Optimal control of two scale stochastic systems in infinite dimensions: the BSDE approach

0.2 MB·

by  Giuseppina Guatteri

Optimal control of two scale stochastic systems in infinite dimensions: the
BSDE approach
Giuseppina Guatteri,
Dipartimento di Matematica,
Politecnico di Milano,
7
Piazza Leonardo da Vinci 32,
1
0 20133 Milano, Italia.
2 e-mail: [email protected]

n
a
Gianmario Tessitore,
J
Dipartimento di Matematica e Applicazioni
4
Universit´a Milano-Bicocca,

]
C via R. Cozzi 53 - Edificio U5,
O 20125 Milano, Italia.
. e-mail: [email protected]
h
t
a
m
Abstract
[

In this paper we study, by probabilistic techniques, the convergence of the value function for a two-
1
v scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds
5 diverges. The value function is represented as the solution of a backward stochastic differential equation
6 (BSDE) that it is shown to converge towards a reduced BSDE. The noise is assumed to be additive both
1
in the slow and the fast equations for the state. Some non degeneracycondition on the slow equation are
1
required. The limit BSDE involves the solution of an ergodic BSDE and is itself interpreted as the value
0
. function of an auxiliary stochastic control problem on a reduced state space.
1
0
7
1 Introduction
1
:
v
In this paper we study the convergence of the value function of an optimal control problem for a singularly
i
X
perturbed infinite dimensional state equation as
r
a
dX = AXε,α+b(Xε,α,Qε,α,α )dt +Rd 1, Xε,α = x ,
t t t t t Wt 0 0
(εdQεt,α = (BQεt,α+F(Xtε,α,Qεt,α)dt+Gρ(αt)dt+ε1/2GdWt2, Qε0 = q0,
wherethestateprocessesX andQareHilbertvalued,AandB areunboundedlinear operators, αrepresents
the control, ( 1) , ( 2) are infinite dimensional cylindrical Wiener processes, b, F, ρ are functions
Wt t≥0 Wt t≥0
satisfying suitable assumptions. We notice that the presence of the constant ε in the second equation
corresponds to the fact that Q evolves with a speed which is larger by a factor 1/ε then the speed of
evolution of the component X. In other words the above equation is a good model for a so called two scale
system. The optimal control problem is then completed by a standard cost functional of the form :
1
Jε(x ,q ,α) := E l(Xε,α,Qε,α,α )dt+h(Xε,α)
0 0 t t t 1
(cid:18)Z0 (cid:19)
Several authors have studied the convergence of singular stochastic control problems in finite dimensions,
see for instance [1], [2], [13], [14], [16]. In particular [1] has been an inspiration for the present work. In that
1
paper authors represent the value function of a singular stochastic control problem, in finite dimensions, by
the solution, in viscosity sense, of an Hamilton Jacobi Bellman equation. Then they show, by PDE methods
their convergence towards the solution, again in viscosity sense, of a reduced parabolic PDE with smaller
state space and a new nonlinearity usually called effective Hamiltonian. Such analysis is performed in the
case of periodic boundary conditions. Although PDE techniques perfectly fit the finite dimensional case
allowing to cover general situations, including state equations with control dependent diffusions that require
introduction of fully non-linear H.J.B. equations, they seem not to be adaptable to the infinite dimensional
case, and consequently to the case of two scale stochastic control problems for stochastic PDEs. The reason
essentially is the difficulty of handling, by analytic tools and viscosity solutions, parabolic equations in
infinite variables, see the discussion in the Introduction of [11].
The purpose of the present paper is twofold. On one side we wish to show that Backward Stochastic
Differential Equations (BSDEs) are, in general, an efficient way to represent the limit of the value functions
of twoscale systems whentheratiobetween thetwo evolutions’ speeddiverge. Ontheotherwewishtoshow
that, in such a way, we can cover the case of infinite dimensional state equations (that is the case of two
scale systems described by stochastic PDEs) that, at our best knowledge, was not considered in the existing
letterature. As a counterpart we notice that we consider state equation in which the control only affects the
drift and in which the noise of the slow component is assumed to be non-degenerate. Such restrictions seem
not to be intrinsic in the BSDEs approach but allows essential technical simplifications.
To be more specific our main result will be to prove that if
vε(x ,q ) := infJε(x ,q ,α)
0 0 0 0
α
then
vε(x ,q ) Y
0 0 0
→
where (X,Y¯,Z¯) is the unique solution of the following decoupled forward backward system of stochastic
differential equations
dX = AX dt+RdW1,
t t t
dY¯ = λ(X ,Z¯ )dt Z¯dW1,
− t t t − t
X = x , Y¯ = h(X ).
0 0 1 1
It is important to notice that the ‘reduced nonlinearity’ λ is itself a component of the unique solution

