Quadratic BSDEs with random terminal time and elliptic PDEs in infinite
  dimension
arXiv:0704.1223v2  [math.PR]  21 Jun 2007
Quadratic BSDEs with random terminal time and
elliptic PDEs in infinite dimension.
Philippe Briand
IRMAR, Université Rennes 1, 35042 Rennes Cedex, FRANCE
philippe.briand@univ-rennes1.fr
Fulvia Confortola
Dipartimento di Matematica e Applicazioni,
Università di Milano-Bicocca
Via R. Cozzi 53 - Edificio U5 - 20125 Milano, Italy
fulvia.confortola@unimib.it
Mathematics Subject Classification: 60H20; 60H30.
Abstract
In this paper we study one dimensional backward stochastic differential equations (BSDEs)
with random terminal time not necessarily bounded or finite when the generator F (t, Y, Z) has
a quadratic growth in Z. We provide existence and uniqueness of a bounded solution of such
BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained
results are then applied to prove existence and uniqueness of a mild solution to elliptic partial
differential equations in Hilbert spaces. Finally we show an application to a control problem.
1 Introduction
Let τ be a stopping time which is not necessarily bounded or finite. We look for a pair of
processes (Yt, Zt)t≥0 progressively measurable which satisfy ∀t ≥ 0,∀T ≥ t
Yt∧τ = YT∧τ +
∫ T∧τ
F (s, Ys, Zs)−
∫ T∧τ
ZsdWs
Yτ = ξ on {τ <∞}
where W is a cylindrical Wiener process in some infinite dimensional Hilbert space Ξ and
the generator F has quadratic growth with respect to the variable z. Moreover the terminal
condition ξ is Fτ -measurable and bounded. We limit ourselves to the case in which (Yt)t≥0 is
one-dimensional and we look for a solution (Yt, Zt)t≥0 such that (Yt)t≥0 is a bounded process
and (Zt)t≥0 is a process with values in the space of the Hilbert-Schmidt operator from Ξ to R
such that E
(∫ t∧τ
<∞,∀t ≥ 0.
BSDEs with random terminal time have been treated by several authors (see for instance [21],
[6], [3], [23]) when the generator is Lipschitz, or monotone and with suitable growth with respect
to y, but Lipschitz with respect to z. Kobylanski [18] deals with a real BSDE with quadratic
http://arxiv.org/abs/0704.1223v2
generator with respect to z and with random terminal time. She requires that the stopping time
is bounded or P-a.s finite. We generalize in a certain sense the result of Kobylanski, but, to
obtain the existence and uniqueness of the solution to (1) for a general stopping time, we have
to require stronger assumption on the generator. In particular it has to be strictly monotone
with respect to y.
We follow the techniques introduced by Briand and Hu in [3], and used successively by
Royer [23], based upon an approximation procedure and on Girsanov transform. We can use
this strategy even if, under our assumptions, the generator is not Lipschitz with respect to
z. The main idea is to exploit the theory of BMO-martingales. It is indeed known that if
(Y,Z) solves a quadratic BSDE with bounded (or P-a.s.) finite final time then
Zs dWs is a
BMO–martingale (see [16]).
Then the result on BSDE is exploited to study existence and uniqueness of a mild solution
(see Section 5 for the definition) to the following elliptic partial differential equation in Hilbert
space H
Lu(x) + F (x, u(x),∇u(x)σ) = 0, x ∈ H, (2)
where F is a function from H ×R×Ξ∗ to R strictly monotone with respect the second variable
and with quadratic growth in the gradient of the solution and L is the second order operator:
Lφ(x) =
Trace(σσ∗∇2φ(x)) + 〈Ax,∇φ(x)〉 + 〈b(x),∇φ(x)〉.
H is an Hilbert space, A is the generator of a strongly continuous semigroup of bounded linear
operators (etA)t≥0 in H, b is a function with values in H and σ belongs to L(Ξ,H)- the space
of linear bounded operator from Ξ to K satisfying appropriate Lipschitz conditions.
Existence and uniqueness of a mild solution of equation (2) in infinite dimensional spaces
have been recently studied by several authors employing different techniques (see [5], [14], [9]
and [10]).
In [13] (following several papers dealing with finite dimensional situations, see, for instance
[4], [6] and [20]) the solution of equation (2) is represented using a Markovian forward-backward
system of equations
dXs = AXsds+ b(Xs)ds + σ(Xs)dWs, s ≥ 0
dYs = −F (Xs, Ys, Zs)ds + ZsdWs, s ≥ 0
X0 = x
where F is Lipschitz with respect to y and z and monotone in y, but with monotonicity constant
large. A such limitation has then been removed under certain conditions in [17], still assuming
F Lipschitz with respect to z, strictly monotone and with arbitrary growth with respect to y.
We follow the same approach to deal with mild solution to (2) when the coefficient F is strictly
monotone in the second variable (there are not conditions on its monotonicity constant) and has
quadratic growth in the gradient of the solution. The main technical point here will be proving
differentiability of the bounded solution of the backward equation in system (3) with respect to
the initial datum x of the forward equation. To obtain this result we follow [17]. The proof is
based on an a-priori bound for suitable approximations of the equations for the gradient of Y
with respect to x. We use again classical result on BMO-martingales.
In the last part of the paper we apply the above result to an optimal control problem with
state equation:
dXτ = AXτdτ + b(Xτ )dτ + σr(Xτ , uτ )dτ + σdWτ ,
X0 = x ∈ H,
where u denotes the control process, taking values in a given closed subset U of a Banach space
U . The control problem consists of minimizing an infinite horizon cost functional of the form
J(x, u) = E
e−λσg(Xuσ , uσ)dσ.
We suppose that r is a function with values in Ξ∗ with linear growth in u and g is a given real
function with quadratic growth in u. λ is any positive number. We assume that neither U nor
r is bounded: in this way the Hamiltonian corresponding to the control problem has quadratic
growth in the gradient of the solution and consequently the associated BSDE has quadratic
growth in the variable Z. The results obtained on equation (2) allows to prove that the value
function of the above problem is the unique mild solution of the corresponding Hamilton-Jacobi-
Bellman equation (that has the same structure as (2). Moreover the optimal control is expressed
in terms of a feedback that involves the gradient of that same solution to the Hamilton-Jacobi-
Bellman equation. We stress that the usual application of the Girsanov technique is not allowed
(since the Novikov condition is not guaranteed) and we have to use specific arguments both to
prove the fundamental relation and to solve the closed loop equation. We adapt some procedure
used in [11] to our infinite dimensional framework on infinite horizon.
The paper is organized as follows: the next Section is devoted to notations; in Section 3
we deal with quadratic BSDEs with random terminal time; in Section 4 we study the forward
backward system on infinite horizon; in Section 5 we show the result about the solution to PDE.
The last Section is devoted to the application to the control problem.
2 Notations
The norm of an element x of a Banach space E will be denoted |x|E or simply |x|, if no confusion
is possible. If F is another Banach space, L(E,F ) denotes the space of bounded linear operators
from E to F , endowed with the usual operator norm.
The letters Ξ, H, U will always denote Hilbert spaces. Scalar product is denoted 〈·, ·〉, with
a subscript to specify the space, if necessary. All Hilbert spaces are assumed to be real and
separable. L2(Ξ, U) is the space of Hilbert-Schmidt operators from Ξ to U , endowed with the
Hilbert-Schmidt norm, that makes it a separable Hilbert space. We observe that if U = R the
space L2(Ξ,R) is the space L(Ξ,R) of bounded linear operators from Ξ to R. By the Riesz
isometry the dual space Ξ∗ = L(Ξ,R) can be identified with Ξ.
By a cylindrical Wiener process with values in a Hilbert space Ξ, defined on a probability
space (Ω,F ,P), we mean a family {Wt, t ≥ 0} of linear mappings from Ξ to L
2(Ω), denoted
ξ 7→ 〈ξ,Wt〉, such that
(i) for every ξ ∈ Ξ, {〈ξ,Wt〉, t ≥ 0} is a real (continuous) Wiener process;
(ii) for every ξ1, ξ2 ∈ Ξ and t ≥ 0, E (〈ξ1,Wt〉 · 〈ξ2,Wt〉) = 〈ξ1, ξ2〉Ξ t.
(Ft)t≥0 will denote, the natural filtration of W , augmented with the family of P-null sets.
