,m ψM (Yrt,x,n ) ||ψM (Yrt,x,n )||f (r, Xrt,x , 0, 0)|ρ−1 (x)dxdr|Fs ].
Taking the limit as M → ∞ first, then the limit as N → ∞, by the monotone convergence
theorem and Young inequality, we have
Z
Z TZ
m
Ks
t,x,n m −1
e |Ys
| ρ (x)dx + KE[
eKr |Yrt,x,n | ρ−1 (x)dxdr|Fs ]
Rd
s
Rd
Z TZ
m(m − 1)
m−2
E[
eKr |Yrt,x,n |
|Zrt,x,n |2 ρ−1 (x)dxdr|Fs ]
+
2
s
Rd
Z
≤ E[
eKT |h(XTt,x )|m ρ−1 (x)dx|Fs ]
Rd
Z
+mLE[
Z
+mE[
T
s
Rd
T Z
Rd
s
Z
≤ E[
Rd
Z
eKr |Yrt,x,n |m−2 |Yrt,x,n ||Zrt,x,n |ρ−1 (x)dxdr|Fs ]
eKr |Yrt,x,n |m−2 |Yrt,x,n ||f (r, Xrt,x , 0, 0)|ρ−1 (x)dxdr|Fs ]
Z
eKT |h(XTt,x )|m ρ−1 (x)dx|Fs ] + m(L2 + 1)E[
T
s
Z
Rd
Z TZ
m
E[
eKr |Yrt,x,n |m−2 |Zrt,x,n |2 ρ−1 (x)dxdr|Fs ]
4
d
s
R
Z TZ
m m−2
+ ·
E[
eKr |Yrt,x,n |m ρ−1 (x)dxdr|Fs ]
4
m
s
Rd
Z TZ
m 2
+ · E[
eKr |f (r, Xrt,x , 0, 0)|m ρ−1 (x)dxdr|Fs ].
4 m
s
Rd
eKr |Yrt,x,n |m ρ−1 (x)dxdr|Fs ]
+
(3.5)
Here and in the following, Cp is a generic constant. Therefore, taking K > m(L2 + 1) + m−2
4 ,
we have
Z T
Z
Z TZ
m−2
E[
sup
|Yst,x,n |m ρ−1 (x)dxds] + sup E[
|Yst,x,n |
|Zst,x,n |2 ρ−1 (x)dxds]
t
n
Rd
n
t
Rd
10
Q. Zhang and H.Z. Zhao
Z
≤ Cp E[
Rd
Z
|h(XTt,x )|m ρ−1 (x)dx] + Cp E[
|h(x)|m ρ−1 (x)dx + Cp
≤ Cp
Rd
Z
T
Z
T
Z
t
Z
|f0 (s, Xst,x )|m ρ−1 (x)dxds]
Rd
|f0 (s, x)|m ρ−1 (x)dxds < ∞.
Rd
t
In particular, taking m = 2p, then the lemma follows.
Taking m = 2 in the proof of Lemma 3.3, we know
Z
E[
T
Z
sup
n
t
Z
|Yst,x,n |2 ρ−1 (x)dxds
Rd
T
Z
+ sup E[
n
Rd
t
|Zst,x,n |2 ρ−1 (x)dxds] < ∞.
(3.6)
Also we have
Z
T
Z
sup E[
n
Rd
t
Z
T
Z
≤ sup E[
n
Rd
t
|fn (s, Xst,x , Yst,x,n , Zst,x,n )|2 ρ−1 (x)dxds]
C(|f0 (s, Xst,x )|2 + |Yst,x,n |2p + |Zst,x,n |2 )ρ−1 (x)dxds] < ∞.
The last inequality follows from the equivalence of norms principle and Lemma 3.3. Define
Ust,x,n = fn (s, Xst,x , Yst,x,n , Zst,x,n ), s ≥ t, then
T
Z
sup E[
n
Z
Rd
t
(|Yst,x,n |2 + |Zst,x,n |2 + |Ust,x,n |2 )ρ−1 (x)dxds] < ∞.
(3.7)
Therefore by using the Alaoglu lemma, we know that there exists a subsequence, still denoted by
(Yst,x,n , Zst,x,n , Ust,x,n ), s.t. (Yst,x,n , Zst,x,n , Ust,x,n ) converges weakly to the limit (Yst,x , Zst,x , Ust,x )
in L2ρ (Ω ⊗ [t, T ] ⊗ Rd ; R1 ⊗ Rd ⊗ R1 ) (or equivalently L2 (Ω ⊗ [t, T ]; L2ρ (Rd ; R1 ) ⊗ L2ρ (Rd ; Rd ) ⊗
L2ρ (Rd ; R1 )). Now we take the weak limit in L2ρ (Ω ⊗ [t, T ] ⊗ Rd ; R1 )) to BSDEs (3.2), we can
verify that (Yst,x , Zst,x , Ust,x ) satisfies the following BSDE:
Yst,x
=
h(XTt,x )
Z
+
s
T
Urt,x dr
Z
−
T
hZrt,x , dWr i.
(3.8)
s
For this, we will check the weak convergence term by term. The weak convergence to the first
term is deduced by the definition of Yst,x . The weak convergence to the second term is trivial
since h(XTt,x ) is independent of n. We then check the weak convergence to the last two terms.
