Introduction

Let us consider a controlled Itô process satisfying the following stochastic differential equation (SDE) dy(t) = f (t, y(t), u(t), ω)dt + σ(t, y(t), u(t), ω)dW (t), on [0, T ] × Ω,

y(0) = y 0 ∈ R n . (1)
In the notation above y(t) ∈ R n denotes the state function and u(t) ∈ R m the control. Associated to a pair state-control (y, u) we define its cost J(y, u) by J(y, u) := E T 0 (t, y(t), u(t))dt + φ(y(T )) .

(

) 2
Precise definitions of the suitable spaces for (y, u) and assumptions over the data (f, σ, , φ), will be provided in the next section. For the dynamics (1) and the cost (2), we will study two types of problems. In the first one, we suppose that we are given a nonempty closed and convex subset U of the space L 2 F of adapted square integrable process, and we analyze the following stochastic optimal control problem with control constraints Min (y,u) J(y, u) s.t. [START_REF] Bensoussan | Lectures on stochastic control[END_REF] holds and u ∈ U.

(SP)

As in the case of deterministic optimal control problems, there are two main approaches to study problem (SP) when U is defined by local constraints, i.e. for a given nonempty closed and convex subset U ⊆ R n , U := u ∈ L 2 F ; u(t, ω) ∈ U, for almost all (t, ω) ∈ [0, T ] × Ω .

The first approach is the global one, based on Bellman's dynamic programming principle, which yields that the value function of (SP) is the unique viscosity solution of an associated second order Hamilton-Jacobi-Bellman equation. For a complete account of this point of view, widely used in practical computations, we refer the reader to Lions [START_REF] Lions | Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications[END_REF][START_REF] Lions | Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 2: viscosity solutions and uniqueness[END_REF] and to the books [START_REF] Fleming | Controlled Markov processes and viscosity solutions[END_REF][START_REF] Pham | Optimisation et contrôle stochastique appliqués à la finance[END_REF][START_REF] Yong | Stochastic controls, Hamiltonian systems and HJB equations[END_REF]. The second approach is the variational one, which consists in the local behavior analysis of the value function under small perturbations of a local minimum. Using this technique, Kushner [START_REF] Kushner | On the existence of optimal stochastic controls[END_REF][START_REF] Kushner | On the stochastic maximum principle: Fixed time of control[END_REF][START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF], Bensoussan [START_REF] Bensoussan | Lectures on stochastic control[END_REF][START_REF] Bensoussan | Stochastic maximum principle for distributed parameter system[END_REF], Bismut [START_REF] Bismut | Analyse convexe et probabilités[END_REF][START_REF] Bismut | Théorie probabiliste du contrôle des diffusions[END_REF][START_REF] Bismut | An introductory approach to duality in optimal stochastic control[END_REF] and Haussmann [START_REF] Haussmann | General necessary conditions for optimal control of stochastic systems[END_REF] obtained natural extensions of Pontryagin maximum principle to the stochastic case, that were generalized by Peng [START_REF] Peng | A general stochastic maximum principle for optimal control problems[END_REF] to the case where U is not necessarily convex and by Cadenillas and Karatzas [START_REF] Cadenillas | The stochastic maximum principle for linear convex optimal control with random coefficients[END_REF] to the non-Markovian case. Relations between the global and variational approach are studied in [START_REF] Zhou | The connection between the maximum principle and dynamic programming in stochastic control[END_REF][START_REF] Zhou | A unified treatment of maximum principle and dynamic programming in stochastic controls[END_REF]. Nevertheless, to the best of our knowledge, nothing has been said about second order optimality conditions for (SP). Using the variational technique we are able to obtain first and second order expansions for the cost function, which are expressed in terms of the derivatives of the Hamiltonian of problem (SP). The main tool is a kind of generalization of Gronwall's lemma for the SDEs (proposition 1) obtained by Mou and Yong [START_REF] Mou | A variational formula for stochastic controls and some applications[END_REF], which allows to expand the cost with respect to directions belonging to a more regular space than the control space. Note that the idea of using a more regular space that the original one was already used [START_REF] Bonnans | On the choice of the function spaces for some state-constrained control problems[END_REF] in the context of deterministic state constrained optimal control problems. By a density argument, we establish first order optimality conditions, which in the case of local constraints are a consequence of the maximum principles obtained in the above references. However, we can also deal with constraints sets that are not necessarily local. The main novelty of our work is that under a polyhedricity assumption (see [START_REF] Haraux | How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities[END_REF][START_REF] Mignot | Contrôle dans les inéquations variationnelles[END_REF]) over the set U, we are also able to provide second order necessary conditions which are new for the stochastic case and are natural extensions of their deterministic counterparts.

In the second type of problem we suppose that we are given functions g i , h j with i ∈ {1, ..., n g }, j ∈ {1, ..., n h }, and we study the following optimal control problem with finitely many equality and inequality constraints Min (y,u) J(y, u) s.t. [START_REF] Bensoussan | Lectures on stochastic control[END_REF] holds and E g i (y(T )) = 0, E h j (y(T )) ≤ 0. (SP ) First order optimality conditions for (SP ), in a maximum principle form, have been obtained in [START_REF] Peng | A general stochastic maximum principle for optimal control problems[END_REF][START_REF] Cadenillas | The stochastic maximum principle for linear convex optimal control with random coefficients[END_REF]. Under a standard qualification condition over g i , h j , the techniques employed for (SP) allow us recover particular cases of the results in [START_REF] Peng | A general stochastic maximum principle for optimal control problems[END_REF][START_REF] Cadenillas | The stochastic maximum principle for linear convex optimal control with random coefficients[END_REF], but in addition we are also able to prove second order necessary conditions for (SP ).

The article is organized as follows: After introducing standard notations and assumptions in section 2, we obtain in section 3 first and second order expansions for the state and cost function. The proof of technical lemmas of this section are provided in the Appendix. In section 4, first and second order necessary conditions are proved for problem (SP) with explicit results for the case of box constraints over the control. A discussion about a non gap second order sufficient condition is also provided. Finally, in section 5 first and second order necessary conditions are derived for problem (SP ).

Notations, assumptions and problem statement

Let us first fix some standard notation. For x belonging to an Euclidean space we will write x i for its i-th coordinate and |x| for its Euclidean norm. Let T > 0 and consider a filtered probability space (Ω, F, F, P), on which a d-dimensional

(d ∈ N * ) Brownian motion W (•) is defined. We suppose that F = {F t } 0≤t≤T
is the natural filtration, augmented by all P-null sets in F, associated to W (•). Let (X, • X ) be a Banach space and for

β ∈ [1, ∞) set L β (Ω; X) := v : Ω → X; v is F -measurable and E v(ω) β X < ∞ , L ∞ (Ω; X) := {v : Ω → X; v is F -measurable and ess sup ω∈Ω v(ω) X < ∞} . For β, p ∈ [1, ∞] and m ∈ N let us define L β,p F := v ∈ L β (Ω; L p ([0, T ]; R m )) ; (t, ω) → v(t, ω) := v(ω)(t) is F-adapted .
We endow these space with the norms

v β,p := E v(ω) β L p ([0,T ];R m ) 1 β and v ∞,p := ess sup ω∈Ω v(ω) L p ([0,T ];R m ) .
For the sake of clarity, when the context is clear, the statement "for a.a. t ∈ [0, T ], a.s. ω ∈ Ω (P-a.s.)" will be abbreviated to "for a.a. (t, ω)". We will write L p F := L p,p F and • p := • p,p . The spaces L β,p F endowed with the norms • β,p are Banach spaces and for the specific case p = 2 the space L 2 F is a Hilbert space. We will denote by •, • 2 the obvious scalar product. Evidently,

for β ∈ [1, ∞] and 1 ≤ p 1 ≤ p ≤ p 2 ≤ ∞, there exist positive constants c β,p1 , c β,p2 , c p1,β , c p2,β such that c β,p1 v β,p1 ≤ v β,p ≤ c β,p2 v β,p2 , c p1,β v p1,β ≤ v p,β ≤ c p2,β v p2,β . For a function [0, T ] × R n × R m × Ω (t, y, u, ω) → ψ(t, y, u, ω) ∈ R n such that for a.a. (t, ω) the function (y, u) → ψ(t, y, u, ω) is C 2 , set ψ y (t, y, u, ω) := D y ψ(t,
y, u, ω) and ψ u (t, y, u, ω) := D u ψ(t, y, u, ω). As usual, when the context is clear, we will systematically omit the ω argument in the defined functions. Now, let z ∈ R n and v ∈ R m be variations associated with y and u respectively. The second derivatives of ψ are written in the following form d). These maps will define the dynamics for our problem. Let us assume that: (H1) [Assumptions for the dynamics] The maps ψ = f, σ i satisfy:

ψ yy (t, y, u)z 2 := D 2 yy ψ(t, y, u)(z, z), ψ uu (t, y, u)v 2 := D 2 uu ψ(t, y, u)(v, v), ψ yu (t, y, u)zv := D 2 yu ψ(t, y, u)(z, v), ψ (y,u) 2 (t, y, u)(z, v) 2 := ψ yy (t, y, u)z 2 + 2ψ yu (t, y, u)zv + ψ uu (t, y, u)v 2 . Consider the maps f, σ i : [0, T ] × R n × R m × Ω → R n (i = 1, ...,
(i) The maps are B([0, T ] × R n × R m ) ⊗ F T -measurable. (ii) For all (y, u) ∈ R n × R m the process [0, T ] t → ψ(t, y, u) ∈ R n is F- adapted.
(iii) For almost all (t, ω) ∈ [0, T ] × Ω the mapping (y, u) → ψ(t, y, u, ω) is C 3 . Moreover, we assume that there exists a constant L 1 > 0 such that for almost all (t, ω)

       |ψ(t, y, u, ω)| ≤ L 1 (1 + |y| + |u|) , |ψ y (t, y, u, ω)| + |ψ u (t, y, u, ω)| ≤ L 1 , |ψ yy (t, y, u, ω)| + |ψ yu (t, y, u, ω)| + |ψ uu (t, y, u, ω)| ≤ L 1 |ψ (y,u) 2 (t, y, u, ω) -ψ (y,u) 2 (t, y , u , ω)| ≤ L 1 (|y -y | + |u -u |) . (4)
Let us define σ(t, y, u) := (σ 1 (t, y, u), ..., σ d (t, y, u)) ∈ R n×d . For variations z ∈ R n and v ∈ R m , associated with y and u, set

σ y (t, y, u)z := (σ 1 y (t, y, u)z, ..., σ d y (t, y, u)z), σ yy (t, y, u)z 2 := (σ 1 yy (t, y, u)z 2 , ..., σ d yy (t, y, u)z 2 ), (5)
and σ u (t, y, u)v, σ yu (t, y, u)zv, σ uu (t, y, u)v 2 , σ (y,u) 2 (t, y, u)(z, v) 2 are analogously defined.