The filtration (Ft) satisfies the usual conditions. All the concepts of measurably for stochastic
processes refer to this filtration. By B(Λ) we mean the Borel σ-algebra of any topological space
We also recall notations and basic facts on a class of differentiable maps acting among Ba-
nach spaces, particularly suitable for our purposes (we refer the reader to [12] for details and
properties). We notice that the use of Gâteaux differentiability in place of Fréchet differentia-
bility is particularly suitable when dealing with evaluation (Nemitskii) type mappings on spaces
of summable functions.
Let now X, Z, V denote Banach spaces. We say that a mapping F : X → V belongs to
the class G1(X,V ) if it is continuous, Gâteaux differentiable on X, and its Gâteaux derivative
∇F : X → L(X,V ) is strongly continuous.
The last requirement is equivalent to the fact that for every h ∈ X the map ∇F (·)h : X → V
is continuous. Note that ∇F : X → L(X,V ) is not continuous in general if L(X,V ) is endowed
with the norm operator topology; clearly, if this happens then F is Fréchet differentiable on
X. It can be proved that if F ∈ G1(X,V ) then (x, h) 7→ ∇F (x)h is continuous from X × X
to V ; if, in addition, G is in G1(V,Z) then G(F ) belongs to G1(X,Z) and the chain rule holds:
∇(G(F ))(x) = ∇G(F (x))∇F (x).
When F depends on additional arguments, the previous definitions and properties have
obvious generalizations.
3 Quadratic BSDEs with random terminal time
Let τ be an Ft-stopping time. It is not necessarily bounded or P-a.s. finite. We work with a
function F defined on Ω× [0,∞) ×R× Ξ∗ which takes its values in R and such that F (·, y, z)
is a progressively measurable process for each (y, z) in R × Ξ∗. We define the following sets of
Ft-progressively measurable processes (ψt)t≥0 with values in a Hilbert space K:
M2,−2λ(0, τ ;K) =
ψ : E
e−2λs|ψs|
M2loc(0, τ ;K) =
ψ : E
(∫ t∧τ
<∞ ∀t ≥ 0
We want to construct an adapted process (Y,Z)t≥0 which solves the BSDE
− dYt = 1t≤τ (F (t, Yt, Zt)dt− ZtdWt), Yτ = ξ on {τ <∞}. (5)
We assume that:
Assumption A1. There exist C ≥ 0 and α ∈ (0, 1) such that
1. |F (t, y, z)| ≤ C
1 + |y|+ |z|2
2. F (t, ·, ·) is G1,1(R× L2(Ξ,R);R);
3. |∇zF (t, y, z)| ≤ C (1 + |z|);
4. |∇yF (t, y, z)| ≤ C (1 + |z|)
Moreover we suppose that there exist two constants K ≥ 0 and λ > 0 such that dP ⊗ dt
a.e.:
5. F is monotone in y in the following sense:
∀y, y′ ∈ R, z ∈ Ξ∗, < y − y′, F (t, y, z) − F (t, y′, z) >≤ −λ|y − y′|2;
6. |F (t, 0, 0)| ≤ K;
7. ξ is a Fτ -measurable bounded random variable; we denote by M some real such that
|ξ| ≤M P-a.s.
We call solution of the equation a pair of progressively measurable processes (Yt, Zt)t≥0 with
values in R× Ξ∗ such that
1. Y is a bounded process and Z ∈ M2loc(0, τ ; Ξ
2. On the set {τ <∞}, we have Yτ = ξ and Zt = 0 for t > τ ;
3. ∀T ≥ 0, ∀t ∈ [0, T ] we have Yt∧τ = YT∧τ +
∫ T∧τ
F (s, Ys, Zs)ds −
∫ T∧τ
ZsdWs.
Before giving the main result of this section we prove a lemma which we use in the sequel.
The proof involves the Girsanov transform and results of the bounded mean oscillation (BMO,
for short) martingales theory.
Here we recall a few well-known facts from this theory following the exposition in [15]. Let
M be a continuous local (P,F)-martingale satisfying M0 = 0. Let 1 ≤ p <∞. Then M is in the
normed linear space BMOp if
||M ||BMOp = sup
∣∣∣E[|MT −Mτ |p|Fτ ]1/p
where the supremum is taken over all stopping time τ ≤ T . By Corollary 2.1 in [15], M is a
BMOp-martingale if and only if it is a BMOq-martingale for every q ≥ 1. Therefore, it is simply
called a BMO-martingale. In particular, M is a BMO-martingale if and only if
||M ||BMO2 = sup
∣∣∣E[〈M〉T − 〈M〉τ |Fτ ]1/2
where the supremum is taken over all stopping time τ ≤ T ; 〈M〉 denotes the quadratic variation
of M . This means that local martingales of the form Mt =
ξsdWs are BMO-martingales if
and only if
||M ||BMO2 = sup
∣∣∣∣∣
∣∣∣∣∣E
||ξs||
∣∣∣Fτ
]1/2∣∣∣∣∣
∣∣∣∣∣
The very important feature of BMO-martingales is the following (see Theorem 2.3 in [15]):
the exponential martingale
E(M)t = Et = exp
0 ≤ t ≥ T
is a uniformly integrable martingale.
Lemma 3.1. Let (U, V ), be solutions to
Ut = ξ +
1s≤τ [asUs + bsVs + ψs ]ds −
Vs dWs (6)
where ξ is Fτ–measurable and bounded and as, bs, ψs are processes such that
1) as ≤ −λ for some λ > 0;
bsdWs is a BMO-martingale;
3) |ψs| ≤ ρ(s) where ρ is a deterministic function.
Moreover we assume that U is bounded. Then we have P-a.s. for all t ∈ [0, T ]
|Ut| ≤ e
−λ(T−t)‖ξ‖∞ +
ρ(s)e−λ(s−t) ds.
Proof. Let (U, V ) be a solution of the BSDE (6) such that U is bounded.
We fix t ∈ R+ and set for s ≥ t es = e
ar dr. By Ito’s formula we have,
Ut = eT ξ +
1s≤τesψsds−
esVs(dWs − bs).
Let QT the probability measure on (Ω,FT ) whose density with respect to P|FT is
ET = exp
bsdWs −
By assumption
bsdWs is a BMO-martingale and the probability measures QT and P|FT are
mutually absolutely continuous and W t = Wt −
br dr for 0 ≤ t ≤ T is a Brownian motion
under QT .
Taking the conditional expectation with respect to Ft we get
|Ut| ≤ E
eT |ξ|+
es|ψs|ds
∣∣∣Ft
, QT a.s.
and thanks to 3)
|Ut| ≤ (Et)
ET eT |ξ|+
ρ(s)esds
∣∣∣ Ft
But from 1) as ≤ −λ and, for all s ≥ t es ≤ e
−λ(s−t) P-a.s., from which we get P-a.s. ∀t ∈ [0, T ]
|Ut| ≤ e
−λ(T−t)||ξ||∞ +
ρ(s)e−λ(s−t)ds.
Corollary 3.2. Let (Y i, Zi), i = 1, 2, be solutions to
Y it = ξ
1s≤τF
i(s, Y is , Z
s) ds−
Zis dWs
where ξi is Fτ–measurable and bounded. We assume that Y
1 and Y 2 are bounded and that the
Zi are such that
ZiddWt are BMO-martingales. Moreover F
1 is −λ-monotone in the following
sense: there exists λ > 0 such that
∀y, y′ ∈ R, z ∈ Ξ∗, < y − y′, F 1(t, y, z)− F 1(t, y′, z) >≤ −λ|y − y′|2;
and verifies
|F 1(t, y, z) − F 1(t, y, z′)| ≤ C |z − z′|
1 + |z|+ |z′|
We assume moreover that
|F 1(t, Y 2t , Z
t )− F
2(t, Y 2t , Z
t )| ≤ ρ(t)
where ρ is a deterministic function. Then we have P-a.s. for all t ∈ [0, T ]
|Y 1t − Y
t | ≤ e
−λ(T−t)‖ξ1 − ξ2‖∞ +
ρ(s)e−λ(s−t) ds.
Proof. Let (Y 1, Z1) and (Y 2, Z2) be solutions of the BSDE with data respectively (ξ1, F 1) and
(ξ2, F 2) such that Y 1 and Y 2 are bounded. We set Y = Y 1−Y 2 and Z = Z1−Z2. It is enough
to write the equation for the difference Y = Y 1 − Y 2
dY t = −1t≤τ [F
1(t, Y 1t , Z
t )− F
2(t, Y 2t , Z
t )dt+ ZtdWt]
dY t = −1t≤τ [(atY t + btZt + ψt)dt+ ZtdWt].
using a linearization procedure by setting
F 1(s, Y 1s , Z
s )− F
1(s, Y 2s , Z
Y 1s − Y
, if Y 1s − Y
s 6= 0
−λ otherwise
F 1(s, Y 2s , Z
s )− F
1(s, Y 2s , Z
|Z1s − Z
(Z1s − Z
s ), if Z
s − Z
s 6= 0
0 otherwise .