(Yˇ,Zˇ,λ) of a parametrized ergodic BSDE (see (4.1) and Theorem 4.2) as they were introduced in [9] (see
[8] and [15] as well). Function λ can itself be interpreted as the optimal cost of an ergodic optimal control
problem. Moreover, as it happensin thefinitedimensionalcase, thespacein whichtheabove reducedBSDE
lives is a subspace of the original one (corresponding to the slow evolution). As a by-product of our main
result, using the Bismut Elworthy formula in in [12] we immediately get that the solution of the reduced
BSDE, and therefore the limit value function, depends on x in a differentiable way and is linked to the
0
unique mild solution of a semilinear parabolic PDE in infinite dimensional spaces:
∂v(t,x) 1
+ Tr[ 2v(t,x)] = λ(x, v(t,x)), t [0,1], x H,
∂t 2 ∇x ∇ ∈ ∈

v(1,x) = h(x).

Finally, exploiting the concavity of λ we give a representation of Y¯t as the value function of an auxiliar
stochastic control problem on a reduced state space.
The paper is organized in the following way. In Section 2 we report some notation and assumptions while
Section 3 contains some estimates on the two scale state equation that will beusefulin the paper. InSection
4 we introduce parametrized ergodic BSDEs and study their regularity with respect to parameters. In
Section 5 we state the form of the limit equations and prove a convergence result for BSDEs that represents
the main technical issue of this paper. In Section 6, we apply our results to a stochastic singular control
problem. Finally, in section 7 weinterpret thesolution of the reducedBSDE in terms of a stochastic optimal
control problem.
2
2 Notation and preliminary results
GivenaBanach spaceE,thenormofitselements xwillbedenotedby x ,orevenby x whennoconfusion
E
| | | |
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. When F = R the dualspace L(E,R) will bedenoted by E∗. The
lettersΞ,H andK willalwaysbeusedtodenoteHilbertspaces. Thescalarproductisdenoted , ,equipped
h· ·i
with a subscript to specify the space, if necessary. All Hilbert spaces are assumed to be real and separable
and the dual of a Hilbert space will never be identified with the space itself. By L (Ξ,H) and L (Ξ,K)
2 2
we denote the spaces of Hilbert-Schmidt operators from Ξ to H and to K, respectively. Finally (K,H)
G
is the space of all Gateaux differentiable mappings φ from K to H such that the map (k,v) φ(k)v is
→ ∇
continuous from K K to H; see [11] for details.
×
Let W1 = (W1) and W2 = (W2) be two independent cylindrical Wiener processes with values in Ξ,
t t≥0 t t≥0
defined on a complete probability space (Ω, ,P). By , t [0,T] we will denote the natural filtration of
t
F {F ∈ }
(W1,W2), augmented withthefamily of P-nullsets of . Obviously, thefiltration ( )satisfies theusual
t
N F F
conditions of right-continuity and completeness. All the concepts of measurability for stochastic processes
will refer to this filtration. By we denote the predictable σ-algebra on Ω [0,T] and by (Λ) the Borel
P × B
σ-algebra of any topological space Λ.
Next we define the following two classes of stochastic processes with values in a Hilbert space V. Given an
arbitrary time horizon T and constant p 1:
≥
Lp(Ω [0,T];V) denotes the space of equivalence classes of processes Y Lp(Ω [0,T];V) admitting
• P × ∈ ×
a predictable version. It is endowed with the norm
T 1/p
Y = E Y pds .
s
| | | |
(cid:16) Z0 (cid:17)
Lp,loc(Ω [0,+ [;V) denotes the set of processes defined on R+ such that their restriction to an
• P × ∞ p
arbitrary [0,T] belongs to L (Ω [0,T];V).
P ×
p
L (Ω;C([0,T];V)) denotes the space of predictable processes Y with continuous paths in V, such that
• P
the norm
Y = (E sup Y p)1/p
p s
k k | |
s∈[0,T]
p
is finite. The elements of L (Ω;C([0,T];V)) are identified up to indistinguishability.
P
Lp,loc(Ω : [0,+ [;V) denotes the set of processes defined on R+ such that their restriction to an
• P C ∞ p
arbitrary [0,T] belongs to L (Ω;C([0,T];V)).
P
Given Φ in L2(Ω [0,T];L (Ξ,V)), the Itˆo stochastic integrals tΦ dW1 and tΦ dW2, t [0,T], are
P × 2 0 s s 0 s s ∈
V-valued martingales belonging to L2(Ω;C([0,T];V)). The previous definitions have obvious extensions to
P R R
processes defined on the entire positive real line R+.
3 The forward system
For arbitrarily fixed x H and q K we consider the following system of controlled stochastic differential
0 0
∈ ∈
equations in H K:
×
dX = AX dt+RdW1, X = x , t 0,
t t t 0 0 ≥
(3.1)