For every β ∈ [1, ∞), let us define the space Y β as

Y β := y ∈ L β (Ω; C([0, T ]; R n )) ; (t, ω) → y(t, ω) := y(ω)(t) is F-adapted , endowed with the norm • β,∞ . Let y 0 ∈ R n , under (H1) we have that for every u ∈ L β,2 F the SDE dy(t) = f (t, y(t), u(t))dt + σ(t, y(t), u(t))dW (t), y(0) = y 0 , (6)
is well posed. In fact (see [24, Proposition 2.1]):

Proposition 1. Suppose that (H1) holds. Then, there exists C > 0 such that for every u ∈ L β,2

F (β ∈ [1, ∞)) equation (6) admits a unique solution y ∈ Y β and y β β,∞ ≤ C |y 0 | β + f (•, 0, u(•)) β β,1 + σ(•, 0, u(•)) β β,2 . (7)
Remark 1. Note that by the first condition in (4), the right hand side of (7) is finite.

Now, let us consider maps : [0, T ] × R n × R m × Ω → R and φ : R n × Ω → R.
These maps will define the cost function of our problem. We assume: (H2) [Assumptions for cost] (i) The maps and φ are respectively

B([0, T ]× R n × R m ) ⊗ F T and B(R n ) ⊗ F T measurable. (ii) For all (y, u) ∈ R n × R m the process [0, T ] t → (t, y, u) ∈ R is F-adapted. (iii)
For almost all (t, ω) the maps (y, u) → (t, y, u, ω) and y → φ(y, ω) are C 2 . In addition, there exists L 2 > 0 such that:

               | (t, y, u, ω)| ≤ L 2 (1 + |y| + |u|) 2 , |φ(y, ω)| ≤ L 2 (1 + |y|) 2 , | y (t, y, u, ω)| + | u (t, y, u, ω)| ≤ L 2 (1 + |y| + |u|) , | yy (t, y, u, ω)| + | yu (t, y, u, ω)| + | uu (t, y, u, ω)| ≤ L 2 , | (y,u) 2 (t, y, u, ω) -(y,u) 2 (t, y , u , ω)| ≤ L 2 (|y -y | + |u -u |) , |φ y (y, ω)| ≤ L 2 (1 + |y|) |φ yy (y, ω)| ≤ L 2 , |φ yy (y, ω) -φ yy (y , ω)| ≤ L 2 (|y -y |) . (8)
Remark 2. The above assumptions include the important case when the cost function is quadratic in (y, u).

In order to provide second order conditions, which will be natural extensions of well known deterministic results, it will be useful to strengthen (H1) and (H2).

[Lipschitz cost] There exists C , C φ > 0 such that for almost all (t, ω) ∈ [0, T ] × Ω and for all (y, u), (y , u ) ∈ R n × R m we have

| (t, y, u, ω) -(t, y , u , ω)| ≤ C (|u -u | + |y -y |) , |φ(y, ω) -φ(y , ω)| ≤ C φ |y -y |. (9)
[Affine dynamics] For ψ = f, σ i and for almost all (t, ω) ∈ [0, T ] × Ω, we have

(y, u) ∈ R n × R m → ψ(t, y, u, ω) is affine. ( 10
)
In our main results we will assume:

(H3) At least one of the following assumptions hold: (H3.i) Condition (9) holds and σ uu ≡ 0. (H3.ii) Condition (10) holds.

For every u ∈ L 2 F denote by y u ∈ Y 2 the solution of [START_REF] Bonnans | On the choice of the function spaces for some state-constrained control problems[END_REF]. Let us define the cost function J : L 2 F → R by

J(u) = E T 0 (t, y u (t), u(t))dt + φ(y u (T )) . (11)
Note that, in view of the first condition in [START_REF] Cadenillas | The stochastic maximum principle for linear convex optimal control with random coefficients[END_REF] and estimate (7) the function J is well defined.

3 Expansions for the state and cost function

From now on we fix ū ∈ L 2 F and set ȳ := y ū. We also suppose that assumptions (H1) and (H2) hold. Our aim in this section is to obtain first and second order

expansions for v ∈ L ∞ F → y ū+v ∈ Y 2 and v ∈ L ∞ F → J(ū + v) ∈ R around v = 0.
The main tool for obtaining the expansion for the state is the following corollary of proposition 1, whose proof is straightforward.

Corollary 2. Let A 1 , A 2 ∈ L ∞ F ([0, T ]; R n×n ), B 1 i ∈ L β,2 F ([0, T ]; R n ) and B 2 i ∈ L ∞ F ([0, T ]; R n×d ) for i = 1, 2. Assume that there exists a constant K > 0 such that B 1 1 β,1 ≤ K B 1 2 β,2 . (12)
Then, omitting time from function arguments, for every w ∈ L β,2 , the SDE

dz = A 1 z + B 1 1 + B 2 1 w dt + A 2 z + B 1 2 + B 2 2 w dW (t), z(0) = 0, ( 13
)
has a unique solution in Y β and

z β β,∞ =    O max B 1 2 β β,2 , w β β,1 if B 2 2 ≡ 0, O max B 1 2 β β,2 , w β β,2
otherwise.

Remark 3. Note that the estimates given in corollary 2 are sharp. In fact, suppose that d = 1 and let w ∈ L 2 ([0, T ]; R) (deterministic). Consider the process z(t) defined by z(t) := t 0 w(s)dW (s) for all t ∈ [0, T ].

By definition z β β,∞ ≥ E(|z(T )| β ) = w β 2 E(|Z| β )
, where Z is an standard normal random variable. Since, in this specific case, w β β,2 = w β 2 , the conclusion follows.

Regarding the expansion for the cost J, the following lemma, which is a consequence of It's lemma for multidimensional It process (see e.g. [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF][START_REF] Yong | Stochastic controls, Hamiltonian systems and HJB equations[END_REF]), will be useful. For the reader convenience we provide the short proof. Lemma 3. Let Z 1 and Z 2 be R n -valued continuous process satisfying

dZ 1 (t) = b 1 (t)dt + σ 1 (t)dW (t) for all t ∈ [0, T ], dZ 2 (t) = b 2 (t)dt + σ 2 (t)dW (t) for all t ∈ [0, T ], ( 14
)
where b 1 , b 2 ∈ L 2 (Ω, L 2 ([0, T ], R n )) and σ 1 , σ 2 ∈ L 2 (Ω, L 2 ([0, T ], R n×d ))
are F-adapted process. Also, let us suppose that P-a.s. we have that Z 1 (0) = 0. Then

E (Z 1 (T ) • Z 2 (T )) = E T 0 Z 1 (t) • b 2 (t) + Z 2 (t) • b 1 (t) + d i=1 σ i 1 (t) • σ i 2 (t) dt .
Proof. It's lemma implies that

Z 1 (T ) • Z 2 (T ) = T 0 Z 1 (t) • b 2 (t) + Z 2 (t) • b 1 (t) + d i=1 σ i 1 (t) • σ i 2 (t) dt + M (T ), (15)
where M (t) is a continuous local martingale given by

M (t) := d i=1 t 0 Z 1 (s) • σ i 2 (s) + Z 2 (s) • σ i 1 (s) dW i (s).
By the Burkholder-Davis-Gundy inequality (see e.g [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]) we have the existence of a constant K > 0 such that for all i = 1, ..., d

E sup t∈[0,T ] t 0 Z 1 (s) • σ i 2 (s) + Z 2 (s) • σ i 1 (s) dW i (s) ≤ K Z 1 • σ i 2 + Z 2 • σ i 1 1,2 .
The Cauchy Schwarz inequality yields that (assuming that n = 1 for notational convenience)

Z 1 • σ i 2 1,2 = E   T 0 |Z 1 (t)| 2 σ i 2 (t) 2 dt 1/2   ≤ ||Z 1 || 2,∞ σ i 2 2 < +∞,
with an analogous estimate for Z 2 • σ i 1 1,2 . Therefore, by [27, Theorem 51], we have that M (t) is a martingale with null expectation. The result follows.

For ψ = , f, σ and t ∈ [0, T ], let us define ψ y (t) = ψ y (t, ȳ(t), ū(t)); ψ u (t) = ψ u (t, ȳ(t), ū(t)), ψ yu (t) = ψ yu (t, ȳ(t), ū(t)); ψ yy (t) = ψ yy (t, ȳ(t), ū(t)); ψ uu (t) = ψ uu (t, ȳ(t), ū(t)),

ψ (y,u) 2 (t) = ψ (y,u) 2 (t, ȳ(t), ū(t)).
As usual in optimal control theory, the expansions for J, with respect to variations of the control variable in the uniform norm • ∞ , will be written in terms of an adjoint state and the derivatives of an associated Hamiltonian. Let us define the adjoint state

(p, q) ∈ L 2 F ([0, T ]; R n ) × (L 2 F ([0, T ]; R n )) d
as the unique solution of the following backward stochastic differential equation (BSDE) (see e.g. [START_REF] Bensoussan | Lectures on stochastic control[END_REF][START_REF] Bismut | An introductory approach to duality in optimal stochastic control[END_REF])

dp(t) = -y (t) + f y (t) p(t) + m i=1 σ i y (t) q i (t) dt + q(t)dW (t), p(T ) = φ y (ȳ(T )) . (16)
In the notation above σ i and q i denote respectively the ith column of σ and q.