ψs = F
1(s, Y 2s , Z
s )− F
2(s, Y 2s , Z
Now we can state the main result of this section, concerning the existence and uniqueness
of solutions of BSDE (5).
Theorem 3.3. Under assumption A1 there exists a unique solution (Y,Z) to BSDE (5) such
that Y is a continuous and bounded process and Z belongs to M2loc(0, τ ; Ξ
Proof. Existence. We adopt the same strategy as in [3] and [23], with some significant modi-
fications.
Denote by (Y n, Zn) the unique solution to the BSDE
Y nt = ξ1τ≤n +
1s≤τF (s, Y
s , Z
s )ds−
Zns dWs, 0 ≤ t ≤ n. (7)
We know from results of [18] that under A1-1,2,3,4 the BSDE (7) has a unique bounded
solution and that ∥∥∥supt∈[0,τ∧n] |Yt|n
≤ (||ξ||∞ + Cn)e
and there exists a constant C = Cn, which depends on
∥∥∥supt∈[0,τ∧n] |Y nt |
, such that
Zns · dWs
≤ Cn.
Now we study the convergence of the sequence of processes (Y n, Zn).
(i) First of all we prove that, thanks to the assumptions of boundedness and monotonicity
A1-5,6, Y n is a process bounded by a constant independent on n. Applying the Corollary 3.2
we have that P-a.s. ∀n ∈ N, ∀t ∈ [0, n]
|Y nt | ≤ e
−λ(n−t)||ξ1τ≤n||∞ +
e−λ(s−t)|F (s, 0, 0)| ds ≤M +
. (8)
Moreover we can show that for each ǫ > 0
e−ǫs|Zns |
2ds) <∞. (9)
To obtain this estimate we take the function ϕ(x) =
e2Cx − 2Cx− 1
/(2C2) which has the
following properties:
ϕ′(x) ≥ 0 if x ≥ 0,
ϕ′′(x)−Cϕ′(x) = 1.
Thanks to (8) we can say that there exist a constant K0 such that ∀s ∈ [0, T ], Y
s + K0 ≥ 0,
P-a.s. Now, if we calculate the Ito differential of e−ǫtϕ(Y nt +K0), using the previous properties,
we have (9).
(ii) Now we prove that the sequence (Y nt )n≥0 converges almost surely. We are going to show
that it is an almost definite Cauchy sequence.
We define Y n and Zn on the whole time axis by setting
Y nt = ξ1τ≤n, Z
t = 0, if t > n.
Fix t ≤ n ≤ m and set Ŷ = Y m − Y n, Ẑ = Zm − Zn and F̂ (s, y, z) = 1s≤nF (s, y, z). We
get, from Ito’s formula
Ŷt = Ŷm +
1s≤τ (F (s, Y
s , Z
s )− F̂ (s, Y
s , Z
s ))ds −
ẐsdWs.
We note that
|F (s, Y ns , Z
s )− F̂ (s, Y
s , Z
s ))| = |1s>nF (s, ξ1τ≤n, 0)| ≤ C(1 +M)1s>n.
Hence, we can apply the Corollary 3.2 with ξ1 = ξ1τ≤m and ξ
2 = ξ1τ≤n, F
1 = F and F 2 = F̂ ,
ρ(t) = C(1 +M)1s>n and state that ∀n,m ∈ N, with n ≤ m and ∀t ∈ [0, n], P-a.s.
|Y mt − Y
t | ≤ e
−λ(m−t)||ξ1τ≤m − ξ1τ≤n||∞ +
C(1 +M)e−λ(s−t) ds ≤
C(1 +M)
e−λ(n−t). (10)
The previous inequality implies that for each t ≥ 0 the sequence of random variable Y nt is a
Cauchy sequence in L∞(Ω), hence converges to a limit, which we denote Yt. If m goes to infinity
in the last inequality, it comes that P-a.s., ∀ 0 ≤ t ≤ n
|Y nt − Yt| ≤ βe
−λ(n−t), where β =M +
C(1 +M)
. (11)
This inequality implies that the sequence of continuous processes (Y n)n∈N converges almost
surely to Y uniformly with respect to t on compact sets. The limit process Y is also continuous
and from (8) we have that ∀t ∈ R+ |Yt| ≤M +
(iii) We show that the sequence (Yn)n also converges in the space M
2,−2λ(0, τ ;R). Indeed
we have
e−2λt|Y nt − Yt|
[∫ n∧τ
e−2λt|Y nt − Yt|
e−2λt|Y nt − Yt|
and using the inequality (11) for the first term, we get that
[∫ n∧τ
e−2λt|Y nt − Yt|
≤ β2ne−2λn.
In addition, from the definition of Y nt on R+, we know that ∀t > n Y
t = ξ1τ≤n. Hence,
e−2λt|Y nt − Yt|
e−2λt|Yt − ξ1n<τ |
)2 ∫ τ
e−2λtdt
e−2λn.
Finally we have
e−2λt|Y nt − Yt|
≤ e−2λn
nβ2 +
Hence (Y n) converges to Y in M2,−2λ(0, τ ;R).
(iv) To continue, we show that the sequence (Zn)n is a Cauchy sequence in the space
M2,−2(λ+ǫ)(0, τ ; Ξ∗).
Fix t ≤ n ≤ m and set, as before, Ŷ = Y m−Y n, Ẑ = Zm−Zn and F̂ (s, y, z) = 1s≤nF (s, y, z).
We write
F (s, Y ms , Z
s )− F̂ (s, Y
s , Z
s ) = a
s Ŷs + b
s Ẑs + 1s>nF (s, ξ1τ≤n, 0)
where
an,ms =
F (s, Y ms , Z
s )− F (s, Y
s , Z
Y ms − Y
, if Y ms − Y
s 6= 0
−λ otherwise
bn,ms =
F (s, Y ns , Z
s )− F (s, Y
s , Z
|Zms − Z
(Zms − Z
s ), if Z
s − Z
s 6= 0
0 otherwise .
From Ito’s formula we get
|Ŷ0|
∫ τ∧m
e−2(λ+ǫ)s|Ẑs|
2 ds+
∫ τ∧m
2e−2(λ+ǫ)sŶsẐsdWs =
= e−2(λ+ǫ)τ∧m|Ŷτ∧m|
∫ τ∧m
e−2(λ+ǫ)s2(λ + ǫ)|Ŷs|
∫ τ∧m
2e−2(λ+ǫ)sŶs[a
s Ŷs + b
s Ẑs]ds+
∫ τ∧m
2e−2(λ+ǫ)sŶsF (s, ξ1τ≤n, 0)ds
and taking the expectation we have
∫ τ∧m
e−2(λ+ǫ)s|Ẑs|
2ds ≤ Ee−2(λ+ǫ)τ∧m|Ŷτ∧m|
2 + E
∫ τ∧m
e−2(λ+ǫ)s2ǫ|Ŷs|
∫ τ∧m
2e−2(λ+ǫ)sŶsb
s Ẑsds+ E
∫ τ∧m
2e−2(λ+ǫ)sŶsF (s, ξ1τ≤n, 0)ds.
Using the fact that
2e−2(λ+ǫ)sŶsb
s Ẑs ≤ 2|Ŷs|
2e−2(λ+ǫ)s|bn,ms |
e−2(λ+ǫ)s|Ẑs|
we get
∫ τ∧m
e−2(λ+ǫ)s|Ẑs|
2 ds ≤ 2Ee−2(λ+ǫ)τ∧m|Ŷτ∧m|
2 + 2E
∫ τ∧m
e−2(λ+ǫ)s2ǫ|Ŷs|
∫ τ∧m
|Ŷs|
2 e−2(λ+ǫ)s|bn,ms |
2ds+ E
∫ τ∧m
4e−2(λ+ǫ)s|Ŷs||F (s, ξ1τ≤n, 0)|ds ≤
≤M2e−2(λ+ǫ)n + β2e−2λn
1 + 4
∫ τ∧m
e−2ǫs|bn,ms |
2ds + 4C(1 +M)E
∫ τ∧m
e−2λs|Ŷs|
We note that
|bn,ms |
2 ≤ C(1 + |Zns |
2 + |Zms |
and by (9) supn≥1E
e−2ǫs|Zns |
2ds <∞. Finally we obtain
∫ τ∧m
e−2(λ+ǫ)s|Ẑs|
2 ds ≤ β′(1 + n)e−2λn
where β′ depends on M,λ,K. Moreover we have that
e−2(λ+ǫ)s|Ẑs|
hence
e−2(λ+ǫ)s|Ẑs|
≤ β′(1 + n)e−2λn.