εdQε = (BQε+F(Xε,Qε))dt+ε1/2GdW2, Qε = q , t 0
t t t t t 0 0 ≥


3
where the “slow” variable X takes its values in H and the “fast” variable Qε takes its values in K, ε ]0,1]
∈
is a small parameter.
Finally A : D(A) H H and B : D(B) K K are unbounded linear operators generating C -
0
⊂ → ⊂ →
semigroups etA and etB over H and K, respectively, while R and G are linear bounded operators
t≥0 t≥0
{ } { }
from Ξ to H (respectively to K).
Moreover, we make the following, standard assumptions:
Hypothesis 3.1 A : D(A) H H is the generator of a semigroup, etA , such that etA
t≥0 L(H,H)
⊂ → { } | | ≤
MeωAt,t 0 for some positive constants MA and ωA. B : D(B) K K is a linear, unbounded operator
≥ ⊂ →
that generates a C0- semigroup etB t≥0 such that etB L(K,K) MBeωBt,t 0 for some MB > 0.
{ } | | ≤ ≥
Moreover there exist constants L > 0 and γ [0, 1[, s.t.:
∈ 2
L
esA + esB ,
| |L2(Ξ,H) | |L2(Ξ,K) ≤ (1 s)γ
∧
We also assume that R L(Ξ;H) and admits a bounded right inverse R−1 L(H;Ξ).
∈ ∈
Hypothesis 3.2 F :H K K is bounded and there exists a constant L for which:
F
× →
F(x,y) F(u,v) L (x u + y v )
K F H K
| − | ≤ | − | | − |
for every x,u H, y,v K. Moreover we assume that for every x H, F(x, ) is Gateaux differentiable,
∈ ∈ ∈ ·
more precisely, F(x, ) 1(K,K).
· ∈ G
Hypothesis 3.3 We ask B +F to be dissipative i.e. there exists some µ >0 such that:
Bq+F(x,q) (Bq′+F(x,q′)),q q′ µ q q′ 2
h − − i≤ − | − |
for all x H,q,q′ D(B).
∈ ∈
Hypothesis 3.4 G (Ξ,K).
∈ L
GivenanycylindricalWienerprocess(β ) withvaluesinΞwedenoteby(βB) thestochasticconvolution
t t≥0 t t≥0
s
βB = e(s−ℓ)Bdβ .
s ℓ
Z0
In the following we shall assume as in [9] that:
Hypothesis 3.5 : sup E βB 2 < .
s>0 | s | ∞
Remark 3.6 Notice that since (β ) is a centered gaussian process this implies that, p 1 it holds
t
∀ ≥
sup E βB p < . Moreover Hypothesis 3.5 is verified whenever B is a strongly dissipative operator.
s>0 | s | ∞
Remark 3.7 Under Hypothesis 3.1 equation (3.1) admits a unique mild solution (Xx0) that has continuous
t
trajectories and satisfies E(sup Xx0 p) c (1+ x p). See [11] for the proof.
t∈[0,1]| t | ≤ p | 0|
Lemma 3.8 Let (Γ ) be a given, K-valued, predictable process with Γ Lp,loc(Ω [0, [,H) and let
s s≥0 ∈ P × ∞
p,loc
(g) be a given, Ξ-valued, process with g L (Ω [0,+ [,K). We introduce the following equation:
s≥0 ∈ P × ∞
dQ = (BQ +F(Γ ,Q ))ds+g ds+Gdβ , s 0, Q = q , (3.2)
s s s s s s 0 0
≥
p,loc
that admits a unique mild solution Q L (Ω;C[0,+ [,K).
∈ P ∞
4
Under hypotheses (3.1)–(3.4), for all p 1, there exists a constant k (independent on T) such that for all
p
≥
T > 0:
sup E Q p k (1+ q p+ sup E Γ p+ sup E βB p+ sup E g p) (3.3)
| s| ≤ p | 0| | s| | s | | s|