RT
RT R
Let η ∈ L2ρ (Ω ⊗ [t, T ] ⊗ Rd ; R1 ). Then noticing t supn E[ s Rd |Urt,x,n |2 ρ−1 (x)dxdr]ds < ∞
due to (3.7), by Lebesgue’s dominated convergence theorem, we have
Z
T
Z
T
Z
(Urt,x,n − Urt,x )drη(s, x)ρ−1 (x)dxds]|
|E[
Rd
t
Z
T
Z
T
s
Z
= |E[
t
Z
≤
t
T
Rd
s
Z
|E[
s
T
Z
Rd
(Urt,x,n − Urt,x )η(s, x)ρ−1 (x)dxdrds]|
(Urt,x,n − Urt,x )η(s, x)ρ−1 (x)dxdr]|ds −→ 0,
as n → ∞.
On the other hand we know for fixed s and x, η(s, x) ∈ L2 (Ω). So there exists ϕ(s, x, r) s.t.
RT
η(s, x) = E[η(s, x)] + t hϕ(s, x, r), dWr i. It is easy to see that for a.e. s ∈ [t, T ], ϕ(s, ·, ·) ∈
Weak Solutions of PDEs with p-Growth Coefficients
11
RT
RT R
L2 (Ω ⊗ [t, T ] ⊗ Rd ; R1 ). Noticing that t supn E[ s Rd |Zrt,x,n |2 ρ−1 (x)dxdr]ds < ∞ due to
(3.7) and using Lebesgue’s dominated convergence theorem again, we obtain
Z
T
Z
T
Z
hZrt,x,n − Zrt,x , dWr iη(s, x)ρ−1 (x)dxds]|
|E[
Z
Rd
t
T Z
Rd
t
T
Z
hϕ(s, x, r), dWr i)]ρ−1 (x)dxds|
hZrt,x,n − Zrt,x , ϕ(s, x, r)idr]ρ−1 (x)dxds|
s
T
Z
t
T
t
Z
hZrt,x,n − Zrt,x , ϕ(s, x, r)iρ−1 (x)dxdr]|ds −→ 0,
|E[
≤
Z
T
Z
E[
Rd
t
T
hZrt,x,n − Zrt,x , dWr i(E[η(s, x)] +
s
=|
Z
T
Z
E[
=|
Z
s
Rd
s
as n → ∞.
Needless to say, if we can show BSDE (3.2) is indeed BSDE (2.2), then we can say (Yst,x , Zst,x )
is a solution of BSDE (2.2). The key is to prove that Ust,x = f (s, Xst,x , Yst,x , Zst,x ) for a.a.
s ∈ [t, T ], x ∈ Rd a.s. However, the weak convergence of Y n , U n and Z n are not enough to this.
The crucial point in this analysis is to establish the strong convergence of Y n and Z n , which
will be done in next section.
4 The strong convergence and the identification of the limiting
BSDEs
In this section, we will show that the combination of methods of weak convergence and strong
convergence of a subsequence (Yst,x,n , Zst,x,n ) gives an effective way to prove that the limit
(Yst,x , Zst,x ) satisfies BSDE (2.2). In contrast, the direct proof that BSDE (3.2) converges
strongly to BSDE (2.2) by using the strongly convergent subsequence (Yst,x,n , Zst,x,n ) without the weak convergence argument will encounter some complications. This is due to that the
dominated convergence theorem does not seem to apply immediately to BSDE (3.2). We start
from an easy lemma.
Lemma 4.1 Under the conditions of Theorem 2.3, if un (t, x) is the weak solution of PDE
RT R
(3.3), then supn 0 Rd |un (s, x)|2p ρ−1 (x)dxds < ∞. Furthermore,
Z
T
Z
lim sup
N →∞ n
0
|un (s, x)|2 IUN c (x)ρ−1 (x)dxds = 0,
Rd
where UN c = {x ∈ Rd : |x| > N }.
Proof. The L2p
of norm principle and
ρ integrability of un follows directly from the equivalence
R
−1
Lemma 3.3. Let’s prove the second part of this lemma. Since Rd ρ (x)dx < ∞,
T
Z
Z
lim sup
N →∞ n
0
|un (s, x)|2 IUN c (x)ρ−1 (x)dxds
Rd
Z TZ
Z TZ
1
p−1
p
sup
|un (s, x)|2p ρ−1 (x)dxds p
|IUN c (x)| p−1 ρ−1 (x)dxds p
N →∞
d
n
0
Rd
Z 0 R
p−1
≤ lim Cp
IUN c (x)ρ−1 (x)dx p = 0.
≤ lim
N →∞
Rd
12
Q. Zhang and H.Z. Zhao
The following two theorems quoted in [21] will be used in this section.
Theorem 4.2 (c.f. [21]) Let X ⊂⊂ H ⊂ Y be Banach spaces, with X reflexive. Here X ⊂⊂ H
means X is compactly embedded in H. Suppose that un is a sequence that is uniformly bounded
in L2 ([0, T ]; X), and dun /dt is uniformly bounded in Lp (0, T ; Y ), for some p > 1. Then there
is a subsequence that converges strongly in L2 ([0, T ]; H).
Theorem 4.3 (Rellich-Kondrachov Compactness Theorem c.f. [21]) Let B be a bounded C 1
domain in Rd . Then H 1 (B) is compactly embedded in L2 (B).