Hence (Zn) is a Cauchy sequence in M2,−2(λ+ǫ)(0, τ ; Ξ∗) and converges to the process Z in this
space.
(v) It remains to show that the process (Y,Z) satisfies the BSDE (5).
We already know that Y is continuous and bounded and Z belongs to M2,−2(λ+ǫ)(0, τ ; Ξ∗).
By definition ∀n ∈ N, ∀T, t such that 0 ≤ t ≤ T ≤ n we have
Y nt∧τ − Y
T∧τ =
∫ T∧τ
F (s, Y ns , Z
∫ T∧τ
Zns dWs. (12)
Fix t and T . We shall pass to the limit in L1 in the previous equality. The sequence Y nt∧τ
converges almost surely to Yt and is bounded by M +
uniformly in n. From Lebesgue’s
theorem we get that the sequence converges to Yt∧τ in L
1. Moreover,
∫ T∧τ
Zns dWs converges in
∫ T∧τ
ZsdWs in L
2 since
(∫ T∧τ
Zns dWs −
∫ T∧τ
ZsdWs
≤ e2(λ+ǫ)TE
∫ T∧τ
e−2(λ+ǫ)s|Zns − Zs|
We can note that
∫ T∧τ
F (s, Y ns , Z
s )ds converges to
∫ T∧τ
F (s, Ys, Zs)ds in L
1. Indeed
∫ T∧τ
F (s, Y ns , Z
s )ds−
∫ T∧τ
F (s, Ys, Zs)ds
∣∣∣∣ ≤ E
|F (s, Y ns , Z
s )ds− F (s, Ys, Zs)|ds
and, by the growth assumption on F , the map (Y,Z) → F (·, Y, Z) is continuous from the
space L1(Ω;L1([0, T ];R)) × L2(Ω;L2([0, T ]; Ξ∗) to L1(Ω;L1([0, T ];R)). (By classical result on
continuity of evaluation operators, see e.g. [1]). Hence, passing to the limit in the equation (12),
we obtain ∀t, T such that t ≤ T
Yt∧τ − YT∧τ =
∫ T∧τ
F (s, Ys, Zs)−
∫ T∧τ
ZsdWs.
So to conclude the proof, it only remains to check the terminal condition. Let ω ∈ {τ < ∞},
and n ∈ N such that n ≥ τ(ω). Then
|Yτ − ξ1t≤2n|(ω) = |Yn∧τ − ξ1t≤2n|(ω) ≤ |Yn∧τ − Y
n∧τ |(ω) + |Y
n∧τ − ξ1t≤2n|(ω) ≤
≤ βeλ(n∧τ)(ω)e−2λn + |Y 2nn∧τ − ξ1t≤2n|(ω) ≤ βe
since Y 2nn∧τ = Y
τ = Y
2n = ξ1t≤2n Then, Yτ = ξ P-a.s. on the set {τ < ∞}, and the process
(Y,Z) is solution for BSDE (5).
Uniqueness.
Suppose that (Y 1, Z1) and (Y 2, Z2) are both solutions of the BSDE (5) such that Y 1 and Y 2
are continuous and bounded and Z1 and Z2 belong to M2loc(0, τ ; Ξ
∗). It follows directly from
the Corollary 3.2 that ∀t ≥ 0
Y 1t − Y
t = 0 P-a.s.
and then, by continuity, Y 1 = Y 2.
Applying Ito’s formula we have that dP⊗ dt-a.e. Z1t = Z
4 The forward-backward system on infinite horizon
In this Section we use the previous result to study a forward-backward system on infinite horizon,
when the backward equation has quadratic generator.
We introduce now some classes of stochastic processes with values in a Hilbert space K which
we use in the sequel.
• Lp(Ω;L2(0, s;K)) defined for s ∈]0,+∞] and p ∈ [1,∞), denotes the space of equivalence
classes of progressively measurable processes ψ : Ω× [0, s[→ K, such that
Lp(Ω;L2(0,s;K))
Elements of Lp(Ω;L2(0, s;K)) are identified up to modification.
• Lp(Ω;C(0, s;K)), defined for s ∈]0,+∞[ and p ∈ [1,∞[, denotes the space of progressively
measurable processes {ψt, t ∈ [0, s]} with continuous paths in K, such that the norm
Lp(Ω;C([0,s];K))
= E sup
r∈[0,s]
is finite. Elements of Lp(Ω;C(0, s;K)) are identified up to indistinguishability.
• L2loc(Ω;L
2(0,∞;K)) denotes the space of equivalence classes of progressively measurable
processes ψ : Ω× [0,∞) → K such that
∀t > 0 E
2dr <∞.
Now we consider the Itô stochastic equation for an unknown process {Xs, s ≥ 0} with values
in a Hilbert space H:
Xs = e
e(s−r)Ab(Xr)dr +
e(s−r)AσdWr, s ≥ 0. (13)
Our assumptions will be the following:
Assumption A2. (i) The operator A is the generator of a strongly continuous semigroup etA,
t ≥ 0, in a Hilbert space H. We denote by m and a two constants such that |etA| ≤ meat for
t ≥ 0.
(ii) b : H → H satisfies, for some constant L > 0,
|b(x)− b(y)| ≤ L|x− y|, x, y ∈ H.
(iii) σ belongs to L(Ξ,H) such that etAσ ∈ L2(Ξ,H) for every t > 0, and
|etAσ|L2(Ξ,H) ≤ Lt
−γeat,
for some constants L > 0 and γ ∈ [0, 1/2).
(iv) We have b(·) ∈ G1(H,H).
(v) Operators A+ bx(x) are dissipative (that is 〈Ay, y〉 + 〈bx(x)y, y〉 ≤ 0 for all x ∈ H and
y ∈ D(A)).
Remark 4.1. We note we need of assumptions (iv) − (v) to obtain a result of regularity of the
process X with respect to initial condition x.
We start by recalling a well known result on solvability of equation (13) on a bounded
interval, see e.g. [12].
Proposition 4.2. Under the assumption A2, for every p ∈ [2,∞) and T > 0 there exists a
unique process Xx ∈ Lp(Ω;C(0, T ;H)) solution of (13). Moreover, for all fixed T > 0, the map
x→ Xx is continuous from H to Lp(Ω;C(0, T ;H)).
E sup
r∈[0,T ]
p ≤ C(1 + |x|)p,
for some constant C depending only on q, γ, T, L, a and m.
We need to state a regularity result on the process X. The proof of the following lemma can
be found in [17].
Lemma 4.3. Under Assumptions A2 the map x→ Xx is Gâteaux differentiable (that is belongs
to G(H,Lp(Ω, C(0, T ;H))). Moreover denoting by ∇xX
x the partial Gâteaux derivative, then
for every direction h ∈ H, the directional derivative process ∇xX
xh, t ∈ R, solves, P− a.s., the
equation
t h = e
eσA∇xF (X
σ )∇xX
σhdσ, t ∈ R
Finally, P-a.s., |∇xX
t h| ≤ |h|, for all t > 0.
The associated BSDE is:
Y xt = Y
F (Xxσ , Y
σ , Z
σ)dσ −
ZxσdWσ, 0 ≤ t ≤ T <∞. (14)
Here Xx is the unique mild solution to (13) starting from X0 = x. Y is real valued and Z takes
values in Ξ∗, F : H × R× Ξ∗ → R is a given measurable function.
We assume the following on F :
Assumption A3. There exist C ≥ 0 and α ∈ (0, 1) such that
1. |F (x, y, z)| ≤ C
1 + |y|+ |z|2
2. F (·, ·, ·) is G1,1,1(H ×R× Ξ∗;R) ;
3. |∇xF (x, y, z)| ≤ C;
4. |∇zF (x, y, z)| ≤ C (1 + |z|);
5. |∇yF (x, y, z)| ≤ C (1 + |z|)
6. λ > 0 and F is monotone in y in the following sense:
x ∈ H, y, y′ R, z ∈ Ξ∗ < y − y′, F (x, y, z) − F (x, y′, z) >≤ −λ|y − y′|2.
Applying Theorem 3.3, we obtain:
Proposition 4.4. Let us suppose that Assumptions A2 and A3 hold. Then we have:
(i) For any x ∈ H, there exists a solution (Y x, Zx) to the BSDE (14) such that Y x is a contin-
uous process bounded by K/λ, and Z ∈ L2loc(Ω;L
2(0,∞; Ξ)) with E
e−2(λ+ǫ)s|Zs|
2ds <
∞. The solution is unique in the class of processes (Y,Z) such that Y is continuous and
bounded, and Z belongs to L2loc(Ω;L
2(0,∞; Ξ)).