s∈[0,T] s∈[0,T] s∈[0,T] s∈[0,T]
Moreover if (Γ′) isanother K-valued, predictable processes inL2(Ω [0,T],H) and Q′ isthe mild solution
s s≥0 P ×
of equation:
dQ′ = (BQ′ +F(Γ′,Q′ ))ds+g ds+Gdβ , s 0, Q = q ,
s s s s s s ≥ 0 0
then, for all T > 0,
T
Q Q′ K e−µ(T−ℓ) Γ Γ′ dℓ P-a.s.
| T − T|≤ | ℓ− ℓ|
Z0
where again K does not depend on T.
Proof. For the reader’s convenience we briefly report the argument which is a slight modification of the
one in [7], Section 6.3.2.
Let Z = eµs(Q βB). By Ito rule (going through Yosida approximations) we deduce that (Z) is the mild
s s− s
solution of the following equation
dZ = µZ +BZ +eµsF(Γ ,e−µsZ +βB))ds+eµsg ds.
s s s s s s s
Differentiating Z 2+ε (going, once more, through Yosida approximations), using dissipativity of B+F
s
| |
as in Hypothesis 3.3 we obtain
p
s s
Z Z 2+ε q2+ε+ eµℓ F(Γ ,βB)+g dℓ+µ Z 2+ε Z dℓ
| s|≤ | s| ≤ 0 ℓ ℓ ℓ | ℓ| −| ℓ|
p q Z0 (cid:12) (cid:12) Z0 hp i
Letting ε 0, by dominated convergence we obt(cid:12)ain: (cid:12)
→
s
Z q + eµℓ F(Γ ,βB)+g dℓ.
| s|≤ | 0| ℓ ℓ ℓ
Z0
(cid:12) (cid:12)
Recalling the definition of Z we conclude (cid:12) (cid:12)
s
Q βB +e−µt q + e−µ(s−ℓ) F(Γ ,βB)+g dℓ
| s|≤ | s | | 0| ℓ ℓ ℓ
Z0
(cid:12) (cid:12)
(cid:12) (cid:12)
and by Holder inequality (for the last term)
s p/p∗ s
|Qs|p ≤ 3p|βsB|p +3pe−pµt|q0|p+ e−p∗µ2(s−ℓ)dℓ 3p e−pµ2(s−ℓ)|F(Γℓ,βℓB)+gℓ|pdℓ
(cid:18)Z0 (cid:19) Z0
The claim then follows by Hypothesis 3.2
The proof of the last statement is similar (and easier) noticing that
d (Q Q′)= B(Q Q′)ds+[F(Γ ,Q ) F(Γ′,Q′)]ds
s s− s s− s s s − s s
and arguing as before
As a special case of equation (3.2), for all x H and q K we denote by Qˆx,q0 the unique mild solution
0
∈ ∈
of equation:
dQˆx,q0 = (BQˆx,q0 +F(x,Qˆx,q0))ds+dWˆ 2, s 0, Qˆx,q0 = q . (3.4)
s s s s ≥ 0 0
where Wˆ 2 = ε−1/2W2 is a cylindrical Wiener process.
s εs
5
4 The ergodic BDSE parametrized
We introduce a function ψ : H K Ξ∗ Ξ∗ R and assume the following
× × × →
Hypothesis 4.1 Function ψ is measurable and there exist L ,L ,L ,L > 0 such that q,q′ K, ξ, ξ′
q ξ x z
∀ ∈ ∈
Ξ∗, x,x′ H, z,z′ Ξ∗:
∈ ∈
ψ(x,q,z,ξ) ψ(x′,q′,z′,ξ′) L (1 + z )x x′ + L z z′ + L (1 + z )q q′ + L ξ ξ′ .
x z q ξ
| − | ≤ | | | − | | − | | | | − | | − |
Moreover we assume that sup ψ(x,q,0,0) < +
x∈H,q∈K| | ∞
Thenextresultstates existenceofasolutiontothesocalledergodic backward stochastic differential equation.
dYˇ = [ψ(x,Qˆx,q0,z,Ξˇ ) λ(x,z)]dt Ξˇ dW2, t s (4.1)
− t t − − t t ∀ ≥
Its proof is in large part contained in [9] Theorem 4.4 and Corollary 5.9.
Theorem 4.2 Under Hypotheses 3.1, 3.2, 3.3, 3.4, 3.5 and 4.1 there exist measurable functions vˇ : H
K Ξ∗ R, ζˇ: H K Ξ∗ R, λ : H Ξ∗ R with ×
× → × × → × →
vˇ(x,q,z) c(1+ z )q (4.2)
| | ≤ | | | |