Lemma 4.4 Under the conditions of Theorem 2.3, if (Yst,x,n , Zst,x,n ) is the solution of BSDEs
(3.2) and Yst,x is the weak limit of Yst,x,n in L2ρ (Ω ⊗[t, T ]⊗Rd ; R1 ), then there is a subsequence of
Yst,x,n , still denoted by Yst,x,n , converges strongly to Yst,x in L2 (Ω ⊗[t, T ]; L2ρ (Rd ; R1 )). Moreover,
we have
s,Xst,x
Yst,x = Ys
and E[
RT R
t
Rd
s,Xst,x
, Zst,x = Zs
for any s ∈ [t, T ], a.a. x ∈ Rd a.s.
(4.1)
|Yst,x |2p ρ−1 (x)dxds] < ∞.
Proof. Let un (s, x) = Yss,x,n . Then by Proposition 3.1, un (s, Xst,x,n ) = Yst,x,n , (σ ∗ ∇un )(s, Xst,x,n )
= Zst,x,n for a.a. s ∈ [t, T ], x ∈ Rd a.s. We claim that un (s, x) satisfies the following PDE in
∗
Hρ1 (Rd ; R1 ):
dun (s, x)/ds = −L un (s, x) − fn s, x, un (s, x), (σ ∗ ∇un )(s, x) ,
0 ≤ s ≤ T.
(4.2)
To prove this claim, first note that un are uniformly bounded in L2 ([0, T ]; Hρ1 (Rd ; R1 )) by the
equivalence of norm principle and the uniform ellipticity condition of σ:
Z
T
Z
(|un (s, x)|2 + |∇un (s, x)|2 )ρ−1 (x)dxds
sup
n
Rd
0
Z
T
Z
≤ Cp sup
n
0
Z
≤ Cp sup E[
n
0
(|un (s, x)|2 + |(σ ∗ ∇un )(s, x)|2 )ρ−1 (x)dxds
Rd
T Z
Rd
(|Ys0,x,n |2 + |Zs0,x,n |2 )ρ−1 (x)dxds] < ∞.
(4.3)
∗
Then we can deduce that dun /ds are uniformly bounded in L2 ([0, T ]; Hρ1 (Rd ; R1 )). For this, we
∗
need to prove that L un and fn ∈ L2 ([0, T ]; Hρ1 (Rd ; R1 )) are uniformly bounded respectively.
First note that for i = 1, 2, · · · , d,
|
∂ρ−1 (x)
−qxi
q
|=|
|≤
≤ qρ−1 (x).
q+1
∂xi
(1 + |x|) |x|
(1 + |x|)q+1
Moreover, recalling the form of L and noticing the conditions on b and σ in (H.5), we can see
that aij and bi are uniformly bounded for all i, j. So for arbitrary s ∈ [0, T ], ψ ∈ Cc∞ (Rd ; R1 ),
we have
Weak Solutions of PDEs with p-Growth Coefficients
Z
13
L un (s, x) · ψ(x)ρ−1 (x)dx
Rd
d
d
1 X ∂un (s, x) ∂(aij ψρ−1 )(x) X
∂(bi ψρ−1 )(x) −
dx
un (s, x)
2 i,j=1
∂xi
∂xj
∂xi
i=1
Z
−
=
Rd
Z
≤
(
Rd
d
d
d
X
X
X
∂(aij ψρ−1 )(x)
∂(bi ψρ−1 )(x)
∂un (s, x)
| + |un (s, x)|)(
|+
|)dx
|
|
|
∂xi
∂xj
∂xi
i,j=1
i=1
i=1
d
d
X
X
∂ψ(x)
∂un (s, x)
| + |un (s, x)|)(
| + |ψ(x)|)ρ−1 (x)dx
|
|
∂xi
∂x
j
Rd i=1
j=1
v
v
uZ
uZ
d
d
u
X
X
u
∂u
(s,
x)
∂ψ(x)
n
t
2
−1
≤ Cp
| + |un (s, x)|) ρ (x)dxt
| + |ψ(x)|)2 ρ−1 (x)dx
(
(
|
|
∂xi
∂xj
Rd i=1
Rd j=1
Z
≤ Cp
(
≤ Cp kun (s, x)kHρ1 (Rd ;R1 ) kψkHρ1 (Rd ;R1 ) .
As Cc∞ (Rd ; R1 ) is dense in Hρ1 (Rd ; R1 ), therefore for arbitrary s ∈ [0, T ], it follows that
kL un (s, ·)kHρ1 ∗ (Rd ;R1 ) ≤ Cp kun (s, ·)kHρ1 (Rd ;R1 ) and by (4.3), we have
sup kL un k2L2 ([0,T ];Hρ1 ∗ (Rd ;R1 )) ≤ Cp sup
n
n
Z
0
T
Z
(|un (s, x)|2 + |∇un (s, x)|2 )ρ−1 (x)dxds < ∞.
Rd
Also using Lemma 3.3 and the equivalence of norm principle again, we obtain
T
Z
0
kfn (s, ·, un (s, ·), (σ ∗ ∇un )(s, ·))k2L2ρ (Rd ;R1 ) ds
T
Z
Z
(|f0 (s, x)|2 + |Ys0,x,n |2p + |Zs0,x,n |2 )ρ−1 (x)dxds] < ∞.
≤ Cp E[
Rd
0
∗
∗
Hence fn ∈ L2 ([0, T ]; L2ρ (Rd ; R1 )) ⊂ L2 ([0, T ]; Hρ1 (Rd ; R1 )) and
sup kfn k2L2 ([0,T ];Hρ1 ∗ (Rd ;R1 ))
n
Z
T
kfn (s, ·, un (s, ·), (σ ∗ ∇un )(s, ·))k2L2ρ (Rd ;R1 ) ds
≤ Cp sup
n
0
Z
T
Z
≤ Cp sup E[
n
0
Rd
(|f0 (s, x)|2 + |Ys0,x,n |2p + |Zs0,x,n |2 )ρ−1 (x)dxds] < ∞.