(ii) For all T > 0 and p ≥ 1, the map x→ (Y x
[0,T ]
[0,T ]
) is continuous from H to the space
Lp(Ω;C(0, T ;R)) × Lp(Ω;L2(0, T ; Ξ)).
Proof. Statement (i) is an immediate consequences of Theorem 3.3. Let us prove (ii). Denoting
by (Y n,x, Zn,x) the unique solution of the following BSDE (with finite horizon):
F (Xxσ , Y
σ , Z
σ )dσ −
Zn,xσ dWσ, (15)
then, from Theorem 3.3again, |Y
t | ≤
and the following convergence rate holds:
t − Y
t | ≤
exp{−λ(n− t)}.
Now, if x′m → x as m→ +∞ then
T − Y
T | ≤ |Y
T − Y
n,x′m
T |+ |Y
T − Y
T |+ |Y
n,x′m
T − Y
exp{−λ(n− T )}+ |Y
n,x′m
T − Y
Moreover for fixed n, Y
n,x′m
T → Y
T in L
p(Ω,FT ,P;R) for all p > 1, by Proposition 4.2 in [2]
Thus Y
T → Y
T in L
p(Ω,FT ,P;R).
Now we can notice that (Y x
[0,T ]
[0,T ]
) is the unique solution of the following BSDE (with
finite horizon):
Y xt = Y
F (Xxσ , Y
σ , Z
ZxσdWσ,
and the same holds for (Y x
[0,T ]
[0,T ]
). By similar argument as in [2] we have
t∈[0,T ]
|Y xt − Y
]1∧1/p
[(∫ T
|Zxt − Z
)p/2]1∧1/p
≤ C E
[∣∣∣Y xT − Y
[(∫ T
∣∣∣F (s,Xxs , Ys, Zs)− F (s,Xx
s , Ys, Zs)
∣∣∣ ds
)p+1] 1p+1
and we can conclude that (Y x
[0,T ]
[0,T ]
) → (Y x
[0,T ]
[0,T ]
) in Lp(Ω;C(0, T ;R)) ×
Lp(Ω;L2(0, T ; Ξ)).
We need to study the regularity of Y x. More precisely, we would like to show that Y x0 belongs
to G1(H,R).
We are now in position to prove the main result of this section.
Theorem 4.5. Under Assumption the map x → Y x0 belongs to G
1(H,R). Moreover |Y x0 | +
0 | ≤ c, for a suitable constant c.
Proof. Fix n ≥ 1, let us consider the solution (Y n,x, Zn,x) of (15). Then, see [2], Proposition
4.2, the map x → (Y n,x(·), Zn,x(·)) is Gâteaux differentiable from H to Lp(Ω, C(0, T ;R)) ×
Lp(Ω;L2(0, T ; Ξ∗)), ∀p ∈ (1,∞). Denoting by (∇xY
n,xh,∇xZ
n,xh) the partial Gâteaux deriva-
tives with respect to x in the direction h ∈ H, the processes {∇xY
t h,∇xZ
t h, t ∈ [0, n]}
solves the equation, P− a.s.,
t h =
∇xF (X
σ , Y
σ , Z
σ )∇xX
σ hdσ
∇yF (X
σ , Y
σ , Z
σ )∇xY
σ hdσ (16)
∇zF (X
σ , Y
σ , Z
σ )∇xZ
σ hdσ −
σ hdWσ .
We note that we can write the generator of the previous equation as
φnσ(u, v) = ψ
σ + a
σu+ b
setting
ψnσ = ∇xF (X
σ , Y
σ , Z
σ )∇xX
anσ = ∇yF (X
σ , Y
σ , Z
σ ) b
σ = ∇zF (X
σ , Y
σ , Z
By Assumption A3 and Lemma 4.3, we have that for all x, h ∈ H the following holds P-a.s.
for all n ∈ N and all σ ∈ [0, n]:
|ψnσ | =
∣∣∣∇xF (Xxσ , Y
σ , Z
σ )∇xX
∣∣∣ ≤ C|h|,
anσ = ∇yF (X
σ , Y
σ , Z
σ ) ≤ −λ ≤ 0, |b
∣∣∣∇zF (Xxσ , Y
σ , Z
∣∣∣ ≤ C(1 + |Zn,xσ |).
Therefore
σ dWσ is a BMO-martingale. Hence
bsdWs is also a BMO-martingale and
by Lemma 3.1, we obtain:
t∈[0,n]
t | ≤ C|h|, P− a.s.;
and applying Itô’s formula to e−2λt|∇xY
2 and arguing as in the proof of Theorem 3.3, points
(iii) and (iv), tanks to the (9), we get:
e−2λt(|∇xY
2 + |∇xZ
2)dt ≤ C1|h|
Fix x, h ∈ H, there exists a subsequence of {(∇xY
n,xh,∇xZ
n,xh,∇xY
0 h) : n ∈ N} which
we still denote by itself, such that (∇xY
n,xh,∇xZ
n,xh) converges weakly to (U1(x, h), V 1(x, h))
in M2,−2λ(0,∞;R × Ξ∗) and ∇xY
0 h converges to ξ(x, h) ∈ R.
Now we write the equation (16) as follows:
t h = ∇xY
∇xF (X
σ , Y
σ , Z
σ )∇xX
(∇yF (X
σ , Y
σ , Z
σ ))∇xY
σ hdσ (17)
∇zF (X
σ , Y
σ , Z
σ )∇xZ
σ hdσ +
σ hdWσ
and define an other process U2t (x, h) by
U2t (x, h) = ξ(x, h) −
∇xF (X
σ , Y
σ , Z
σ )∇xX
σ hdσ
(∇yF (X
σ , Y
σ , Z
σ(x, h)dσ (18)
∇zF (X
σ , Y
σ , Z
σ (x, h)dσ +
V 1σ (x, h)dWσ ,
where (Y x, Zx) is the unique bounded solution to the backward equation (14), see Proposition
4.4. Passing to the limit in the equation (17) it is easy to show that ∇xY
t h converges to
U2t (x, h) weakly in L
1(Ω) for all t > 0.
Thus U2t (x, h) = U
t (x, h), P-a.s. for a.e. t ∈ R
+ and |U2t (x, h)| ≤ C|h|.
Now consider the following equation on infinite horizon
U(t, x, h) = U(0, x, h) −
∇xF (X
σ , Y
σ , Z
σ )∇xX
(∇yF (X
σ , Y
σ , Z
σ))U(t, x, h)dσ (19)
∇zF (X
σ , Y
σ , Z
σ )V (σ, x, h)dσ +
V (σ, x, h)dWσ .
We claim that this equation has a solution.
For each n ∈ N consider the finite horizon BSDE (with final condition equal to zero):
Un(t, x, h) =
∇xF (X
σ , Y
σ , Z
σ )∇xX
(∇yF (X
σ , Y
σ , Z
σ ))Un(t, x, h)dσ
∇zF (X
σ , Y
σ , Z
σ)Vn(σ, x, h)dσ −
Vn(σ, x, h)dWσ ,
By the result in [2] we know that this equation has a unique solution (Un(·, x, h), Vn(·, x, h)) ∈
Lp(Ω;C(0, n;R)) × Lp(Ω;L2(0, n; Ξ∗)). The generator of this equation can be rewrite as
φt(u, v) = ψt + atu+ btv
where ψt = ∇xF (X
t , Y
t , Z
t )∇xX
t and |ψt| ≤ C|h|, at = ∇yF (X
σ , Y
σ , Z
σ) ≤ −λ, bt =
∇zF (X
σ , Y
σ , Z
σ ) and |bt| ≤ C(1+ |Z
t |). On the interval [0, n] the process
Zxs dWs is a BMO-
martingale. Hence, from the Lemma 3.1 it follows that P-a.s. ∀n ∈ N, ∀t ∈ [0, n] |Unt | ≤
and as in the proof of existence in the Theorem 3.3, we can conclude that
1. for each t ≥ 0 Un(t, x, h) is a Cauchy sequence in L∞(Ω) which converges to a process U
and P-a.s., ∀t ∈ [0, n]
|Un(t, x, h)− U(t, x, h)| ≤
|h|e−λ(n−t);
2. V n(·, x, h) is a Cauchy sequence in L2loc(Ω;L
2([0,∞); Ξ∗);
3. The processes limit (U(·, x, h), V (·, x, h) satisfy the BSDE (19).
Moreover still from Lemma 3.1 we get that the solution is unique.