(where c > 0 depends only on the constants introduced in the above mentioned Hypotheses) such that the
following holds. If we set:
Yˇx,q0,z = vˇ(x,Qˆx,q0,z), Ξˇx,q0,z = ζˇ(x,Qˆx,q0,z) (4.3)
t t t t
then Ξˇx,q0,z is in L2,loc([0,+ [,Ξ∗) and (Yˇx,q,z,Ξˇx,q,z,λ(x,z)) is a solution to equation (4.1).
∞ t
Moreover we have:
λ(x,z) λ(x′,z′) L1(1+ z )x x′ +L1 z z′ . (4.4)
| − | ≤ x | | | − | z| − |
for some posive constants L1 and L1.
x z
Proof. Fix x H and z Ξ∗; in [9] Theorem 4.4, authors prove existence of the function vˇ(x, ,z),
ζˇ(x, ,z) and λ∈(x,z) such th∈at (4.2) holds and, if Yˇx,q0,z, Ξˇx,q0,z are defined as in (4.3), then Ξˇx,q0,z ·is in
L2,loc·([0,+ [,Ξ∗) and (Yˇx,q0,z,Ξˇx,q0,z,λ(x,z)) is a solution to equation (4.1).
Measurably∞of vˇ, ζˇand λ twith respect to all parameters follows by their construction (see again [9] Theorem
4.4).
Weonlyneedtoprove(4.4). Fixedx,x′ H andz,z′ Ξ∗ wesetλ = λ(x,z) λ(x′,z′),Y = Yx,0,z Yx′,0,z′,
Ξ = Ξˇx,0,z Ξˇx′,0,z′, ∈ ∈ − −
−
e e
e ψ(x,Qˆxr,0,z,Ξˇrx,0,z)−ψ(x,Qˆxr,0,z,Ξˇrx′,0,z′)(Ξˇx,0,z Ξˇx′,0,z′), if Ξˇx,0,z = Ξˇx′,0,z′
θt =  0 |Ξˇrx,0,z −Ξˇrx′,0,z′|2K∗ r − r elsewrhere6 r
and 
f = ψ(x,Qˆx,0,z,Ξˇx′,0,z′) ψ(x′,Qˆx′,0,z′,Ξˇx′,0,z′)
t r r − r r
Then we have
T T
Y +λT = Y + f dr Ξ (θ dt+dWˆ 2), T t 0
0 T r − r t r ∀ ≥ ≥
Zt Zt
So,byGirsanovtheoreme,thereexiestsaprobabilityP(meeanvaluedenotedbyE)suchthatW = tθ dℓ+Wˆ 2,
t 0 ℓ t
t 0, is a cylindrical Wiener process. We notice that
≥ R
e e f
T T
λT = Y Y + f dr Ξ dW , T t 0
T 0 r r r
− − ∀ ≥ ≥
Z0 Z0
e e e f
6
and consequently:
T
λ T−1 Y +T−1EY +T−1 E f ds (4.5)
0 T s
| | ≤ | | | | | |
Z0
Thanks to Hypothesis 4.1 we get that for eall t 0: e e e
≥
f L (1+ z )x x′ +L z z′ +L (1+ z )Qˆx,0 Qˆx′,0 , P a.s.
| t|≤ x | | | − | z| − | q | | | t − t | −
We notice that with respect to (W ) processes Qˆx,0 and Qˆx′,0 satisfy respectively
t
dQˆx,0 =f(BQˆx,0+F(x,Qˆx,0))ds+θ ds+dW , s 0,
s s s s s ≥
dQˆx′,0 = (BQxˆ′,0 +F(x′,Qˆx′,0))ds+θ ds+dW , s 0,
s s s s fs ≥
and Lemma 3.8 yields Qˆxs,0 Qˆxs′,0 (K/µ)x x′ thus f
| − | ≤ | − |
f (L +L K/µ)(1+ z )x x′ +L z z′ P a.s. for all t 0. (4.6)
t x q z
| |≤ | | | − | | − | − ≥
x,0 x′,0
Moreover under Hypothesis 3.5 Lemma 3.8 also yields sup E(Q + Q ) < . Thus by (4.2) we
t∈[0,∞[ | t | | t | ∞
get sup E(Y ) < . Consequently T−1E(Y ) 0 as T and the claim follows by (4.5) and
t∈[0,∞[ t T
| | ∞ | | → → ∞
(4.6) letting T . e
→ ∞
e e e e
5 Limit equation and convergence of singular BSDEs
We are interested in the following forward-backward system for t [0,1]
∈
dX = AX +RdW1,
t t t
εdQε = (BQε+F(Xε,Qε)dt+ε1/2GdW2,
t t t t t (5.1)
 dYε = ψ(Xε,Qε,Zε,Ξε/√ε)dt ZεdW1 ΞεdW2,
− t t t t t − t t − t t
Xε = x Qε = q , Yε = h(X ),
0 0 0 0 1 1