∗
Therefore we conclude that dun /ds are uniformly bounded in L2 ([0, T ]; Hρ1 (Rd ; R1 )).
Noticing Theorem 4.3 and applying Theorem 4.2 with X = Hρ1 (U1 ; R1 ), H = L2ρ (U1 ; R1 )
∗
and Y = Hρ1 (U1 ; R1 ), we are able to extract a subsequence of un (s, x), denoted by u1n (s, x),
which converges strongly in L2 ([0, T ]; L2ρ (U1 ; R1 )). It is obvious that this u1n (s, x) satisfies the
conditions in Theorem 4.2. Applying Theorem 4.2 again, we are able to extract a subsequence
of u1n (s, x), denoted by u2n (s, x), that converges strongly in L2 ([0, T ]; L2ρ (U2 ; R1 )). Actually we
can do this procedure for all Ui , i = 1, 2, · · ·. Now we pick up the diagonal sequence uii (s, x),
i = 1, 2, ··· and still denote this sequence by un for convenience. It is easy to see that un converges
strongly in all L2 ([0, T ]; L2ρ (Ui ; R1 )), i = 1, 2, · · ·. For arbitrary ε > 0, noticing Lemma 4.1, we
can find j(ε) large enough such that
14
Q. Zhang and H.Z. Zhao
T
Z
Z
2|un (s, x)|2 ρ−1 (x)dxds <
sup
n
0
Uj(ε)
c
ε
.
3
For this j(ε), there exists n∗ (ε) > 0 s.t. when m, n ≥ n∗ (ε), we know
kum − un k2L2 ([0,T ];L2ρ (Uj(ε) ;R1 )) =
T
Z
Z
0
|um (s, x) − un (s, x)|2 ρ−1 (x)dxds <
Uj(ε)
ε
.
3
Therefore as m, n ≥ n∗ (ε),
kum − un k2L2 ([0,T ];L2ρ (Rd ;R1 ))
Z TZ
Z
2 −1
≤
|um (s, x) − un (s, x)| ρ (x)dxds +
0
Uj(ε)
0
T
Z
(2|um (s, x)|2 + 2|un (s, x)|2 )ρ−1 (x)dxds
Uj(ε) c
< ε.
That is to say un converges strongly in L2 ([0, T ]; L2ρ (Rd ; R1 )). Now using the equivalence of
norm principle, we know as m, n → ∞,
kYst,x,m − Yst,x,n k2L2 (Ω⊗[t,T ];L2ρ (Rd ;R1 ))
Z TZ
= E[
|um (s, Xst,x ) − un (s, Xst,x )|2 ρ−1 (x)dxds]
Rd
t
T
Z
Z
|um (s, x) − un (s, x)|2 ρ−1 (x)dxds −→ 0.
≤ Cp
(4.4)
Rd
t
So the claim that Yst,x,n converges strongly in L2 (Ω ⊗ [t, T ]; L2ρ (Rd ; R1 )) follows. But we know
that Yst,x is the weak limit of Yst,x,n in L2 (Ω ⊗ [t, T ]; L2ρ (Rd ; R1 )), therefore Yst,x,n converges
strongly to Yst,x in L2 (Ω ⊗ [t, T ]; L2ρ (Rd ; R1 )).
To see (4.1), first notice that in BSDE (3.8), h(·) ∈ L2ρ (Rd ; R1 ) and U·t,· ∈ M 2 ([t, T ]; L2ρ (Rd ; R1 ))
are given, so there exists (Y·t,· , Z·t,· ) ∈ M 2 ([t, T ]; L2ρ (Rd ; R1 )) ⊗ M 2 ([t, T ]; L2ρ (Rd ; Rd )) satisfying the spatial integral form of BSDE (3.8). By Lemma 3.3 and Proposition 3.4 in [24],
(Yst,x , Zst,x ) ∈ S 2 ([t, T ]; L2ρ (Rd ; R1 )) ⊗ M 2 ([t, T ]; L2ρ (Rd ; Rd )) is the unique solution of BSDE
s,X t,x
s,X t,x
(3.8) and Yst,x = Ys s , Zst,x = Zs s for any s ∈ [t, T ], a.a. x ∈ Rd a.s. If we define
Yss,x = u(s, x), then we can prove the strong limit of un (s, x) in L2 ([0, T ]; L2ρ (Rd ; R1 )) is u(s, x)
and Yst,x = u(s, Xst,x ) for a.a. s ∈ [t, T ], x ∈ Rd a.s. To prove this, we only need to see that, by
the equivalence of norm principle,
Z TZ
lim
|un (s, x) − u(s, x)|2 ρ−1 (x)dxds
n→∞
Rd
0
T
Z
Z
≤ lim Cp E[
n→∞
Rd
0
T
Z
Z
= lim Cp E[
n→∞
Rd
0
s,Xs0,x 2 −1
|un (s, Xs0,x ) − Ys
| ρ
(x)dxds]
|Ys0,x,n − Ys0,x |2 ρ−1 (x)dxds] = 0,
and
Z
E[
t
T
Z
Rd
|Yst,x − u(s, Xst,x )|2 ρ−1 (x)dxds]
(4.5)
Weak Solutions of PDEs with p-Growth Coefficients
T
Z
= E[
Z
s,Xst,x
|Ys
Rd
t
Z
T
Z
≤ Cp
t
Rd
15
− u(s, Xst,x )|2 ρ−1 (x)dxds]
|Yss,x − u(s, x)|2 ρ−1 (x)dxds = 0.