The following estimates hold (see [24, Proposition 3.1]):

Proposition 4. Assume that (H1), (H2) hold and that ū ∈ L β,2

F (β ∈ [1, ∞)). Then there exists C > 0 such that p β β,∞ + d i=1 qi β β,2 ≤ C 1 + ū β β,2 .
The

Hamiltonian H : [0, T ] × R n × R m × R n × R n×d × Ω → R is defined as H(t, y, u, p, q, ω) := (t, y, u, ω) + p • f (t, y, u, ω) + d i=1 q i • σ i (t, y, u, ω). ( 17
)
For notational convenience, omitting the dependence on ω, we set

H u (t) := H u (t, ȳ(t), ū(t), p(t), q(t)), H (y,u) 2 (t) := H (y,u) 2 (t, ȳ(t), ū(t), p(t), q(t)). ( 18
)
3.1 First order expansions

Let β ∈ [1, ∞] and v ∈ L β,2 F . We consider the linearized mapping v ∈ L β,2 F → y 1 [ū, v] ∈ Y β , where y 1 [ū, v] is the unique solution of dy 1 (t) = [f y (t)y 1 (t) + f u (t)v(t)]dt + [σ y (t)y 1 (t) + σ u (t)v(t)]dW (t), y 1 (0) = 0. ( 19
)
The second assumption in (4) and proposition 1 yields that y 1 [ū, v] is well defined. If the context is clear, for notational convenience we will write

y 1 = y 1 [ū, v]
. Corollary 2 will be the main tool for establishing the following useful estimates:

Lemma 5. Let v ∈ L 2β,4 F with β ∈ [1, ∞) and set δy = δy[ū, v] := y ū+v -ȳ, d 1 = d 1 [ū, v] := δy -y 1 .
Then, the following estimates hold:

δy β β,∞ + y 1 β β,∞ = O( v β β,1 ) if σ u ≡ 0, O( v β β,2 ) otherwise. ( 20
)
d 1 β β,∞ = O( v 2β 2β,2 ) if σ uu ≡ 0, O( v 2β 2β,4 ) otherwise. (21)
Proof. See the appendix.

Now we can prove the following proposition.

Proposition 6. The map v ∈ L ∞ F → ŷ(v) := y ū+v ∈ Y 2 is differentiable with a Lipschitz derivative given by Dŷ(v)v = y 1 [ū + v, v] for all v, v ∈ L ∞ F . (22)
Proof. By estimate [START_REF] Lions | Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications[END_REF] in lemma 5 we have that δy -

y 1 2,∞ = O( v 2 ∞ ), implying that (22) holds. Let us prove that Dŷ is Lipschitz. For notational convenience, we assume that n = m = d = 1. Let v 1 , v 2 ∈ L ∞ F and write δv := v 1 -v 2 . For ψ = f, σ, i = 1, 2 and v ∈ L ∞ F with v ∞ = 1
, by an abuse of notation we set

y (i) := y ū+vi , y (i) 1 := y 1 [ū + v i , v ], δy := y (1) -y (2) , δy 1 := y (1) 1 -y (2) 1 ψ (i) y (t) := ψ y (t, y (i) (t), ū(t) + v i (t)), ψ (i) u (t) := ψ u (t, y (i) (t), ū(t) + v i (t)), δψ y (t) := ψ (1) y (t) -ψ (2) y (t), δψ u (t) := ψ (1) u (t) -ψ (2) u (t).
A straightforward calculation shows that

dδy 1 (t) = [f (2) y (t)δy 1 (t) + δf y (t)y (1) 1 (t) + δf u (t)v (t)]dt +[σ (2) y (t)δy 1 (t) + δσ y (t)y (1) 1 (t) + δσ u (t)v (t)]dW (t), δy 1 (0) = 0. ( 23
)
Using (H1) and corollary 2 we obtain that

δy 1 2 2,∞ = O δy 2 2,∞ + δv 2 2 , uniformly in v ∈ L ∞ F with v ∞ = 1.
The result follows from [START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF].

Now we focus our attention on the cost function J. Let us define Υ 1 :

L 2 F → R by Υ 1 (v) := E T 0 H u (t) v(t)dt . (24)
In view of proposition 4, with β = 2, Υ 1 is well defined. Lemma 3 yields the following well known alternative expression for Υ 1 .

Lemma 7. For every v ∈ L 2 F we have that:

Υ 1 (v) = E T 0 [ y (t)y 1 (t) + u (t)v(t)] dt + φ y (ȳ(T ))y 1 (T ) . (25)
Proof. Noting that

φ y (ȳ(T ))y 1 (T ) = p(T ) • y 1 (T ) -p(0) • y 1 (0), lemma 3, applied to Z 1 = y 1 and Z 2 = p, yields E (φ y (ȳ(T ))y 1 (T )) = I 1 +I 2 +I 3 , where
I 1 := -E T 0 y 1 (t) • y (t) + f y (t) p(t) + d i=1 σ i y (t) qi (t) dt , I 2 := E T 0 p(t) • [f y (t)y 1 (t) + f u (t)v(t)] dt , I 3 := d i=1 E T 0 qi (t) • σ i y (t)y 1 (t) + σ i u (t)v(t) dt .
Plugging the expressions of I 1 , I 2 and I 3 introduced above into the right hand side of ( 25) yields the result.

The expression above for Υ 1 allows to obtain a first order expansion of J around ū. Proposition 8. Assume that (H1), (H2) hold and let v ∈ L 4 F . Then,

J(ū + v) = J(ū) + Υ 1 (v) + r 1 (v), with Υ 1 (v) = O( v 2 ); r 1 (v) = O v 2 4,2 if σ uu ≡ 0, O v 2 4 otherwise. ( 26
)
If in addition (9) holds, then

Υ 1 (v) = O( v 1 ) if σ u ≡ 0, O( v 1,2 ) otherwise, ; r 1 (v) = O v 2 2 if σ uu ≡ 0, O v 2 2,4 otherwise. ( 27
)
Proof. Let us denote δJ := J(ū + v) -J(ū). By definition

δJ = E T 0 [ (t, y ū+v (t), ū(t) + v(t)) -(t, ȳ(t), ū(t))] dt + φ(y ū+v (T )) -φ(ȳ(T )) .
Using (H2), a Taylor expansion for and φ implies that

δJ = E T 0 y (t)δy(t) + u (t)v(t)dt + φ y (ȳ(T ))δy(T ) + O( δy 2 2,∞ + v 2 2 ). Since δy = y 1 + d 1 , lemma 7 implies that δJ = Υ 1 (v) + r 1 (v) where r 1 (v) = z 1 (v) + z 2 (v) with z 1 (v) := E T 0 y (t)d 1 (t)dt + φ y (ȳ(T ))d 1 (T ) , z 2 (v) := O δy 2 2,∞ + v 2 2 .
Now, we estimate Υ 1 (v) using [START_REF] Peng | A general stochastic maximum principle for optimal control problems[END_REF]. By assumption (H2) and the Cauchy

Schwarz inequality

E T 0 u (t)v(t)dt = O ( v 2 )
. On the other hand, by ( 20)

E T 0 y (t)y 1 (t)dt + φ y (ȳ(T ))y 1 (T ) = O    E sup t∈[0,T ] |y 1 (t)| 2 1 2    = O( v 2 ). Thus Υ 1 (v) = O( v 2 ). If (9) holds, then E T 0 u (t)v(t)dt = O ( v 1 )
, and

E T 0 y (t)y 1 (t)dt + φ y (ȳ(T ))y 1 (T ) = O E sup t∈[0,T ] |y 1 (t)| .
Thus, estimates for Υ 1 (v) in [START_REF] Protter | Stochastic Integration and Differential Equations , Second Edition[END_REF] follow from [START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF] with β = 1. Let us estimate r 1 (v). Assumption (H2) and [START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF]

imply that z 2 (v) = O( v 2
2 ). On the other hand, by (H2) and the Cauchy Schwarz inequality

z 1 (v) = O    E sup t∈[0,T ] |d 1 (t)| 2 1 2    .
Thus [START_REF] Pham | Optimisation et contrôle stochastique appliqués à la finance[END_REF] follows from estimates [START_REF] Lions | Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications[END_REF] with β = 2. If in addition (9) holds, then

z 1 (v) = O E sup t∈[0,T ] |d 1 (t)|
and the estimates for r 1 (v) in [START_REF] Protter | Stochastic Integration and Differential Equations , Second Edition[END_REF] follows from ( 21) with β = 1.

Remark 4. The above proof shows that the hypotheses for the perturbation v can be weakened. For example, if (9) holds and

σ uu = 0, for all v ∈ L 2 F we have that J(ū + v) = J(ū) + Υ 1 (v) + r 1 (v) with Υ 1 (v) = O( v 1 ) and r 1 (v) = O( v 2
2 ). Analogously, if (10) holds, then d 1 ≡ 0 and we have that

J(ū + v) = J(ū) + Υ 1 (v) + r 1 (v) with Υ 1 (v) = O( v 2 ) and r 1 (v) = O( v 2
2 ). Therefore, if (H3) holds, the function J is differentiable at ū.

The following corollary is an inmediate consequence of the proposition above.