that, as we will see in the sequel, is associated to a controlled multiscale dynamics. Function h : H R
→
satisfies:
Hypothesis 5.1 h is Lipschitz continuous with constant L > 0.
We have that:
Theorem 5.2 Assume 3.1–3.4, 4.1 and 5.1.
Then for every ε > 0 (5.1) has a unique solution (X,Qε,Yε,Zε,Ξε), with X L2(Ω;C([0,1];H)), Qε
∈ P ∈
L2(Ω;C([0,1];K)), Yε L2(Ω;C([0,1];R)), Zε L2(Ω [0,1];Ξ∗), Ξε L2(Ω [0,1];Ξ∗).
P P P P
∈ ∈ × ∈ ×
Proof. The proof is contained in [11], Propositions 3.2 and 5.2, we justnotice that the system is decoupled,
so once the forward equation is solved then it becomes a parameter in the backward equation.
We remark that if we slow down time, that is, for s [0,1/ε[ we set: Qˆε = Qε , Yˆε = Yε, Ξˆε = ε−1/2Ξε
∈ s εs s εs s εs
then the following holds:
dQˆε =(BQˆε+F(X ,Qˆε)dt+ GdWˆ 2,
s s εs s s
dYˆε = ψ(X ,Qˆε,Zˆε,Ξˆε)ds √εZˆε dWˆ 1 ΞˆεdWˆ 2, (5.2)
− s εs s s s − εs s − s s
X = x Qˆε = q , Yˆε = h(X ).
0 0 0 0 1/ε 1