Coming back to equation (18), we have that (U2(x, h), V 1(x, h)) is solution in R+ of the
equation (19).
In particular we notice that U(0, x, h) = ξ(x, h) is the limit of ∇xY
0 h (along the cho-
sen subsequence). The uniqueness of the solution to (19) implies that in reality U(0, x, h) =
limn→∞∇xY
0 h along the original sequence.
Now let xm → x.
|U(0, x, h) − U(0, xm, h)| ≤ |U(0, x, h) − U
n(0, x, h)| + |Un(0, x, h) − Un(0, xm, h)|+ (20)
+|Un(0, xm, h) − U(0, xm, h)| ≤
e−λn|h|+ |Un(0, x, h) − Un(0, xm, h)|,
where we have used the (1). We now notice that ∇xF , ∇yF , ∇zF are, by assumptions, con-
tinuous and |∇xF | ≤ C, |∇yF | ≤ C(1 + |Z|)
2α, |∇zF | ≤ C(1 + |Z|) . Moreover the following
statements on continuous dependence on x hold:
maps x → Xx, x → ∇xX
xh are continuous from H → L
P(Ω;C(0, T ;H)) (see [12] Proposition
3.3);
the map x→ Y x
[0,T ]
is continuous from H to L
P(Ω;C(0, T ;R)) (see Proposition 4.4 here);
the map x→ Zx
[0,T ]
is continuous from H to L
P(Ω;L
2(0, T ; Ξ)) (see Proposition 4.4 here ).
We can therefore apply to (20) the continuity result of [12] Proposition 4.3 to obtain in
particular that Un(0, x
m, h) → Un(0, x, h) for all fixed n as m → ∞. And by (20) we can
conclude that U(0, x′m, h) → U(0, x, h) as m→ ∞.
Summarizing U(0, x, h) = limn→∞∇xY
0 h exists, moreover it is clearly linear in h and
verifies |U(0, x, h)| ≤ C|h|, finally it is continuous in x for every h fixed.
Finally, for t > 0,
[Y x+th0 − Y
0 ] = lim
n,x+th
0 − Y
0 ] = lim
n,x+θth
0 hdθ
= lim
U(0, x+ θth)hdθ = U(0, x)h
and the claim is proved.
5 Mild Solution of the elliptic PDE
Now we can proceed as in [13]. Let us consider the forward equation
Xs = e
e(s−r)Ab(Xr)dr +
e(s−r)AσdWr, s ≥ 0. (21)
Assuming that Assumption A2 holds, we define in the usual way the transition semigroup
(Pt)t≥0, associated to the process X:
Pt[φ](x) = E φ(X
t ), x ∈ H,
for every bounded measurable function φ : H → R. Formally, the generator L of (Pt) is the
operator
Lφ(x) =
Trace
σσ∗∇2φ(x)
+ 〈Ax+ b(x),∇φ(x)〉.
In this section we address solvability of the non linear stationary Kolmogorov equation:
Lv(x) + F (x, v(x),∇v(x)σ) = 0, x ∈ H, (22)
when the coefficient F verifies Assumption A3. Note that, for x ∈ H, ∇v(x) belongs to H∗, so
that ∇v(x)σ is in Ξ∗.
Definition 5.1. We say that a function v : H → R is a mild solution of the non linear stationary
Kolmogorov equation (22) if the following conditions hold:
(i) v ∈ G1(H,R) and ∃C > 0 such that |v(x)| ≤ C, |∇xv(x)h| ≤ C |h|, for all x, h ∈ H;
(ii) the following equality holds, for every x ∈ H and T ≥ 0:
v(x) = e−λT PT [v](x) +
e−λt Pt
·, v(·),∇v(·)σ
+ λv(·)
(x) dt. (23)
where λ is the monotonicity constant in Assumption A3.
Together with equation (21) we also consider the backward equation
Yt − YT +
ZsdWs =
F (Xs, Ys, Zs)ds 0 ≤ t ≤ T <∞ (24)
where F : H ×R×Ξ∗ → R is the same occurring in the nonlinear stationary Kolmogorov equa-
tion. Under the Assumptions A2, A3, Propositions 4.2-4.4 give a unique solution {Xxt , Y
t , Z
for t ≥ 0, of the forward-backward system (21)-(24). We can now state the following
Theorem 5.2. Assume that Assumption A2, Assumption A3 and hold then equation (22) has
a unique mild solution given by the formula
v(x) = Y x0 .
where {Xxt , Y
t , Z
t , t ≥ 0} is the solution of the forward-backward system (21)-(24). Moreover
the following holds:
Y xt = v(X
t ), Z
t = ∇v(X
t )σ.
Proof. Let us recall that for s ≥ 0, Y xs is measurable with respect to F[0,s] and Fs; it follows
that Y x0 is deterministic (see also [7]). Moreover, as a byproduct of Proposition 4.5, the function
v defined by the formula v(x) = Y x0 has the regularity properties stated in Definition 5.1. The
proof that the equality (23) holds true for v is identical to the proof of Theorem 6.1 in [13].
6 Application to optimal control
We wish to apply the above results to perform the synthesis of the optimal control for a general
nonlinear control system on an infinite time horizon. To be able to use non-smooth feedbacks
we settle the problem in the framework of weak control problems. Again we follow [13] with
slight modifications. We report the argument for reader’s convenience.
As above by H, Ξ we denote separable real Hilbert spaces and by U we denote a Banach
space.
For fixed x0 ∈ H an admissible control system (a.c.s) is given by (Ω,F , (Ft)t≥0,P, {Wt, t ≥
0}, u) where
• (Ω,F ,P) is a complete probability space and (Ft)t≥0 is a filtration on it satisfying the
usual conditions.
• {Wt : t ≥ 0} is a Ξ-valued cylindrical Wiener process relatively to the filtration (Ft)t≥0
and the probability P.
• u : Ω×[0,∞[→ U is a predictable process (relatively to (Ft)t≥0) that satisfies the constraint:
ut ∈ U , P-a.s. for a.e. t ≥ 0, where U is a fixed closed subset of U .
To each a.c.s. we associate the mild solution X ∈ LrP(Ω;C(0, T ;H)) (for arbitrary T > 0 and
arbitrary r ≥ 1) of the state equation:
dXτ = (AXτ + b(Xτ ) + σr(Xτ , uτ )) dτ + σ dWτ , τ ≥ 0,
X0 = x ∈ H,
and the cost:
J(x, u) = E
e−λtg(Xt, ut) dt, (26)
where g : H × U → R. Our purpose is to minimize the functional J over all a.c.s. Notice the
occurrence of the operator σ in the control term: this special structure of the state equation is
imposed by our techniques.
We work under the following assumptions.
Assumption A4. 1. The process W is a Wiener process in Ξ, defined on a complete prob-
ability space (Ω,F ,P) with respect to a filtration (Ft) satisfying the usual conditions.
2. A, b verify Assumption A2.
3. σ satisfies Assumption A2 (iii) with γ = 0;
4. The set U is a nonempty closed subset of U .
5. The functions r : H × U → Ξ, g : H × U → R are Borel measurable and for all x ∈ H,
r(x, ·) and g(x, ·) are continuous functions from U to Ξ and from U to R, respectively.
6. There exists a constant C ≥ 0 such that for every x, x′ ∈ H , u ∈ K it holds that
|r(x, u)− r(x′, u)| ≤ C(1 + |u|)|x − x′|,
|r(x, u)| ≤ C(1 + |u|), (27)
0 ≤ g(x, u) ≤ C(1 + |u|2), (28)
7. There exist R > 0 and c > 0 such that for every x ∈ H u ∈ U satisfying |u| ≥ R,
g(x, u) ≥ c|u|2. (29)
We will say that an (Ft)-adapted stochastic process {ut, t ≥ 0} with values in U is an
admissible control if it satisfies
e−λt|ut|
2dt <∞. (30)
This square summability requirement is justified by (29): a control process which is not
square summable would have infinite cost.
Now we state that for every admissible control the solution to (25) exists.
Proposition 6.1. Let u be an admissible control. Then there exists a unique, continuous,
(Ft)-adapted process X satisfying E supt∈[0,T ] |Xt|
2 <∞, and P-a.s., t ∈ [0, T ]
Xt = e
e(t−s)Ab(Xs)ds+
e(t−s)AσdWs +
e(t−s)Aσr(Xs, us)ds.
Proof. The proof is an immediate extension to the infinite dimensional case of the Proposition
2.3 in [11].