Corollary 9. Assume that (H1), (H2) hold and let

v ∈ L ∞ F . Then, Υ 1 (v) = O( v 2 ) and J(ū + v) = J(ū) + Υ 1 (v) + r 1 (v) with r 1 (v) = O( v 2 ∞ ).

Second order expansions

In this subsection we obtain second order expansions for the state and the cost function. As in the precedent subsection we begin with the analysis for the state mapping. The second order

linearization of v ∈ L ∞ F → ŷ(v) = y ū+v ∈ Y 2 around v ≡ 0 in the direction v ∈ L ∞ F is defined as the unique solution y 2 = y 2 [ū, v] of dy 2 (t) = f y (t)y 2 (t) + f yy (t)y 1 (t) 2 + 2f yu (t)y 1 (t)v(t) + f uu (t)v(t) 2 dt + σ y (t)y 2 (t) + σ yy (t)y 1 (t) 2 + 2σ yu (t)y 1 (t)v(t) + σ uu (t)v(t) 2 dW (t); y 2 (0) = 0.
(28) Note that by the third assumption in (4) and proposition 1, we have that y 2 is well defined. We give now some useful bounds over y 2 and the rest

d 2 = d 2 [ū, v] := δy -y 1 -1 2 y 2 . ( 29
)
As for y 1 and d 1 , when the context is clear, we omit the arguments of y 2 and d 2 .

Lemma 10. For v ∈ L ∞ F and β ∈ [1, ∞) the following estimates hold:

y 2 β β,∞ = O( v 2β 2β,2 ) if σ uu ≡ 0, O( v 2β 2β,4 ) otherwise. (30)
d 2 β β,∞ = O( v β 2β,2 v 2β 4β,4 ) if σ uuu ≡ 0, O( v β 2β,2 v 2β 4β,4 + v 3β 3β,6 ) otherwise. (31)
Proof. See the appendix. Now, we study J. Let us define Υ 2 :

L ∞ F → R by Υ 2 (v) := E T 0 H (y,u) 2 (t)(y 1 (t), v(t)) 2 dt + φ yy (ȳ(T ))(y 1 (T )) 2 . ( 32
)
As for Υ 1 , a useful alternative expression for Υ 2 holds.

Lemma 11. For every v ∈ L ∞ F we have that:

Υ 2 (v) = E T 0 y (t)y 2 (t) + (y,u) 2 (t)(y 1 (t), v(t)) 2 dt + E φ y (ȳ(T ))y 2 (T ) + φ yy (ȳ(T ))(y 1 (T )) 2 . (33)
Proof. By definition of y 2 and p, we have that

φ y (ȳ(T ))y 2 (T ) = p(T ) • y 2 (T ) -p(0) • y 2 (0). Lemma 3 yields E (φ y (ȳ(T ))y 2 (T ))) = I 1 + I 2 + I 3 ,
where

I 1 := -E T 0 y 2 (t) • y (t) + f y (t) p(t) + d i=1 σ i y (t) qi (t) dt , I 2 := E T 0 p(t) • f y (t)y 2 (t) + f (y,u) 2 (t)(y 1 (t), v(t)) 2 dt , I 3 := d i=1 E T 0 qi (t) • σ i y (t)y 2 (t) + σ i (y,u) 2 (t)(y 1 (t), v(t)) 2 dt .
Plugging the expressions of I 1 , I 2 and I 3 introduced above into the right hand side of (33) yields the result.

In order to obtain the main result of this section, the following lemma is useful.

Lemma 12. Assume that (H1), (H2) hold and let v ∈ L ∞ F . Then,

J(ū + v) = J(ū) + Υ 1 (v) + 1 2 Υ 2 (v) + ζ(v) + z(v), (34)
where

z(v) = O v ∞ v 2 2 and, recalling (29)
,

ζ(v) := E T 0 y (t)d 2 (t)dt + φ y (ȳ(T ))d 2 (T ) . ( 35
)
Proof. See the appendix.

Now we are able to obtain a second order expansion of J around ū.

Proposition 13. Assume that (H1), (H2) hold and let v ∈ L ∞ F . Then,

J(ū + v) = J(ū) + Υ 1 (v) + 1 2 Υ 2 (v) + r 2 (v), ( 36
)
and the following estimates hold:

Υ 2 (v) = O( v 2 4,2 ) if σ uu ≡ 0, O( v 2 4 ) otherwise, r 2 (v) = O( v ∞ v 2 4,2 ) if σ uuu ≡ 0, O( v ∞ v 2 4 )
otherwise.

(37) If in addition (9) holds then

Υ 2 (v) = O( v 2 2 ) if σ uu ≡ 0, O( v 2 2,4 ) otherwise, r 2 (v) = O( v ∞ v 2 2 ) if σ uuu ≡ 0, O( v ∞ v 2 2,4 ) otherwise. ( 38
)
Proof. Let us first estimate Υ 2 (v) by using the expression obtained in lemma 11 and the bounds obtained in lemmas 5 and 10. By [START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF] with β = 2,

E sup t∈[0,T ] |y 1 (t)| 2 + T 0 |v(t)| 2 dt = O( v 2 2 ). ( 39
)
In view of assumption (H2) and (39) we obtain that

E T 0 (y,u) 2 (t)(y 1 (t), v(t)) 2 dt + φ yy (ȳ(T ))(y 1 (T )) 2 = O( v 2 2 ). ( 40
)
On the other hand, assumption (H2) and the Cauchy Schwarz inequality yield

E T 0 y (t)y 2 (t)dt + φ y (ȳ(T ))y 2 (T ) = O    E( sup t∈[0,T ] |y 2 | 2 ) 1 2    , (41)
and the estimate for Υ 2 (v) in (37) follows from [START_REF] Zhou | The connection between the maximum principle and dynamic programming in stochastic control[END_REF]. If (9) holds, then

E T 0 y (t)y 2 (t)dt + φ yy (ȳ(T ))y 2 (T ) = O E sup t∈[0,T ] |y 2 | , (42)
and the estimate for Υ 2 (v) in (38) follows from [START_REF] Zhou | The connection between the maximum principle and dynamic programming in stochastic control[END_REF].

In order to conclude the proof, letting r 2 (v) := ζ(v)+z(v), lemma 12 implies that it suffices to estimate ζ(v). By (H2) and the Cauchy Schwarz inequality

ζ(v) = O    E sup t∈[0,T ] |d 2 (t)| 2 1 2    .
Hence, using [START_REF] Zhou | A unified treatment of maximum principle and dynamic programming in stochastic controls[END_REF]

with β = 2, ζ(v) = O( v 4,2 v 2 8,4 ) if σ uuu ≡ 0, O( v 4,2 v 2 8,4 + v 3 6,6 ) otherwise. Since O( v 4,2 v 2 8,4 ) = O( v ∞ v 2 4,2 ) and O( v 3 6,6 ) = O( v ∞ v 2 4
), the estimate for r 2 (v) in (37) follows. If in addition assumption (9) holds, then by [START_REF] Zhou | A unified treatment of maximum principle and dynamic programming in stochastic controls[END_REF]

with β = 1, ζ(v) = O E sup t∈[0,T ] |d 2 (t)| = O( v 2 v 2 4 ) if σ uuu ≡ 0, O( v 2 v 2 4 + v 3 3,6 ) otherwise. Since O( v 2 v 2 4 ) = O( v ∞ v 2 2 ) and O( v 3 3,6 ) = O( v 2 2,4
), the estimate for r 2 (v) in (38) follows.

Remark 5. The proof of proposition 13 shows that the estimates

Υ 2 (v) = O( v 2 2 ) and r 2 (v) = O( v ∞ v 2
2 ) also hold in the case when f and σ are affine mappings, since in this case y 2 = d 2 = 0. Therefore if (H3) holds, then we find the natural extension of the well known expansion for the deterministic case, i.e.

J(ū

+ v) = J(ū) + Υ 1 (v) + 1 2 Υ 2 (v) + O( v ∞ v 2 2 ) with Υ 1 (v) and Υ 2 (v) = O( v 2 2 ).
Since Υ 2 is a quadratic form and, for every

β, p ∈ [1, ∞], the space L ∞ F is dense in L β,p F , we have that: If Υ 2 (v) = O( v β,p
) then Υ 2 admits a unique continuous extension in L β,p . This fact, together with the following corollary, will allow us to prove, in the next sections, second order necessary condition for the problems explained in the introduction.

Corollary 14. Assume that (H1)-(H3) hold. Then,

J(ū + v) = J(ū) + Υ 1 (v) + 1 2 Υ 2 (v) + r(v) for all v ∈ L ∞ F , (43)
where

Υ 1 (v) = O( v 2 ), Υ 2 (v) = O( v 2 2 ) and r(v) = O( v ∞ v 2 2 ).

Optimality conditions: The case of control constraints

Let U be a nonempty closed and convex subset of L 2 F and consider the problem Min J(u) subject to u ∈ U.

(SP)

The asymptotic expansions obtained for J in section 3 allow us to obtain first and second order necessary conditions at a local optimum ū ∈ L 2 F for the control constrained problem (SP). We first obtain first order optimality conditions using the procedure explained in the introduction: According to the regularity of the data of (SP) and the dependence on u of the σ-term, a perturbation in an appropriate space is taken. Then, the results of the previous section yield a positivity condition of Υ 1 over a certain cone which is extended, by a density argument, to a larger one. Similar considerations apply in order to establish second order necessary conditions.

Let us first fix some notations which are standard in optimization theory. Consider a Banach space (X, • X ) and a nonempty closed convex set C ⊆ X. For x, x ∈ X define the segment [x, x ] := {x + λ(x -x) ; λ ∈ [0, 1]}. The radial, the tangent and the normal cone to C at x are defined respectively by

R C (x) := {h ∈ X ; ∃ θ > 0 such that [x, x + θh] ⊆ C}, T C (x) := {h ∈ X ; ∃ x(θ) = x + θh + o(θ) ∈ C, θ > 0, ||o(θ)/θ|| X → 0}, N C (x) := {x * ∈ X * ; x * , x -x X * ,X ≤ 0, for all x ∈ C} , (44)
where X * denotes the dual topological space of X and •, • X * ,X is the duality product. Recall that, since C is a closed convex set, the tangent cone T C (x) is the closure of the radial cone R C (x) in X, and that N C (x) is the polar cone of T C (x), i.e., We have the following first order conditions for (SP).