7
where Wˆ ℓ = ε−1/2Wℓ , ℓ = 1,2.
s εs
The purpose of our work is to study the limit behaviour of Yε as ε tends to 0.
We introduce the candidate limit equation, that turns out to be a forward-backward system on the finite
horizon [0,1] and on the reduced state space H.
dX = AX dt+RdW1,
t t t
dY¯ = λ(X ,Z¯ )dt Z¯ dW1, (5.3)
− t t t − t t
X = x , Y¯ = h(X ).
0 0 1 1

where λ is defined in Theorem 4.2. Thanks to (4.4) one has that
Theorem 5.3 System (5.3)hasonlyonesolution(X,Y,Z)withX Lp(Ω;C([0,1];H)), Y Lp(Ω;C([0,1];R)),
Z Lp(Ω [0,1];Ξ∗). ∈ P ∈ P
∈ P ×
Proof. Thank to (4.4) the proof of existence and uniqueness of the solution to equation (5.3) is standard
(see, for instance [11]).
We can now state our main result:
Theorem 5.4 Under Hypothesis 3.1—3.5, 4.1 and 5.1, if Y¯ is the solution to equation (5.3) and Yε is the
t
solution to equation (5.1), we have that the following holds
limYε = Y¯ (5.4)
0 0
ε→0
Proof. We must compare
1 1
Yε Y¯ = h(X ) h(X )+ (ψ(X ,Qε,Zε,Ξε/√ε) λ(X ,Z¯ ))dt (Zε Z¯ )dW1
0 − 0 1 − 1 t t t t − t t − t − t t
Z0 Z0
1
ΞεdW2.
− t t
Z0
By adding and subtracting we get:
1 1
(ψ(X ,Qε,Zε,Ξε/√ε) λ(X ,Z¯ ))dt = [(ψ(X ,Qε,Zε,Ξε/√ε) ψ(X ,Qε,Z¯ ,Ξε/√ε)]dt
t t t t − t t t t t t − t t t t
Z0 Z0
1
+ (ψ(X ,Qε,Z¯ ,Ξε/√ε) λ(X ,Z¯ )dt = I +I . (5.5)
t t t t − t t 1 2
Z0
We leave I for the moment
1
As far as I is concerned we have to use a discretization argument.
2
Letusnowintroduceforevery N positiveinteger, apartitionoftheinterval [0,1]oftheformt = k2−N, k =
k
0,1,...,2N and define a couple of step processes XN and ZN defined as follows:
XN(t) = X(t ), t [t ,t [, k = 0,...2N 1, (5.6)
k k k+1 e
∈ −
tk
ZN(t)= 2N Z¯ dℓ, for t [t ,t [, k = 1,...2N 1, Z = 0 for t [0,t [, (5.7)
ℓ k k+1 0 1
∈ − ∈
Ztk−1
where X,Z¯ aree part of the solution of (5.3). By construction e
1
lim E ZN Z¯ 2dt (5.8)
N→∞ Z0 | t − t|
We fix N, then for k =0,1,...,2N 1 we consider thee following, iteratively defined, class of forward SDE:
−
dˆN,k = (B ˆN,k +F(X , ˆN,k))dt+GdWˆ 2, s t /ǫ, ˆN,k = ˆN,k−1, (5.9)
Qs Qs tk Qs s ≥ k Qtk/ǫ Qtk/ǫ
8
Moreover we define
YˇN,k = vˇ(X , ˆN,k,ZN), ΞˇN,k = ζˇ(X , ˆN,k,ZN), for s t /ǫ
s tk Qs tk s tk Qs tk ≥ k
so that the triplet ((YˇN,k) ,λ(X ,ZeN),(ΞˇN,k) ) verifies: e
s s≥tk/ε tk tk s s≥tk/ε
dYˇN,k = [ψ(X , ˆN,k,eZN,ΞˇN,k) λ(X ,ZN)]ds ΞˇN,kdWˆ 2, s t /ε (5.10)
− s tk Qs tk s − tk tk − s s ∀ ≥ k
and e e
YˇN,k c(1+ ZN ) ˆN,k for all s t /ǫ P a.s. (5.11)
| t | ≤ | tk| |Qs | ≥ k −
for some positive constant c > 0 independent of k and N.
e
We also set XN = 2N−1X I (t), for t [0,1[ and, for s [0,1/ǫ[:
t k=0 tk [tk,tk+1[ ∈ ∈
P
2N−1 2N−1
ˆN = ˆN,kI (s), ΞˇN = ΞˇN,kI (s).
Qs Qs [tk/ǫ,tk+1/ǫ[ s s [tk/ǫ,tk+1/ǫ[
k=0 k=0
X X
so that, for all N N and k = 0,...,2N 1 have
∈ −
tk+1/ε tk+1/ε
YˇN,k YˇN,k [ψ(XN, ˆN,ZN,ΞˇN) λ(XN,ZN)]ds+ ΞˇN dWˆ 2 = 0. (5.12)
tk/ε − tk+1/ε−Ztk/ε εs Qs εs s − εs εs Ztk/ε s s
e e
So, coming back to the term I , we have, after rescaling of time:
2
2N−1 tk+1/ε
I = ε [ψ(X ,Qˆε,Z¯ ,Ξˆε) λ(X ,Z¯ )]ds
2 εs s εs s − εs εs
k=0 Ztk/ε
X
and, adding the null terms in (5.12) for k = 1,...,2N:
2N−1 tk+1/ε 2N−1 tk+1/ε
I = ε [ψ(X ,Qˆε,Z¯ ,Ξˆε) ψ(XN, ˆN,ZN,ΞˇN)]ds+ε ΞˇN dWˆ 2 (5.13)
2 εs s εs s − εs Qs εs s s s
k=0 Ztk/ε k=0 Ztk/ε
X X
2N−1 tk+1/ε 2N−e1
ε [λ(X ,Z¯ ) λ(XN,ZN)]ds+ε (YˇN,k YˇN,k ).
− k=0 Ztk/ε εs εs − εs εs k=1 tk/ε − tk+1/ε
X X
e
Therefore coming back to our original term Yε Y¯ we have
0 − 0
N 1/ε
Yε Y¯ =ε (YˇN,k YˇN,k )+ε ψ(X ,Qˆε,Zε ,Ξˆε) ψ(X ,Qˆε,Z¯ ,Ξˆε) ds
0 − 0 Xk=1 tk/ε − tk+1/ε Z0 h εs s εs s − εs s εs s i
1/ε
+ε [ψ(X ,Qˆε,Z¯ ,Ξˆε) ψ(XN, ˆN,ZN,ΞˇN)]ds
εs s εs s − εs Qs εs s
Z0
1/ε
ε [λ(X ,Z¯ ) λ(XN,ZN)]ds e
− εs εs − εs εs
Z0
1/ε 1/ε
√ε (Zε Z¯ )dWˆ 1 εe (Ξˆε ΞˇN)dWˆ 2.
− εs− εs s − s− s s
Z0 Z0
9
By lipscitzianity of ψ and λ (see Theorem 4.2 )
1/ε N
Yε Y¯ =ε ε,N ds+ε (YˇN,k YˇN,k )
0 − 0 Rs tk/ε − tk+1/ε
Z0 k=1
X
1/ε
+ε [ψ(X ,Qˆε,Zε ,Ξˆε) ψ(X ,Qˆε,Z¯ ,Ξˆε)]ds
εt s εs s − εs s εs s
Z0
1/ε
+ε [ψ(XN, ˆN,ZN,Ξˆε) ψ(XN, ˆN,ZN,ΞˇN)]ds
εs Qs εs s − εs Qs εs t
Z0
1/ε 1/ε
+ε (ΞˇN Ξˆε)dWˆe2+√ε (Zε Z¯e)dWˆ 1
s − s s εs− εs s
Z0 Z0
where, see Hypothesis 4.1, for a suitable constant c it holds:
ε,N c(1+ Z¯ )X XN +c(1+ Z¯ )Qˆε ˆN +cZ¯ ZN (5.14)
|Rs | ≤ | εs| | εs − εs| | εs| | s −Qs | | εs− εs|
We need to get rid of some terms using a Girsanov argument, thus we introduce: e
[ψ(X ,Qˆε,Zε ,Ξˆε) ψ(X ,Qˆε,Z¯ ,Ξˆε)]
εt s εs s − εs s εs s (Zε Z¯ )∗ if Zε Z¯ = 0
δ1,ε(s) = Zε Z¯ 2 εs− εs | εs− εs|6 (5.15)
 | εs− εs|
0 if |Zεεs−Z¯εs|= 0
and 