By the previous Proposition and the arbitrariness of T in its statement, the solution is defined
for every t ≥ 0. We define in a classical way the Hamiltonian function relative to the above
problem: for all x ∈ H, z ∈ Ξ∗,
F (x, y, z) = inf{g(x, u) + zr(x, u) : u ∈ U} − λy
Γ(x, y, z) = {u ∈ U : g(x, u) + zr(x, u)− λy = F (x, y, z)}.
The proof of the following Lemma can be found in [11] Lemma 3.1.
Lemma 6.2. The map F is a Borel measurable function from H × Ξ∗ to R. There exists a
constant C > 0 such that
− C(1 + |z|2)− λy ≤ F (x, y, z) ≤ g(x, u) + C|z|(1 + |u|)− λy ∀u ∈ U . (32)
We require moreover that
Assumption A5. F satisfies assumption A3 2-3-4.
We notice that the cost functional is well defined and J(x, u) <∞ for all x ∈ H and all a.c.s.
By Theorem 5.2, the stationary Hamilton-Jacobi-Bellman equation relative to the above
stated problem, namely:
Lv(x) + F (x, v(x),∇v(x)σ) = 0, x ∈ H, (33)
admits a unique mild solution, in the sense of Definition 5.1.
6.0.1 The fundamental relation
Proposition 6.3. Let v be the solution of (33). For every admissible control u and for the
corresponding trajectory X starting at x we have
J(x, u) = v(x)+
− F (Xt,∇v(Xt)σ)− λv(Xt) +∇xv(Xt)σr(Xt, ut) + g(Xt, ut)
Proof. We introduce the sequence of stopping times
τn = inf{t ∈ [0, T ] :
2ds ≥ n},
with the convention that τn = T if the indicated set is empty. By (30), for P-almost every ω ∈ Ω,
there exists an integer N(ω) depending on ω such that
n ≥ N(ω) =⇒ τn(ω) = T. (34)
Let us fix u0 ∈ K, and for every n, let us define
unt = ut1t≤τn + u01t>τn
and consider the equation
dXnt = b(X
t )dt+ σ[dWt + r(X
t , u
t )dt], 0 ≤ t ≤ T
Xn0 = x.
Let us define
W nt :=Wt +
r(Xns , u
s )ds 0 ≤ t ≤ T.
From the definition of τn and from (27), it follows that
|r(Xns , u
2ds ≤ C
(1 + |uns |)
2ds ≤ C
(1 + |us|)
2ds +C ≤ C + Cn. (36)
Therefore defining
ρn = exp
−r(Xns , u
s )dWs −
|r(Xns , u
the Novikov condition implies that Eρn = 1. Setting dP
T = ρndP|FT , by the Girsanov theorem
W n is a Wiener process under PnT . Relatively to W
n the equation (35) can be written:
dXnt = b(X
t )dt+ σdW
t , 0 ≤ t ≤ T
Xn0 = x.
Consider the following finite horizon Markovian forward-backward system (with respect to
probability PnT and to the filtration generated by {W
τ : τ ∈ [0, T ]}).
Xnτ (x) = e
e(τ−s)Ab(Xns (x)) ds+
e(τ−s)Aσ dW ns , τ ≥ 0,
Y nτ (x)− v(X
T (x)) +
Zns (x)dW
F (Xns (x), Y
s (x), Z
s (x))ds, 0 ≤ τ ≤ T,
and let (Xn(x), Y n(x), Zn(x)) be its unique solution with the three processes predictable rela-
tively to the filtration generated by {W nτ : τ ∈ [0, T ]} and: E
T supt∈[0,T ] |X
t (x)|
2 < +∞, Y n(x)
bounded and continuous, EnT
|Znt (x)|
2dt < +∞. Moreover, Theorem 5.2 and uniqueness of
the solution of system (38), yields that
Y nt (x) = v(X
t (x)), Z
t (x) = ∇v(X
t (x))G(X
t (x)). (39)
Applying the Itô formula to e−λtY nt (x), and restoring the original noise W we get
e−λτnY nτn(x) = e
−λTY nT (x) +
λe−λtY ns (x)ds −
e−λsZns (x) dWs
e−λs [F (Xns (x), Y
s (x), Z
s (x)) − Z
s (x)r(X
s , u
s )] ds.
We note that for every p ∈ [1,∞) we have
ρ−pn = exp
r(Xns , u
s )dW
|r(Xns , u
· exp
p2 − p
|r(Xns , u
. (41)
By (36) the second exponential is bounded by a constant depending on n and p, while the first
one has Pn-expectation, equal to 1. So we conclude that Enρ
n <∞. It follows that
e−2λt|Znt (x)|
ρ−2n |Z
t (x)|
≤ (Enρ−2n )
|Znt (x)|
We conclude that the stochastic integral in (40) has zero expectation. Using the identification
in (39) and taking expectation with respect to P, we obtain
Ee−λτnY nτn = e
E[v(XnT (x))] + E
λe−λtY ns (x)ds+
e−λs [F (Xns (x), Y
s (x), Z
s (x))− Z
s (x)r(X
s (x), u
s )] ds ≤
≤ e−λTE[v(XnT (x)] + E
λe−λsY ns (x)ds + E
e−λsg(Xns (x), u
s )ds.
Now we let n→ ∞. By Proposition 4.4,
|Y nt | = sup
|v(Xnt )| ≤
; (43)
in particular
λe−λsY ns (x)ds ≤ E
λe−λs
ds ≤ EK(T − τn)
and the right-hand side tends to 0 by (34). By the definition of un and (28),
g(Xns , u
s )ds = E
1s>τng(X
s , u0)ds ≤
1s>τn(1 + |u0|
2)ds ≤ CE(T − τn) (44)
and the right-hand side tends to 0 again by (34). Next we note that, again by (34), for n ≥ N(ω)
we have τn(ω) = T and v(X
T ) = v(X
) = v(Xτn) = v(XT ). We deduce, thanks to (43), that
Ev(XnT ) → Ev(XT ), and from (42) we conclude that
lim sup
Ee−λτnY nτn ≤ e
Ev(XT ).
On the other hand, for n ≥ N(ω) we have τn(ω) = T and e
−λτnY nτn = e
−λTY nT = e
−λT v(XnT ) =
e−λT v(XT ). Since Y
n is bounded, by the Fatou lemma, Ee−λT v(XT ) ≤ lim infn→∞ Ee
−λτnY nτn .
We have thus proved that
Ee−λτnY nτn = e
Ev(XT ). (45)
Now we return to backward equation in the system (38) and write
e−λτnY nτn = Y
−e−λtF (Xnt , Y
t , Z
t )dt+
−λe−λtY nt dt+
e−λtZnt dWt+
e−λtZnt r(X
t , u
t )dt
Arguing as before, we conclude that the stochastic integral has zero P-expectation. Moreover,
we have Y n0 = v(x), and, for t ≤ τn, we also have u
t = ut, X
t = Xt, Y
t = v(X
t ) = v(Xt) and
Znt = ∇xv(Xt). Thus, we obtain
E[e−λτnY nτn ] = v(x)+
− F (Xt, v(Xt),∇xv(Xt)σ)− λv(Xt) +∇xv(Xt)σr(Xt, ut)
dt (46)
e−λtg(Xt, ut)dt+ E[e
−λτnY nτn ] = v(x)+
− F (Xt, v(Xt),∇xv(Xt)σ) − λv(Xt) +∇xv(Xt)σr(Xt, ut) + g(Xt, ut)
dt. (47)
Noting that −F (x, y, z)− λy + zr(x, u) + g(x, u) ≥ 0 and recalling that g(x, u) ≥ 0 by (45) and
the monotone convergence theorem, we obtain for n→ ∞,
e−λtg(Xt, ut)dt+ e
Ev(XT ) = v(x)+
− F (Xt,∇xv(Xt)σ) − λv(Xt) +∇xv(Xt)σr(Xt, ut) + g(Xt, ut)
dt. (48)
Recalling that v is bounded, letting T → ∞, we conclude
J(x, u) = v(x)+
e−λt [−F (Xt, v(Xt),∇v(Xt)σ)− λv(Xt) +∇xv(Xt)σr(Xt, ut) + g(Xt, ut)] dt.
The above equality is known as the fundamental relation and immediately implies that v(x) ≤
J(x, u) and that the equality holds if and only if the following feedback law holds P-a.s. for
almost every t ≥ 0:
F (Xt, v(Xt),∇xv(Xt)σ) = ∇xv(Xt)σ + g(Xt, ut)− λv(Xt)
where X is the trajectory starting at x and corresponding to control u.