(4.6)
RT R
Moreover, we can prove that E[ t Rd |Yst,x |2p ρ−1 (x)dxds] < ∞. For this, by the equivalence
RT R
of norm principle, we only need to prove that 0 Rd |u(s, x)|2p ρ−1 (x)dxds < ∞. To assert
the claim, we first prove that we can find a subsequence of {un (s, x)}∞
n=1 still denoted by
{un (s, x)}∞
,
s.t.
n=1
un (s, x) −→ u(s, x) and sup |un (s, x)|2p < ∞ for a.e. s ∈ [t, T ], x ∈ Rd .
(4.7)
n
RT R
For this, from (4.5), we know that 0 Rd |un (s, x) − u(s, x)|2 ρ−1 (x)dxds −→ 0. Therefore we
may assume without losing any generality that un (s, x) −→ u(s, x) for a.e. s ∈ [0, T ], x ∈ Rd
∞
and extract a subsequence of {un (s, x)}∞
n=1 , still denoted by {un (s, x)}n=1 , s.t.
Z TZ
1
|un+1 (s, x) − un (s, x)|ρ−1 (x)dxds ≤ n .
2
0
Rd
For any n,
|un (s, x)| ≤ |u1 (s, x)| +
n−1
X
|ui+1 (s, x) − ui (s, x)| ≤ |u1 (s, x)| +
∞
X
|ui+1 (s, x) − ui (s, x)|.
i=1
i=1
Then by the triangle inequality of the norm, we have
Z TZ
sup |un (s, x)|ρ−1 (x)dxds
T
Z
≤
|u1 (s, x)|ρ−1 (x)dxds +
Rd
0
T
Z
n
Rd
0
Z
≤
0
∞ Z
X
i=1
T
Z
|ui+1 (s, x) − ui (s, x)|ρ−1 (x)dxds
Rd
0
∞
X
1
|u1 (s, x)|ρ−1 (x)dxds +
< ∞.
i
2
d
R
i=1
Z
Therefore, (4.7) follows from the above. By a similar argument as in Lemma 4.1, for this
subsequence un , we can prove that for any δ > 0,
Z TZ
lim sup
|un (s, x)|2p−δ I{|un (s,x)|2p−δ >N } (s, x)ρ−1 (x)dxds = 0.
N →∞ n
0
Rd
That is to say that |un (s, x)|2p is uniformly integrable. Together with un (s, x) −→ u(s, x) for
a.e. s ∈ [0, T ], x ∈ Rd , we have
Z TZ
Z TZ
|u(s, x)|2p−δ ρ−1 (x)dxds = lim
|un (s, x)|2p−δ ρ−1 (x)dxds
0
Z
Rd
T
n→∞
Z
≤ sup
n
0
Rd
0
Rd
|un (s, x)|2p−δ ρ−1 (x)dxds ≤ Cp sup
n
Z
0
T
Z
2p−δ
|un (s, x)|2p ρ−1 (x)dxds 2p ≤ Cp ,
Rd
where the last Cp < ∞ is a constant independent of n and δ. Then using Fatou lemma to take
RT R
the limit as δ → 0 in the above inequality, we can get 0 Rd |u(s, x)|2p ρ−1 (x)dxds < ∞.
16
Q. Zhang and H.Z. Zhao
Considering the strongly convergent subsequence {Y·t,·,n }∞
n=1 and using a standard argument
to BSDE (3.2), we can prove that for arbitrary m, n
Z
E[
Rd
T
Z
≤ 2L2 E[
Z
Rd
s
s
T
Z
E[
×Cp
Z TZ
1
|Zrt,x,m − Zrt,x,n |2 ρ−1 (x)dxdr]
−
+ E[
2
s
Rd
s
Z TZ
|Yrt,x,m − Yrt,x,n |2 ρ−1 (x)dxdr] + 2 E[
|Yrt,x,m − Yrt,x,n |2 ρ−1 (x)dxdr]
Yst,x,n |2 ρ−1 (x)dx]
|Yst,x,m
s
Z
Rd
(|f0 (r, x)|2 + |Yrt,x,n |2p + |Zrt,x,n |2 )ρ−1 (x)dxdr].
Rd
s
So by Condition (H.2) and Lemma 3.3, we can conclude that the corresponding subsequence
of {Z·t,·,n }∞
n=1 converges strongly as well. Certainly the strong convergence limit should be
identified with the weak convergence limit Z·t,· , hence the following corollary follows without a
surprise.
Corollary 4.5 Let (Y·t,· , Z·t,· ) be the solution to BSDE (2.2) and (Y·t,·,n , Z·t,·,n ) be the subsequence of the solutions to BSDE (3.2), which Y·t,·,n converges strongly to Y·t,· in L2 (Ω ⊗
[t, T ]; L2ρ (Rd ; R1 )), then Z·t,·,n also converges strongly to Z·t,· in L2 (Ω ⊗ [t, T ]; L2ρ (Rd ; Rd )).