N C (x) = {x * ∈ X * ; x * , h X * ,X ≤ 0, for all h ∈ T C (x)} . (45
Lemma 15. Assume that (H1), (H2) hold and let ū ∈ U be a local solution of (SP). Then:

Υ 1 (v) ≥ 0 for all v ∈ clo 2 R U (ū) ∩ L 4,2 F if σ uu ≡ 0, clo 2 R U (ū) ∩ L 4 F otherwise. ( 46
)
If in addition (9) holds then

Υ 1 (v) ≥ 0 for all v ∈ clo 1 (R U (ū)) if σ u ≡ 0, clo 1,2 R U (ū) ∩ L 2,4 F otherwise. ( 47
) Proof. Let v ∈ R U (ū) ∩ L 4 F .
Since ū is a local solution of (SP), proposition 8 implies that, for θ > 0 small enough, we have

0 ≤ J(ū + θv) -J(ū) = θΥ 1 (v) + v 2 4 O(θ 2 ). ( 48
)
Thus, dividing by θ in (48) and letting θ ↓ 0, we have that Υ 1 (v) ≥ 0. Analogously, if σ uu = 0 we have that Υ 1 (v) ≥ 0 for all v ∈ R U (ū) ∩ L 4,2 F . Condition (46) follows from the continuity of Υ 1 . The proof of (47) follows in the same manner, with the obvious modifications.

Note that the results obtained in lemma 15 are rather general, since they include the case of non local constraints. Of course they will give no information if the cones in (46)-(47) reduce to {0}. This may happen for somewhat pathological examples, as we next show.

Example 1. Let u 0 ∈ L 2
F and suppose that u 0 / ∈ L β,p F for any β, p ∈ (2, ∞]. The constraint U := {u = αu 0 / for some α ∈ [0, 1]} is such that, at ū = 0, the radial cone is given by R

U (ū) = {λu 0 / for λ ≥ 0}, but R U (ū) ∩ L β,p F = {0}.
Thus, we will assume the following assumption over the constraint set U:

(H4) For every ū ∈ U we have that

T U (ū) = clo 2 (R U (ū) ∩ L ∞ F ) . (49)
We have the following proposition whose proof is straightforward.

Proposition 16. Assume that (H1), (H2), (H4) hold and let ū be a local solution of (SP). Then

Υ 1 (v) ≥ 0 for all v ∈ T U (ū). (50)
Remark 6. Note that if J(•) is convex, then (50) is a sufficient condition for the (global) optimality of ū.

Clearly, we have that (H4) can hold for non local constraints. As an example, it can be checked that (49) holds for U = u ∈ L 2 F / u 2 ≤ 1 and ū ∈ U. Now we consider the case when U is defined by local constraints. The next lemma extends a well-known result on deterministic local constraints to the case of adapted local constraints. We first need to introduce some basic notions of multifunction theory. We say that the application (t, ω)

∈ [0, T ] × Ω → U (t, ω) ∈ P(R m ) is a B([0, T ]) × F T -measurable multifunction if, for any closed set C ⊂ R m , we have that U -1 (C) ∈ B([0, T ]) × F T .
In addition, we say that it is F -adapted if for all t ∈ [0, T ] we have that U (t, •) -1 (C) ∈ F t . Finally, U is said to be closed-convex-valued if U (t, ω) is a.a. closed and convex.

Given a B([0, T ]) × F T -measurable multifunction U , consider the set

U := u ∈ L 2 F ; u(t, ω) ∈ U (t, ω), a.a. (t, ω) ∈ [0, T ] × Ω . ( 51
)
Lemma 17. Let ū ∈ U, with U closed-convex-valued and F -adapted. Then: (i) Assumption (49) holds at ū. (ii) The tangent cone is given by

T U (ū) = v ∈ L 2 F ; v(t, ω) ∈ T U (t,ω) (ū(t, ω)) a.a. (t, ω) ∈ [0, T ] × Ω . ( 52
)
Proof. (i) By a diagonal argument, it suffices to prove that, for every v ∈ R U (ū), there exists a sequence

v k ∈ R U (ū) ∩ L ∞ F such that v k -v 2 → 0. Indeed, set v k (t, ω) := 1 {|v(t,ω)|≤k} v(t, ω). ( 53
)
Being a product of adapted functions, v k is adapted and, since U (t, ω) is convex, it belongs to R U (ū). Finally, since v k (t, ω) → v(t, ω) for a.a. (t, ω) and

|v k (t, ω)| ≤ |v(t, ω)|, the dominated convergence theorem implies that v k → v in L 2 F . (ii) Let v ∈ T U (ū)
. By definition, for θ > 0 small enough and a.a. (t, ω):

ū(t, ω) + θv(t, ω) + r θ (t, ω) ∈ U (t, ω), (54)
where r θ (•, •)/θ → 0 in L 2 F as θ ↓ 0, and therefore also in L 2 F T . Thus, extracting a subsequence if necessary, we have that r θ (t, ω)/θ → 0 for a.a. (t, ω) from which we deduce that v(t, ω) ∈ T U (t,ω) (ū(t, ω)). Conversely, let v belongs to the r.h.s. of (52) and for ε > 0 set

v ε := ε -1 (P U (ū + εv) -ū) , (55)
where P U (•) denotes the orthogonal projection in L 2 F onto U . By definition of P U (•) we have that v ε ∈ R U (ū). For (t, ω) in [0, T ] × Ω set P U (t,ω) (•) for the orthogonal projection in R m onto U (t, ω). In view of lemma 34 in the appendix, we have that for a.a. (t, ω)

v ε (t, ω) := ε -1 P U (t,ω) (ū(t, ω) + εv(t, ω)) -ū(t, ω) . (56) Clearly, v ε (t, ω) ∈ R U (t,ω) (ū(t, ω)) and for a.a. (t, ω) we have v ε (t, ω) → v(t, ω). Since |v ε (t, ω)| ≤ |v(t, ω)|, the dominated convergence theorem implies that v ε → v in L 2 F T and therefore also in the closed subspace L 2 F . Using that v ε ∈ R U (ū) we obtain that v ∈ T U (ū). Let a, b ∈ R
m with -∞ ≤ a i < b i ≤ +∞ for all i ∈ {1, ..., m} and define

U a,b := x ∈ R m ; a i ≤ x i ≤ b i . ( 57
)
For u ∈ L 2 F and every index i ∈ {1, ..., m}, set

I a i (u) := (t, ω) ∈ [0, T ] × Ω ; u i (t, ω) = a i , I b i (u) := (t, ω) ∈ [0, T ] × Ω ; u i (t, ω) = b i .
The following corollary is a direct consequence of proposition 16 and lemma 17.

Corollary 18. Assume that (H1), (H2) hold suppose that U is in the form (51) with U closed-convex-valued and F -adapted. Let ū ∈ U be a local solution of (SP), then

H u (t, ω)v(t, ω) ≥ 0 for all v ∈ T U (t,ω) (ū(t, ω)), a.e.. (58)
In particular, if U (t, ω) ≡ U a,b (defined in (57)), then for every i ∈ {1, ..., m}, a.e.:

H i u (t, ω) =    ≥ 0 if (t, ω) ∈ I a i (ū), ≤ 0 if (t, ω) ∈ I b i (ū), = 0 elsewhere. ( 59
)
Remark 7. Since (58) is equivalent to (50) when U is in the form (51), we have that if J(•) is convex then (58) is a sufficient condition for the (global) optimality of ū.

Second order necessary conditions

In order to obtain second order necessary conditions for (SP) we proceed as in the previous section, i.e. we prove a general result and after, under some standard assumptions, we yield a more precise characterization for the important case of local constraints. Let us define

Υ ⊥ 1 := v ∈ L 2 F ; Υ 1 (v) = 0 . ( 60
)
We have the following general second order necessary conditions.

Proposition 19. Assume that (H1), (H2) hold and suppose that ū ∈ U is a local solution of (SP). Then, the following second order necessary condition holds:

Υ 2 (v) ≥ 0 for all v ∈ clo 4,2 R U (ū) ∩ L ∞ F ∩ Υ ⊥ 1 if σ uu ≡ 0, clo 4 R U (ū) ∩ L ∞ F ∩ Υ ⊥ 1 otherwise. ( 61
)
If in addition (H3) holds, then

Υ 2 (v) ≥ 0 for all v ∈ clo 2 R U (ū) ∩ L ∞ F ∩ Υ ⊥ 1 if σ uu ≡ 0, clo 2,4 R U (ū) ∩ L ∞ F ∩ Υ ⊥ 1 otherwise. ( 62
) Proof. If v ∈ R U (ū) ∩ L ∞ F ∩ Υ ⊥ 1 , proposition 13 implies that for θ small enough 0 ≤ J(ū + θv) -J(ū) = θ 2 2 Υ 2 (v) + θ 3 O( v 3 ∞ ).
Dividing the above equation by θ and letting θ ↓ 0 yields Υ 2 (v) ≥ 0 and the result follows from the bounds in proposition 13 for the quadratic form Υ 2 .

The critical cone to U at ū are defined by

C(ū) := {v ∈ T U (ū) / Υ 1 (v) ≤ 0} . (63)
In order to obtain more precise second order necessary conditions, we suppose standard assumptions in the second order analysis of problems with convex constraints. The first one is a natural extension of (H4) to the second order case.