ψ(XN, ˆN,ZN,Ξˆε) ψ(XN, ˆN,ZN,ΞˇN)
εs Qs εs s − εs Qs εs s (Ξˆε ΞˇN)∗ if Ξˆε ΞˇN = 0
δ2,ε,N(s) = 0 e |Ξˆεs−ΞˇNs |2 e s− s if |Ξˆsε−ΞˇsN| 6= 0 (5.16)
| s− s |
 1,ε 2,N,ε
We notice that processes (δ ) and (δ ) are bounded uniformly with respect to N and ǫ.
s s∈[0,1/ε] s s∈[0,1/ε]
We have
1/ε 1/ε
Yε Y¯ =ε δ1,ε(s)[Zε Z¯ ]ds+ε δ2,ε,N(s)[ΞˇN Ξˆε]ds
0 − 0 εs− εs s − s
Z0 Z0
1/ε 1/ε
+ε (ΞˇN Ξˇε)dWˆ 2+√ε (Zε Z¯ )dWˆ 1
s − s s εs− εs s
Z0 Z0
1/ε N
+ε ε,N ds+ε (YˇN,k YˇN,k )
Rs tk/ε − tk+1/ε
Z0 k=1
X
and rescaling time (speeding it up this time)
1 1
Yε Y¯ = δ1,ε(t/ε)[Zε Z¯ ]dt+ δ2,ε,N(t/ε)[ΞˇN Ξˆε ]dt
0 − 0 t − t ε−1t− ε−1t
Z0 Z0
1 1
+√ε (ΞˇN Ξˇε )dW2+ (Zε Z¯ )dW1
ε−1t− ε−1t t t − t t
Z0 Z0
1 N
+ ε,N dt+ε (YˇN,k YˇN,k )
Rε−1t tk/ε − tk+1/ε
Z0 k=1
X
We set for t [0,1]:
∈
t
W1 =: δ1,ε(r/ε)dr+W1 (5.17)
t t
Z0
t
Wf2 =:ε−1/2 δ2,ε,N(r/ε)dr+W2 (5.18)
t t
Z0
f
10