From Lemma 4.4, we know that there is a subsequence of Y·t,·,n , still denoted by Y·t,·,n ,
converges strongly to Y·t,· in L2 (Ω ⊗[t, T ]; L2ρ (Rd ; R1 )), i.e. M 2 ([t, T ]; L2ρ (Rd ; R1 )). Indeed, doing
Itˆ
o’s formula to ψM (Yrt,x,n − Yrt,x ) and eKr ϕn,m ψM (Yrt,x ) , with Corollary 4.5 we can further
d
1
prove that Y·t,·,n converges to Y·t,· in S 2 ([t, T ]; L2ρ (Rd ; R1 )) and Y·t,· ∈ S 2p ([t, T ]; L2p
ρ (R ; R ))
respectively (To see similar calculations, one can refer to the argument in the proof of Lemma
3.3 in [24]).
Proposition 4.6 For (Y·t,· , Z·t,· ) and (Y·t,·,n , Z·t,·,n ) given in Corollary 4.5, Y·t,· is the limit of
d
1
Y·t,·,n in S 2 ([t, T ]; L2ρ (Rd ; R1 )). Moreover, Y·t,· ∈ S 2p ([t, T ]; L2p
ρ (R ; R )).
Now we are ready to prove the identification of the limiting BSDEs.
Lemma 4.7 The random field U , Y and Z have the following relation:
Ust,x = f (s, Xst,x , Yst,x , Zst,x ) for a.a. s ∈ [t, T], x ∈ Rd a.s.
(4.8)
Proof. Let K be a set in Ω ⊗ [t, T ] ⊗ Rd s.t. supn |Yst,x,n | + supn |Zst,x,n | + |f0 (s, Xst,x )| < K.
Similar to the proof of (4.7), we can find a subsequence of {(Yst,x,n , Zst,x,n )}∞
n=1 , still denoted by
t,·,n
t,x,n
t,x,n
t,x
t,x
{(Yst,x,n , Zst,x,n )}∞
,
satisfying
(Y
,
Z
)
−→
(Y
,
Z
)
and
sup
|Y
|+supn |Z·t,·,n | <
·
n
s
s
s
s
n=1
∞ for a.e. s ∈ [t, T ], x ∈ Rd a.s. Then it turns out that as K → ∞, K ↑ Ω ⊗[t, T ]⊗Rd . Moreover
it is easy to see that along the subsequence,
Z
E[
T
t
≤ 6C 2 E[
Z
Rd
Z T
2(sup |fn (s, Xst,x , Yst,x,n , Zst,x,n )|2 + |f (s, Xst,x , Yst,x , Zst,x )|2 )IK (s, x)ρ−1 (x)dxds]
n
Z
Rd
t
+6C 2 E[
Z
t
T
(|f0 (s, Xst,x )|2 + sup |Yst,x,n |2p + sup |Zst,x,n |2 )IK (s, x)ρ−1 (x)dxds]
Z
Rd
n
n
(|f0 (s, Xst,x )|2 + |Yst,x |2p + |Zst,x |2 )IK (s, x)ρ−1 (x)dxds] < ∞.
Weak Solutions of PDEs with p-Growth Coefficients
17
Thus, we can apply Lebesgue’s dominated convergence theorem to the following calculation:
T
Z
Z
|fn (s, Xst,x , Yst,x,n , Zst,x,n )IK (s, x) − f (s, Xst,x , Yst,x , Zst,x )IK (s, x)|2 ρ−1 (x)dxds]
lim E[
n→∞
Z
T
t
Rd
Rd
n→∞
Z
= E[
t
T
Z
lim |fn (s, Xst,x , Yst,x,n , Zst,x,n ) − f (s, Xst,x , Yst,x , Zst,x )|2 IK (s, x)ρ−1 (x)dxds]
Z
≤ 2E[
lim |fn (s, Xst,x , Yst,x,n , Zst,x,n ) − f (s, Xst,x , Yst,x,n , Zst,x,n )|2 IK (s, x)ρ−1 (x)dxds]
Rd n→∞
t
Z
T
Z
+2E[
lim |f (s, Xst,x , Yst,x,n , Zst,x,n ) − f (s, Xst,x , Yst,x , Zst,x )|2 IK (s, x)ρ−1 (x)dxds].
Rd n→∞
t
(4.9)
Since Yst,x,n −→ Yst,x for a.e. s ∈ [t, T ], x ∈ Rd a.s., there exists a N (s, x, ω) s.t. when
n ≥ N (s, x, ω), |Yst,x,n | ≤ |Yst,x | + 1. So taking n ≥ max{N (s, x, ω), |Yst,x | + 1}, we have
inf(n,|Yst,x,n |) t,x,n
fn (s, Xst,x , Yst,x,n , Zst,x,n ) = f (s, Xst,x , |Y t,x,n
, Zst,x,n ) = f (s, Xst,x , Yst,x,n , Zst,x,n ).
Ys
|
s
That is to say limn→∞ |fn (s, Xst,x , Yst,x,n , Zst,x,n ) − f (s, Xst,x , Yst,x,n , Zst,x,n )|2 = 0 for a.e. s ∈
[t, T ], x ∈ Rd a.s. On the other hand, limn→∞ |f (s, Xst,x , Yst,x,n , Zst,x,n )−f (s, Xst,x , Yst,x , Zst,x )|2 =
0 for a.e. s ∈ [t, T ], x ∈ Rd a.s. is obvious due to the continuity of (y, z) → f (s, x, y, z).