(H5) For every ū ∈ U and v * ∈ N U (ū) (recall ( 44)), we have that

clo 2 R U (ū) ∩ L ∞ F ∩ (v * ) ⊥ = clo 2 R U (ū) ∩ (v * ) ⊥ . ( 64
)
For our second assumption, we need the following notion of polyhedricity (see [START_REF] Haraux | How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities[END_REF][START_REF] Mignot | Contrôle dans les inéquations variationnelles[END_REF]). The set U is said to be polyhedric at ū

∈ U if for all v * ∈ N U (ū), the set R U (ū) ∩ (v * ) ⊥ is dense in T U (ū) ∩ (v * ) ⊥ with respect to the • 2 norm. If
U is polyhedric at each u ∈ U we say that U is polyhedric.

Remark 8. Note that, if (H1), (H2) and (H4) hold, proposition 16 yields that, at a local minimum, -Υ 1 ∈ N U (ū) and

C(ū) = T U (ū) ∩ Υ ⊥ 1 .
Thus, if U is polyhedric and in addition (H5) holds,

clo 2 R U (ū) ∩ L ∞ F ∩ Υ ⊥ 1 = C(ū). ( 65
)
We state a second order necessary condition which is a natural extension of the deterministic counterpart.

Theorem 20. Let ū be a local solution of (SP) and assume that (i) Assumptions (H1)-(H5) hold.

(ii) The constraint set U is polyhedric.

Then, the following second order necessary condition hold at ū:

Υ 2 (v) ≥ 0 for all v ∈ C(ū). ( 66
)
Proof. As in the proof of proposition 19 we have that Υ

2 (v) ≥ 0 for all v ∈ R U (ū) ∩ L ∞ F ∩ Υ ⊥ 1 .
The result follows as in the proof of proposition 19 since under our assumptions Υ

2 (v) = O( v 2
2 ) and (65) holds.

Now, let us focus our attention in local constraints, i.e. when U is defined by (51).

Lemma 21. Let U be defined by (51) with U closed-convex-valued and Fadapted. Let ū ∈ U, it holds that

(i) The normal cone N U (ū) is given by N U (ū) = v * ∈ L 2 F / v * (t, ω) ∈ N U (t,ω) (ū(t, ω)), a.a. (t, ω) ∈ [0, T ] × Ω . ( 67
) (ii) For every v * ∈ N U (ū) we have that T U (ū) ∩ (v * ) ⊥ = {v ∈ T U (ū) / v * (t, ω) • v(t, ω) = 0, a.a. (t, ω) ∈ [0, T ] × Ω} . ( 68
)
Proof. Since (ii) follows directly from (i) and lemma 17 (ii), it is enough to show (i). By lemma 17(ii) and since N U (t,ω) (ū(t, ω)) is the polar cone of T U (t,ω) (ū(t, ω)), the r.h.s. of (67) is included in N U (ū). To prove the other inclusion, let v * ∈ N U (ū) and consider

v (t, ω) ∈ T U (t,ω) (ū(t, ω)), v (t, ω) ∈ N U (t,ω) (ū(t, ω)) such that v * (t, ω) = v (t, ω) + v (t, ω), v (t, ω) • v (t, ω) = 0.
We know that the above decomposition exists and it is unique. By lemma 17(ii)

we have that v ∈ T U , and thus 0

≥ IE T 0 v * (t, ω) • v (t, ω)dt = v 2 2 proving that v = 0. The result follows.
In order to verify the polyhedricity assumption in the case of local constraints, we will need in fact to assume that for a.a. (t, ω) the set U (t, ω) is a polyhedron. More precisely, let q ∈ N and suppose that there exist mappings Σ : [0, T ] × Ω → P ({1, ..., q}), a

i : [0, T ] × Ω → R m , b i : [0, T ] × Ω → R m
, where i ∈ {1, ..., q}, such that Σ, a i and b i are B([0, T ]) × F T measurable and for each t we have that Σ(t, •), a i (t, •) and b i (t, •) are F t measurable. We suppose that

U (t, ω) = {x ∈ R m / a i (t, ω), x ≤ b i (t, ω), for i ∈ Σ(t, ω) } . ( 69
)
We have

In the deterministic case it is well known that the application u ∈ L 2 ([0, T ]; R m ) → y 1 (u)(T ) ∈ R n is weakly continuous. This allows to verify that the associated quadratic form is a Legendre form iff some form of the strong Legendre condition holds. We show with two examples that u ∈ L

2 F → y 1 (u)(T ) ∈ L 2 F T (R n ) is not weakly continuous.
Example 2 (σ dependent on u). Let us take m = n = 1 and let us consider the dynamics

dy 1 (t) = u(t)dW (t) for t ∈ [0, T ]; y 1 (0) = 0.
Let u n be a (deterministic) orthonormal base of L 2 ([0, T ]; R) and denote y n := y 1 [u n ]. By the dominated convergence theorem it is easy to check that u n converges weakly to 0 in L 2 F , but

E y n (T ) 2 = E   T 0 u n (t)dW (t) 2   = T 0 u 2 n (t)dt = 1.
Example 3 (σ independent on u). We take m = n = 1 and T = 2. Let us consider the dynamics

dy 1 (t) = u(t)dt for t ∈ [0, T ]; y 1 (0) = 0.
Let φ n be an orthonormal base of the Hilbert space L 2 (R) endowed with the scalar product

g, h * := +∞ -∞ g(x)h(x)e -x 2
2 dx.

As a classical example (see e.g [START_REF] Hida | Brownian Motion[END_REF]) we can take φ n (x) = h n (x)/ √ 2πn!, where h n is the nth Hermite polynomial. Consider the sequence u n ∈ L 2 F defined by u n (t) := φ n (W (1))I (1,2] (t) and set

y n := y 1 [u n ]. For every f ∈ L 2 F , we have E 2 0 f (t)u n (t) dt = E φ n (W (1)) 2 1 f (t)dt , = E φ n (W (1))E 2 1 f (t)dt|W (1) → 0,
by definition of φ n . Thus, u n converges weakly to 0 in L 2 F . On the other hand,

E y n (T ) 2 = E 2 0 u n dt 2 = E φ n (W (1)) 2 = 1.

Optimality conditions: The case of final state constraints

In this section we suppose that U = L 2 F and we consider the problem

Min u∈U J(u) subject to E g i (y u (T )) = 0, E h j (y u (T )) ≤ 0, (SP )
for all i ∈ {1, ..., n g }, j ∈ {1, ..., n h }. In the notation above, g i : R n × Ω → R and h j : R n × Ω → R. We make the following assumption:

(H6) The maps g i and h j satisfy the same assumptions made for φ in (H2).

In order to provide first and second order necessary conditions for (SP ) we will need the following lemma:

Lemma 26. Let ū ∈ U and consider a function c : R n × Ω → R satisfying the assumptions for φ in (H2). Then, the map

v ∈ L ∞ F → ĉ(v) := E [c(y ū+v (T ))]
∈ R is differentiable with a Lipschitz derivative and admits the following second order expansion at v = 0

ĉ(v) = ĉ(0) + Υ c 1 (v) + 1 2 Υ c 2 (v) + O( v 3 ∞ ), (75)
where

Υ c 1 (v) := E [c y (y ū(T ))y 1 [ū, v](T )] = O( v 2 ), Υ c 2 (v) := E c y (y ū(T ))y 2 [ū, v](T ) + c yy (y ū(T ))(y 1 [ū, v](T )) 2 = O( v 2 ∞ ).
(76) If in addition c satisfies the assumptions for φ in (H3), then

Υ c 2 (v) := O( v 2 2 ). (77)
For any ȳ, y ∈ Y 2 , we have, for a.a. ω:

c(ȳ + y) = c(ȳ) + c y (ȳ)y + 1 0 [c y (ȳ + θy) -c y (ȳ)] ydθ. (79)
Using (H2) and the Cauchy-Schwarz inequality, it follows that

E [c(ȳ(T ) + y(T ))] = E [c(ȳ(T )) + c y (ȳ(T )) (y(T ) -ȳ(T ))] + O ||y -ȳ|| 2 2,∞ , E [c y (ȳ(T )) (y(T ) -ȳ(T ))] = O ||y -ȳ|| 2,∞ .
Thus, expression (78) holds. We next prove that the derivative is Lipschitz. For ȳ and ỹ in Y 2 , define ∆(ȳ, ỹ) := max

y 2,∞≤1 |IE ([c y (ȳ) -c y (ỹ)] y)| (80)
By (H2) Dc(•, ω) is Lipschitz, hence the Cauchy-Schwarz inequality implies that ∆(ȳ, ỹ) ≤ L 2 IE (|ȳ -ỹ| |y|) ≤ L 2 ȳ -ỹ 2,∞ . (81)
The result follows.

Remark 9. If Υ c 1 (•) is surjective then Graves theorem [START_REF] Graves | Some mapping theorems[END_REF] (see also Dontchev [START_REF] Dontchev | The Graves theorem revisited[END_REF]) implies that ĉ enjoys the metric regularity property. In other words, if ĉ(0) = 0 and v ∞ is small enough, then there exists

v ∈ L ∞ such that ĉ(v ) = 0 and v -v ∞ = O(|ĉ(v)|).

First order necessary condition

Let ū ∈ L 2 F be a local solution of (SP ) and denote by ȳ its associated state. Lemma 26 implies that the maps

v ∈ L ∞ F → gi (v) := E g i (y ū+v (T )) ∈ R, v ∈ L ∞ F → hj (v) := E h j (y ū+v (T )) ∈ R, (82)
are C 1 with derivatives at v = 0 given respectively by

Υ g 1i (•) := E g i y (y ū(T ))y 1 [ū, •](T ) , Υ h 1j (•) := E h j y (y ū(T ))y 1 [ū, •](T ) . (83)
For notational convenience we write g := (g 1 , ..., gng ) and Υ g 1 := (Υ g 11 , ..., Υ g 1ng ). The set I 1 (ū) of active constraints is defined as

I 1 (ū) := j ∈ {1, ..., n h } / E h j (ȳ(T )) = 0 . (84)
We assume the following constraint qualification condition:

(H7) The following assertions hold true:

(i) The application Υ g 1 : L ∞ F → R ng is surjective. (ii) There exists v ∈ (Υ g 1 ) ⊥ ∩ L ∞ F such that Υ h 1j (v) < 0 for all j ∈ I 1 (ū).
We will need the following density lemma, proved in [9, Lemma 1].