6.0.2 Existence of optimal controls: the closed loop equation.
Next we address the problem of finding a weak solution to the so-called closed loop equation.
We have to require the following
Assumption A6. Γ(x, y, z), defined in 31, is non empty for all x ∈ H and z ∈ Ξ∗.
By simple calculation (see [11] Lemma 3.1), we can prove that this infimum is attained in a
ball of radius C(1 + |z|), that is,
F (x, y, z) = min
u∈U ,|u|≤C(1+|z|)
[g(x, u) + zr(x, u)]− λy, x ∈ H, y ∈ R, z ∈ Ξ∗,
F (x, y, z) < g(x, u) + zr(x, u)− λy if |u| > C(1 + |z|). (49)
Moreover, by the Filippov Theorem (see, e.g., [1, Thm. 8.2.10, p. 316]) there exists a measurable
selection of Γ, a Borel measurable function γ : H × Ξ∗ → U such that
F (x, y, z) = g(x, γ(x, z)) + zr(x, γ(x, z)) − λy, x ∈ H, y ∈ R, z ∈ Ξ∗. (50)
By (49), we have
|γ(x, z)| ≤ C(1 + |z|). (51)
We define
u(x) = γ(x,∇xv(Xt)σ) P-a.s. for a.e t ≥ 0.
The closed loop equation is
dXt = AXtdt+ b(Xt)dt+ σ[dWt+ r(Xt, u(Xt))dt] t ≥ 0
X0 = x
By a weak solution we mean a complete probability space (Ω,F ,P) with a filtration (Ft) satisfy-
ing the usual conditions, a Wiener process W in Ξ with respect to P and (Ft), and a continuous
(Ft)-adapted process Xwith values in H satisfying, P-a.s.,
e−λt|u(Xt)|
2dt <∞
and such that (52) holds. We note that by (27) it also follows that
|r(Xt, u(Xt))|
2dt <∞, P− a.s.,
so that (52) makes sense.
Proposition 6.4. Assume that b, σ, g satisfy Assumption A4, F verifies Assumption A5 and
Assumption A6 holds. Then there exists a weak solution of the closed loop equation, satisfying
in addition
e−λt|u(Xt)|
2dt <∞. (53)
Proof. We start by constructing a canonical version of a cylindrical Wiener process in Ξ. An
explicit construction is needed to clarify the application of an infinite-dimensional version of the
Girsanov theorem that we use below. We choose a larger Hilbert space Ξ
⊃ Ξ in such a way
that Ξ is continuously and densely embedded in Ξ
with Hilbert-Schmidt inclusion operator J .
By Ω we denote the space C([0,∞[,Ξ
) of continuous functions ω : [0,∞[→ Ξ
endowed with
the usual locally convex topology that makes Ω a Polish space, and by B its Borel σ-field. Since
JJ ∗ has finite trace on Ξ
, it is well known that there exists a probability P on B such that
the canonical processes W
t (ω) := ω(t), t ≥ 0, is a Wiener process with continuous paths in Ξ
satisfying E[〈W
t , ξ
′ ] = 〈J J ∗ξ
′ (t∧ s) for all ξ
, t, s ≥ 0. This is called
a JJ ∗-Wiener processes in Ξ
in [8], to which we refer the reader for preliminary material on
Wiener processes on Hilbert spaces. Let us denote by G the P-completion of B and by N the
family of sets A ∈ G with P(A) = 0. Let Bt = σ{W
s : s ∈ [0, t]} and Ft = σ(Bt,N ), t ≥ 0, where
as usual σ(·) denotes the σ-algebra in Ω generated by the indicated collection of sets or random
variables. Thus (Ft)t≥0 is the Brownian filtration of W
The Ξ-valued cylindrical Wiener process {W
t : t ≥ 0, ξ ∈ Ξ} can now be defined as follows.
For ξ in the image of J ∗J we take η such that ξ = J ∗J η and define W
s = 〈W
s,J η〉Ξ′ . Then
we notice that E|W
2 = t|J η|2
′ = t|ξ|
Ξ, which shows that the mapping ξ → W
s , defined for
ξ ∈ J ∗J (Ξ) ⊂ Ξ with values in L2(Ω,F ,P), is an isometry for the norms of Ξ and L2(Ω,F ,P).
Consequently, noting that J ∗J (Ξ) is dense in Ξ, it extends to an isometry ξ → L2(ω,F ,P),
still denoted ξ → W
s . An appropriate modification of {W
t : t ≥ 0, ξ ∈ Ξ} gives the required
cylindrical Wiener process. We note that the Brownian filtration of W coincides with (Ft)t≥0.
Now let X ∈ L
(Ω, C(0,+∞;H)) be the mild solution of
dXτ = AXτ dτ + b(Xτ ) dτ + σ dWτ
X0 = x
If together with previous forward equation we also consider the backward equation
Yt − YT +
ZsdWs =
F (Xs, Ys, Zs)ds 0 ≤ t ≤ T <∞ (55)
we know that there exists a unique solution {Xxt , Y
t , Z
t , t ≥ 0} forward-backward system (54)-
(55) and by Proposition 5.2,
v(x) = Y x0 .
is the solution of the of the non linear stationary Kolmogorov equation:
Lv(x) + F (x, v(x),∇v(x)σ) = 0, x ∈ H. (56)
Moreover the following holds:
Yτ (x) = v(Xτ (x)), Zτ (x) = ∇v(Xτ (x))σ (57)
We have
e−(λ+ǫ)t|Zt|
2dt <∞. (58)
and hence
2dt <∞. (59)
By (27) we have
|r(Xt, u(Xt))| ≤ C(1 + |u(Xt)|), (60)
and by (51),
|u(Xt)| = |γ(Xt,∇v(Xt(x))σ)| ≤ C(1 + |∇v(Xt(x))σ|) = C(1 + |Zt|). (61)
Let us define ∀T > 0
MT = exp
〈r(Xs, u(Xs), dWs〉Ξ −
|r(Xs, u(Xs)|
. (62)
Now, arguing exactly as in the proof of Proposition 5.2 in [11], we can prove that EMT = 1, and
M is a P-martingale. Hence there exists a probability P̂T on FT admitting MT as a density with
respect to P, and by the Girsanov Theorem we can conclude that {Ŵt, t ∈ [0, T ]} is a Wiener
process with respect to P and (Ft). Since Ξ
is a Polish space and P̂T+h coincide with P̂T on
BT , T, h ≥ 0, by known results (see [22], Chapter VIII, §1, Proposition (1.13)) there exists a
probability P̂ on B such that the restriction on BT of P̂T and that of P̂ coincide, T ≥ 0. Let
Ĝ be the P̂-completion of B and F̂T be the P̂-completion of BT . Moreover, since for all T > 0,
{Ŵt : t ∈ [0, T ]} is a Ξ-valued cylindrical Wiener process under P̂T and the restriction of P̂T
and of P̂ coincide on BT modifying {Ŵt : t ≥ 0} in a suitable way on a P̂-null probability set we
can conclude that (Ω, Ĝ, {F̂t, t ≥ 0}, P̂, {Ŵt, t ≥ 0}, γ(X,∇v(X)σ(X))) is an admissible control
system. The above construction immediately ensures that, if we choose such an admissible
control system, then (52) is satisfied. Indeed if we rewrite (54) in terms of {Ŵt : t ≥ 0} we get
dXτ = AXτ dτ + b(Xτ ) dτ + σ [r(Xτ , u(Xτ ))dτ + dŴτ ]
X0 = x.
It remains to prove (53). We define stopping times
σn = inf
t ≥ 0 :
e−λt|Zs|
2ds ≥ n
with the convention that σn = ∞ if the indicated set is empty. By (58) for P-a.s. ω ∈ Ω there
exists an integer N(ω) depending on ω such that σn(ω) = ∞ for n ≥ N(ω). Applying the Ito
formula to e−λtYt, with respect to W , we obtain
e−λσnYσn = Y0 −
e−λsZs dWs+
e−λs [−F (Xs, Ys, Zs)− λYs(x)ds + Zsr(Xs, u(Xs))] ds.
from which we deduce that
Ee−λσnYσn + E
e−λsg(Xs, u(Xs))ds = Y0+
e−λs [−F (Xs, Ys, Zs)− λYsds + Zsr(Xs, u(Xs)) + g(Xs, u(Xs))] ds = Y0.
with the last equality coming from the definition of u. Recalling that Y is bounded, it follows
e−λsg(Xs, u(Xs))ds ≤ C
for some constant C independent of n. By (29) and by sending n to infinity, we finally prove
(53).
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