Therefore by (4.9), fn (s, Xst,x , Yst,x,n , Zst,x,n )IK (s, x) = Ust,x,n IK (s, x) converges strongly to
f (s, Xst,x , Yst,x , Zst,x )IK (s, x) in L2ρ (Ω ⊗ [t, T ] ⊗ Rd ; R1 ), but Ust,x,n IK (s, x) converges weakly to
Ust,x IK (s, x) in L2ρ (Ω ⊗ [t, T ] ⊗ Rd ; R1 ), so f (s, Xst,x , Yst,x , Zst,x )IK (s, x) = Ust,x IK (s, x) for a.e.
r ∈ [t, T ], x ∈ Rd a.s. The lemma follows when K → ∞.
Proof of Theorem 2.3. With Lemma 4.7 and Proposition 4.6, the existence of solutions to
BSDE (2.2) is easy to see. Now we prove the uniqueness. If there is another solution (Y˜st,x , Z˜st,x )
to BSDE (2.2), then for a.e. x ∈ Rd , (Yst,x − Y˜st,x , Zst,x − Z˜st,x ) satisfies
Yst,x − Y˜st,x =
T
Z
f (r, Xrt,x , Yrt,x , Zrt,x ) − f (r, Xrt,x , Y˜rt,x , Z˜rt,x ) dr −
s
Z
T
hZrt,x − Z˜rt,x , dWr i.
s
Applying Itˆ
o’s formula to |Yst,x − Y˜st,x |2 , by the stochastic Fubini theorem and Conditions (H.3)
and (H.4), we have
Z
E[
Rd
|Yst,x − Y˜st,x |2 ρ−1 (x)dx] + E[
Z
s
Z
≤ 2L2 E[
s
T
Z
T
Z
Rd
∗
|Zrt,x − Z˜rt,x |2 ρ−1 (x)dxdr]
1
|Yrt,x − Y˜rt,x |2 ρ−1 (x)dxdr] + E[
2
d
R
Z
s
T
Z
Rd
|Zrt,x − Z˜rt,x |2 ρ−1 (x)dxdr].
By Gronwall’s inequality, the uniqueness of the solution to BSDE (2.2) follows immediately. 5 The PDEs
Now we make use of the results for BSDE (2.2) to give the probabilistic representation to PDEs
with p-growth coefficients. Actually the solution of BSDE in the ρ-weighted L2 space gives the
18
Q. Zhang and H.Z. Zhao
unique weak solution of its corresponding PDE (2.4).
Proof of Theorem 2.4. Using Corollary 4.5, we first prove the relationship between (Y, Z)
and u. Since (4.6), we only need to prove that (σ ∗ ∇u)(s, Xst,x ) = Zst,x for a.a. s ∈ [t, T ],
x ∈ Rd a.s. This can be deduced from (4.1) and the strong convergence of Z·t,·,n to Z·t,· in
L2 (Ω ⊗ [t, T ]; L2ρ (Rd ; R1 )) by the similar argument as in Proposition 4.2 in [24].
We then prove that u(t, x) is the unique weak solution of PDE (2.4). We still start from
PDE (3.3). Let un (s, x) be the weak solution of PDE (3.3). Then by the definition for the weak
N
solution of PDE, we know (un , σ ∗ ∇un ) ∈ L2 ([0, T ]; L2ρ (Rd ; R1 )) L2 ([0, T ]; L2ρ (Rd ; Rd )) and
for an arbitrary ϕ ∈ Cc∞ (Rd ; R1 ),
Z
Z
1
un (t, x)ϕ(x)dx −
un (T, x)ϕ(x)dx −
2
d
d
R
R
Z TZ
˜ (x)dxds
−
un (s, x)div (b − A)ϕ
T
Z
Z
∗
(σ ∗ ∇un )(s, x) (σ ∗ ∇ϕ)(x)dxds
Rd
t
fn s, x, un (s, x), (σ ∗ ∇un )(s, x) ϕ(x)dxds.
=
t
T
Rd
t
Z
Z
(5.1)
Rd
We can prove along a subsequence that each term of (5.1) converges to the corresponding term
of (2.5). By (4.5), we know that un converges strongly to u in L2ρ ([0, T ] ⊗ Rd ; R1 ), thus un also
˜ (x)|) < ∞ and ρ is a continuous functional
converges weakly. Moreover, supx∈Rd (|div (b − A)ϕ
in Rd , so it is obvious that
Z
T
Z
lim
n→∞
t
˜ (x)dxds =
un (s, x)div (b − A)ϕ
Rd
Z
t
T
Z
˜ (x)dxds.
u(s, x)div (b − A)ϕ
Rd
Also it is easy to see that
1
lim
n→∞ 2
T
Z
Z
∗
(σ ∗ ∇un )(s, x) (σ ∗ ∇ϕ)(x)dxds
Rd
t
Z Z
1 T
un (s, x)div(σσ ∗ ∇ϕ)(x)ρ(x)ρ−1 (x)dxds
= lim −
n→∞
2 t Rd
Z Z
1 T
=−
u(s, x)div(σσ ∗ ∇ϕσ)(x)ρ(x)ρ−1 (x)dxds
2 t Rd
Z Z
∗
1 T
(σ ∗ ∇u)(s, x) (σ ∗ ∇ϕ)(x)dxds.