Lemma 27. Let X be a normed vector space. Given a i , i = 1 to q, in X * , and b ∈ R q , define K

:= {x ∈ X / a i , x ≤ b i , i = 1, ..., q} .
If Y is a dense subspace of X, then K ∩ Y is a dense subset of K.

Let us define the cones T 2 (ū), T ∞ (ū) as

T 2 (ū) := v ∈ L 2 F ; Υ g 1 (v) = 0, Υ h 1j (v) ≤ 0, j ∈ I(ū) , T ∞ (ū) := T 2 (ū) ∩ L ∞ F . (85)
Lemma 28. Under (H1),(H2),(H6),(H7), problem

Min v∈U Υ 1 (v) subject to v ∈ T 2 (ū) (86)
admits v = 0 as a solution.

Proof. Let v ∈ T ∞ (ū) and consider ε > 0. Assumption (H7) (ii) implies that for θ > 0:

g(θ(v + εv)) = g(0) + θΥ g 1 (v + εv) + o(θ) = o(θ). Remark 9 implies that there exists o ∞ (θ) ∈ L ∞ F , with o ∞ (θ) ∞ /θ → 0 as θ ↓ 0, such that g(θ(v + εv) + o ∞ (θ)) = 0. Therefore, setting û(θ) := ū + θ(v + εv) + o ∞ (θ)
, we obtain that E g(y û(θ) (T )) = 0 for θ > 0 small enough. On the other hand, for every j ∈ {1, ..., n h },

E h j (û(θ)) = hj (θ(v + εv) + o ∞ (θ)) = h(0) + θΥ h 1j (v + εv) + o(θ).
Thus, if the jth constraint is active at ū, then by (H7)(ii), we have that E(h j (û(θ))) = εθΥ h 1j (v) + o(θ) < 0 for all θ small enough. If the constraint is not active the same conclusion trivially holds. Therefore, there exists θ ε such that for θ ∈ [0, θ ε ), û(θ) is feasible. Thus,

0 ≤ J(û(θ)) -J(ū) = θΥ 1 (v + εv) + o(θ), which implies that Υ 1 (v + εv) ≥ 0. Since ε is arbitrary, we have Υ 1 (v) ≥ 0. Since L ∞ F is dense in L 2
F and Υ 1 is continuous, the result follows from lemma 27.

For λ = (λ g , λ h ) ∈ R ng × R n h , let us set (p λ , q λ ) ∈ Y 2 × L 2
F for the unique solution of dp(t) =y (t) + f y (t) p(t) + m i=1 σ i y (t) q i (t) dt + q(t)dW (t),

p(T ) = φ y (ȳ(T )) + g y (ȳ(T )) λ g + h y (ȳ(T )) λ h . (87)
We say that λ := (λ g , λ h ) ∈ R ng × R n h is a Lagrange multiplier at ū if the following conditions hold H u (t, ȳ(t), ū(t), p λ (t), q λ (t), ω) = 0 for a.a. (t, ω),

λ j h E h j (ȳ(T )) = 0 and λ j h ≥ 0 for all j ∈ {1, ..., n h }.

We denote by Λ(ū) for the set of Lagrange multipliers at ū. Proposition 29. Under (H1),(H2),(H6),(H7) the set Λ(ū) is a nonempty compact subset of R ng × R n h .

Proof. Lemma 28 implies that the linear program (86) has value 0. By a standard duality result for linear programs (affine cost function and constraints) in a Banach space setting (see e.g. [7, Thm 2.202]), the set of (λ g , λ h ) ∈ R ng ×R

I 1 (ū) + such that for all v ∈ L 2 F Υ 1 (v) + ng i=1 λ i g Υ g 1i (v) + j∈I 1 (ū) λ j h Υ h 1j (v) = 0 (90)
is a nonempty compact subset of R ng ×R n h . Letting λ j h = 0 if j / ∈ I 1 (ū) relation (89) trivially holds, while equation (88) follows from (90) and lemma 3, along the lines of the proof of lemma 7. Now we treat the so-called non qualified case. For α ∈ R let us define the generalized Hamiltonian H : R ×

[0, T ] × R n × R m × R n × R n×d × Ω → R by H(α, t, y, u, p, q, ω) := α (t, y, u, ω) + p • f (t, y, u, ω) + d i=1 q i • σ i (t, y, u, ω).
We say that λ := (α, λ g , λ h ) ∈ R × R ng × R n h is a generalized Lagrange multiplier at ū if λ = 0 and the following conditions hold H u (α, t, ȳ(t), ū(t), p λ (t), q λ (t), ω) = 0 for a.a. (t, ω),

λ j h E h j (ȳ(T )) = 0, λ j h ≥ 0 for all j ∈ {1, ..., n h } and α ≥ 0.

We denote by Λ gen (ū) for the set of generalized Lagrange multipliers at ū.

Proposition 30. Under (H1),(H2) the set Λ gen (ū) is nonempty.

Proof. If (H6),(H7) hold then proposition 29 yields that ∅ = Λ(ū) ⊆ Λ gen (ū). If (H6) does not hold there exists λ g = 0 orthogonal to the image of Υ g 1 . Therefore, lemma 3 yields that (0, λ g , 0) is a generalized Lagrange multiplier at ū. If (H6) hold but (H7) does not, then the pair (0, 0

) ∈ R × L ∞ F is a solution of Min z s.t. Υ g 1 (v) = 0, Υ h 1j (v) ≤ z for j ∈ I 1 (ū).
By considering the dual problem, we obtain the existence of a not null (λ g , λh ) ∈

R ng × R I 1 (ū) + such that for all v ∈ L ∞ F ng i=1 λ i g Υ g 1i (v) + j∈I 1 (ū) λj h Υ h 1j (v) = 0. ( 93
)
Define λ h ∈ R n h as λ j h = λj h if j ∈ I 1 (ū) and 0 otherwhise. The result follows, since lemma 3 implies that (0, λ g , λ h ) ∈ Λ gen (ū).

Remark 10. Proposition 30 is a particular case of the Pontryagin maximum principle in [START_REF] Peng | A general stochastic maximum principle for optimal control problems[END_REF].

Second order necessary condition

Lemma 26 implies that the maps gi , hj defined in (82) admit a second order expansion at v = 0. Let us denote respectively by Υ g 2i , Υ h 2j the associated quadratic forms. As for Υ g 1i , we set Υ g 2 := (Υ g 21 , ..., Υ g 21 ). Define the cones C 2 (ū), C ∞ (ū) by

C 2 (ū) := {v ∈ T 2 (ū) ; Υ 1 (v) = 0} , C ∞ (ū) := C 2 (ū) ∩ L ∞ F . (94)
Note that C 2 (ū) = clo 2 (C ∞ (ū)) by lemma 27. For v ∈ C ∞ (ū) let us set

I 2 (ū, v) := j ∈ I(ū) ; Υ h 1j (v) = 0 .
Lemma 31. Under (H1),(H2),(H6),(H7), for every v ∈ C ∞ (ū), the problem

Min w∈L ∞ F Υ 1 (w) + Υ 2 (v) s.t. Υ g 1 (w) + Υ g 2 (v) = 0, Υ h 1j (w) + Υ h 2j (v) ≤ 0 for j ∈ I 2 (ū, v) (QP v )
has a nonempty feasible set and a non-negative value.

Proof. By (H7) the feasible set of (QP v ) is nonempty. Let w ∈ L ∞ F be feasible and v satisfy (H7) (ii). Fix ε > 0 and for θ > 0 set

u(θ) = ū + θv + 1 2 θ 2 (w + εv). (95)
By lemma 12 applied to the equality constraints, we have that

IE[g(y u(θ) (T ))] = 1 2 θ 2 [Υ g 1 (w) + Υ g 2 (v)] + o(θ 2 ) = o(θ 2 ). ( 96
)
Remark 9 implies that there exists a path of the form ũ(θ

) = u(θ) + o ∞ (θ 2 ) ∈ L ∞ F , where o ∞ (θ 2 ) ∞ /θ 2 → 0, such that IE[g(y ũ(θ) (T ))] = 0. At the same time, IE[h j (y ũ(θ) (T ))] = 1 2 θ 2 Υ h 1j (w + εv) + Υ h 2j (v) + o(θ 2
) for all j ∈ I 2 (ū, v).

(97) It follows that, for θ > 0 small enough, ũ(θ) is feasible and therefore by lemma 12

0 ≤ lim θ↓0 J(ũ(θ)) -J(ū) 1 2 θ 2 = Υ 1 (w) + Υ 2 (v) (98)
as was to be proved.

For λ = (λ g , λ h ) ∈ R ng × R n h and z ∈ R n let us set H (y,u) 2 [λ](t) := H (y,u) 2 (t, ȳ(t), ū(t), p λ (t), q λ (t)), Ψ yy [λ](z) 2 := φ yy (z) 2 + ng i=1 λ i g g i yy (z) 2 + n h j=1 λ j h h j yy (z) 2 , (99)
where (p λ , q λ ) is defined by (87). Now we are able to prove a second order necessary condition for the local minimum ū of (SP ).