=
2 t Rd
Also we have proved that fn (s, Xst,x , Yst,x,n , Zst,x,n ) converges weakly to f (s, Xst,x , Yst,x , Zst,x ) in
L2ρ (Ω ⊗ [t, T ] ⊗ Rd ; R1 ). In fact we can follow the same procedure as in the proof of Lemma
4.7 to prove fn s, x, un (s, x), (σ ∗ ∇un )(s, x) converges weakly to f s, x, u(s, x), (σ ∗ ∇u)(s, x)
in L2ρ ([t, T ] ⊗ Rd ; R1 ). So we have
Z
T
Z
lim
n→∞
Z
T
t
Z
=
t
Rd
fn s, x, un (s, x), (σ ∗ ∇un )(s, x) ϕ(x)dxds
Rd
f s, x, u(s, x), (σ ∗ ∇u)(s, x) ϕ(x)dxds.
Weak Solutions of PDEs with p-Growth Coefficients
19
R
R
For any t ∈ [0, T ], limn→∞ Rd un (t, x)ϕ(x)dx = Rd u(t, x)ϕ(x)dx can be proved as follows
using Proposition 4.6:
Z
Z
lim |
(un (t, x) − u(t, x))ϕ(x)dx|2 ≤ lim Cp E[
|un (t, Xt0,x ) − u(t, Xt0,x )|2 ρ−1 (x)dx]
n→∞
n→∞
Rd
Rd
Z
≤ lim Cp E[ sup
|Yt0,x,n − Yt0,x |2 ρ−1 (x)dx] = 0.
n→∞
0≤t≤T
Rd
Therefore we can prove (2.5) is satisfied for all t ∈ [0, T ]. That is to say u(t, x) is a weak solution
of PDE (2.4).
The uniqueness of PDE (2.4) can be derived from the uniqueness of BSDE (2.2). Let u be
a solution of PDE (2.4). Define F (s, x) = f s, x, u(s, x), (σ ∗ ∇u)(s, x) . Since u is the solution,
RT R
so 0 Rd |u(s, x)|2p + |(σ ∗ ∇u)(s, x)|2 ρ−1 (x)dxds < ∞ and
Z
T
Z
|F (s, x)|2 ρ−1 (x)dxds
Rd
T Z
0
Z
|f0 (s, x)|2 + |u(s, x)|2p + |(σ ∗ ∇u)(s, x)|2 ρ−1 (x)dxds < ∞.
≤ Cp
0
(5.2)
Rd
If we define Yst,x = u(s, Xst,x ) and Zst,x = (σ ∗ ∇u)(s, Xst,x ), then by Lemma 3.2,
T
Z
E[
Z
Rd
t
T
Z
(|Yst,x |2p + |Zst,x |2 )ρ−1 (x)dxds]
Z
≤ Cp
t
|u(s, x)|2p + |(σ ∗ ∇u)(s, x)|2 ρ−1 (x)dxds < ∞.
Rd
Using some ideas of Theorem 2.1 in [1], similar to the argument as in Section 4 in [24], we
N 2
d
1
have for t ≤ s ≤ T , (Y·t,· , Z·t,· ) ∈ M 2p ([t, T ]; L2p
M ([t, T ]; L2ρ (Rd ; Rd )) solves the
ρ (R ; R ))
following BSDE:
Z T
Z T
t,x
t,x
t,x
Ys = h(XT ) +
F (r, Xr )dr −
hZrt,x , dWr i.
(5.3)
s
s
Multiply ϕ ∈ Cc0 (Rd ; R1 ) on both sides and then take the integration over Rd . Noting the
definition of F (s, x), Yst,x and Zst,x , we have that (Yst,x , Zst,x ) satisfies the spatial integration form
d
1
of BSDE (2.2). Similar to Proposition 4.6, we can deduce that Y·t,· ∈ S 2p ([t, T ]; L2p
ρ (R ; R ))
t,x
t,x
and therefore (Ys , Zs ) is a solution of BSDE (2.2). If there is another solution u
ˆ to PDE
t,x ˆ t,x
ˆ
(2.4), then by the same procedure, we can find another solution (Ys , Zs ) to BSDE (2.2),
where
Yˆst,x = u
ˆ(s, Xst,x ) and Zˆst,x = (σ ∗ ∇ˆ
u)(s, Xst,x ).
By Theorem 2.3, the solution of BSDE (2.2) is unique. Therefore
Yst,x = Yˆst,x for a.a. s ∈ [t, T ], x ∈ Rd a.s.
In particular, when t = 0,
Ys0,x = Yˆs0,x for a.a. s ∈ [0, T ], x ∈ Rd a.s.
20
Q. Zhang and H.Z. Zhao
By Lemma 3.2 again,
Z
0
T
Z
Z
|u(s, x) − u
ˆ(s, x)|2 ρ−1 (x)dxds ≤ Cp E[
Rd
0
T
Z
Rd
|Ys0,x − Yˆs0,x |2 )ρ−1 (x)dxds] = 0.
So u(s, x) = u
ˆ(s, x) for a.a. s ∈ [0, T ], x ∈ Rd a.s. The uniqueness is proved.
Acknowledgements. We would like to acknowledge useful conversations with K.D. Elworthy, C.R. Feng, J. Lorinczi, K.N. Lu, S.G. Peng and S.J. Tang. QZ would like to thank the
Department of Mathematical Sciences of Loughborough University for appointing him as a Research Associate. He wishes to acknowledge their financial support to the project through the
appointment and partial financial support of the National Basic Research Program of China
(973 Program) with Grant No.2007CB814904. HZ would like to thank K.N. Lu for inviting
him to visit Brigham-Young University under their programme of Special Year on Stochastic
Dynamics and S.J. Tang for inviting him to visit Laboratory of Mathematics for Nonlinear
Science, Fudan University.
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