Theorem 32. Assume that (H1)-(H3),(H6),(H7) hold and that gi , hj satisfy the assumptions for φ in (H3). Then for every v ∈ C 2 (ū) we have that

max λ∈Λ(ū) E T 0 H (y,u) 2 [λ](t)(y 1 (t), v(t)) 2 dt + Ψ yy [λ](ȳ(T ))(y 1 (T )) 2 ≥ 0. (100)
Proof. Let v ∈ C ∞ (ū). Using lemma 31 and considering the dual problem associated to (QP v ), we obtain that max

(λg,λ h )∈ Λ(ū) Υ 2 (v) + ng i=1 λ i g Υ g 2i (v) + j∈I 2 (ū,v) λ j h Υ h 2j (v) ≥ 0, (101)
where

Λ(ū) :=    (λ g , λ h ) ∈ R ng × R I 2 (ū,v) + ; Υ 1 (•) + ng i=1 λ i g Υ g 1i (•) + j∈I 2 (ū,v) λ j h Υ h 1j (•) = 0    .
On the other hand, since v ∈ C ∞ (ū) , for every λ = (λ g , λ h ) ∈ Λ(ū) we have that

j∈I 1 (ū)\I 2 (ū,v) λ j h Υ h 1j (v) = 0.
Since λ h ≥ 0 and v ∈ C ∞ (ū), each term in the above sum is nonpositive, and hence equal to zero since the sum is null. Thus, by definition of I 2 (ū, v), we have that λ j h = 0 for all j ∈ I 1 (ū) \ I 2 (ū, v). Therefore, by (101

) max λ∈Λ(ū) Υ 2 (v) + ng i=1 λ i g Υ g 2i (v) + n h j=1 λ j h Υ h 2j (v) ≥ 0. (102)
Inequality (100) for every v ∈ C ∞ (ū) then follows from lemma 3, along the lines of the proof of lemma 11. Using that C ∞ (ū) is dense in C 2 (ū), estimate (77) and that Λ(ū) is compact, we obtain the result for all v ∈ C 2 (ū). Using the second assumption in (4), estimates [START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF], [START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF] follow from corollary 2 applied to (103) and ( 19) respectively.

We next prove [START_REF] Lions | Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications[END_REF]. We have that

dd 1 (t) = fy (t)δy(t) -f y (t)y 1 (t) + fu (t) -f u (t) v(t) dt + [σ y (t)δy(t) -σ y (t)y 1 (t) + (σ u (t) -σ u (t)) v(t)] dW (t), d 1 (0) = 0. For ψ = f, σ, we have that ψy (t) -ψ y (t) y 1 (t) = O ([ |δy(t)| + |v(t)| ] |y 1 (t)|) . Also, [σ u (t) -σ u (t)] v(t) = O (|δy(t)||v(t)|) if σ uu ≡ 0, O ([ |δy(t)| + |v(t)| ] |v(t)|) otherwise.
Therefore, the following equation holds for d 1 :

dd 1 (t) = fy (t)d 1 (t) + O ([|δy(t)| + |v(t)|][|y 1 (t)| + |v(t)|]) dt + [σ y (t)d 1 (t) + O (D(δy, y 1 , v))] dW (t),
where

D(δy(t), y 1 (t), v(t)) = [|δy(t)| + |v(t)|] [|y 1 (t)| + |v(t)|] -|v(t)| 2 if σ uu ≡ 0, [|δy(t)| + |v(t)|] [|y 1 (t)| + |v(t)|] otherwise.
By [START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF] and the Cauchy Schwarz inequality

|δy||y 1 | β β,2 = E T 0 |δy(t)| 2 |y 1 (t)| 2 dt β 2 = O E sup |δy(t)| β |y 1 (t)| β = O E sup |δy(t)| 2β 1 2 E sup |y 1 (t)| 2β 1 2 = O( v 2β 2β,2 ). (104)
By similar arguments,

|y 1 ||v| β β,2 = E T 0 |y 1 (t)| 2 |v(t)| 2 dt β 2 = O( v 2β 2β,2 ), |δy||v| β β,2 = E T 0 |δy(t)| 2 |v(t)| 2 dt β 2 = O( v 2β 2β,2 ),
and ( 21) follows by corollary 2, since

v 2 β β,1 = v 2β 2β,2 and v 2 β β,2 = v 2β 2β,4
. Proof of lemma 10. As in the proof of lemma 5 we suppose that m = n = d = 1. We will use repeatedly that for every β, p, q ∈ [1, ∞), we have

|v| q β β,p = v qβ qβ,qp for all v ∈ L qβ,qp F .
Proof of (30): Recall that, by (H1), for ψ = f, σ we assume that ψ yy , ψ yu and ψ uu are bounded. Using (20),

y 2 1 β β,2 = E   T 0 |y 1 (t)| 4 dt β 2   = O E sup |y 1 (t)| 2β = O v 2β 2β,2 .
(105) Analogously, the estimates associated with the term

y 1 v is of order v 2β 2β,2 . Estimate (30) follows from corollary 2 since v 2 β β,1 = v 2β 2β,2 and v 2 β β,2 = v 2β
2β,4 . Proof of (31): Recall that d 2 = δy -y 1 -1 2 y 2 . We have, omitting time from the arguments,

dd 2 (t) = f y d 2 + 1 2 f yy [δy] 2 -1 2 f yy [y 1 ] 2 + f yu δyv -f yu y 1 v + r t (f )(δy, v) 2 dt+ σ y (t)d 2 + 1 2 σ yy [δy] 2 -σ yy [y 1 ] 2 + σ yu δyv -σ yu y 1 v + r t (σ)(δy, v) 2 dW (t).
where for ψ = f, σ the map r t (ψ) is defined by

r t (ψ) := 1 0 (1 -θ) [ψ yy (ȳ(t) + θδy(t), ū(t) + θv(t)) -ψ yy (ȳ(t), ū(t))] dθ.
Recall that if Q is a quadratic form and a is the associated symmetric bilinear form, we have the identity Q(y) -Q(x) = a(y + x, y -x). Thus, since Dψ is Lipschitz, we obtain

dd 2 (t) = [f y d 2 + O (|d 1 | {|δy| + |y 1 |} + |d 1 ||v| + α t (f ))] dt+ [σ y d 2 + O (|d 1 | {|δy| + |y 1 |} + |d 1 ||v| + α t (σ))] dW (t) (106)
where, for ψ = f, σ, Proof. First note that (ii) follows directly from (i). In order to prove (i) we essentially reproduce the proof in [START_REF] Rockafellar | Integral functionals, normal integrands and measurable selections[END_REF] to obtain that P a (t, ω) is (B[0, T ]×F T )/B(R m )measurable and, using that U is F-adapted, we prove that P a (t, ω) is F-adapted.

Let us fix a ∈ R m and consider the sequence of multifunctions U k (t, ω) := {v ∈ R m ; dist(v, U (t, ω)) < k -1 ; |v -a| < dist(a, U (t, ω)) + k -1 }.

(109) Let C be a closed subset of R m . Then P a (t, ω) ∈ C iff C ∩ U k (t, ω) = ∅ for all k, thus

P -1 a (C) = ∩ k U -1 k (C). ( 110
)
Next, let D be a countable dense subset of C, which always exists. We claim that

U -1 k (C) = U -1 k (D) = ∪ d∈D U -1 k (d). ( 111
)
The second equality is obvious and since D is a dense subset of C, in order to establish the first equality it suffices to check that if c ∈ C and (t 0 , ω 0 ) ∈ U -1 k (c), then for c close enough to c we have that (t 0 , ω 0 ) ∈ U -1 k (c ). But this follows directly from the definition of U k (t, ω) in (109). Our claim follows.

On the other hand, for any v ∈ R m and α ≥ 0 we have

{(t, ω) ∈ [0, T ] × Ω ; dist(v, U (t, ω)) ≤ α} = U -1 (v + α B), ( 112
)
where B is the unit ball in R m . Thus, since U is B[0, T ] × F T -measurable, so is the process dist(v, U (t)). Similarly, for a.a. (115) Then u k is a sequence of measurable and adapted functions, and since u k is a Castaing representation, |u k (t, ω) -u(t, ω)| → inf a∈U (t,ω) |u(t, ω) -a| fort a.a. (t, ω). Therefore, since U (t, ω) is closed and convex, we obtain that u k (t, ω) → P U (t,ω) u(t, ω) for a.a. (t, ω). Using the dominated convergence theorem, we deduce that u k has a limit in L 2 F equal to P U (t,ω) (u(t, ω)) for a.a. (t, ω) . It follows that the value of problem (114) is less or equal than E T 0 |P U (t,ω) u(t, ω) -u(t, ω)| 2 dt . Since obviously it cannot be less, and the projection problem has a unique solution, the conclusion follows.

) 4 . 1

 41 First order necessary conditionsConsider as in section 3 a fixed ū ∈ L 2 F . For β, p ∈ [1, ∞] and a subset A ⊆ L β,p F we write clo β,p (A) for the closure of A in L β,p F . If A ⊆ L β F we set clo β (A) := clo β,β (A).

Proof.

  Letting ≡ 0 and φ = c in proposition 13 yields (75) as well as the estimates in (76) and (77). It remains to show that the derivative of ĉ(v) is Lipschitz. In view of proposition 6 and the chain rule, it suffices to show that y ∈ Y 2 → c(y) := E [c(y(T ))] is differentiable with a Lipschitz derivative given by Dc(ȳ)(y) = E [c y (ȳ(T ))y(T )] for all ȳ, y ∈ Y 2 .

α 2 =β 2  2 = 2 = 3 .I 2 =

 222232 t (ψ) := |δy(t)| 3 + |v(t)|3 if ψ uuu = 0, |δy(t)| 3 + |δy(t)| |v(t)| 2 if ψ uuu ≡ 0. Now, let us estimate the terms in the dW (t) part of (106),|d 1 ||δy| β β,2 = E T 0 |d 1 (t)| 2 |δy(t)| 2 dt β O E sup |d 1 (t)| β |δy(t)| β = O( v β 2β,2 v 2β 4β,4 ),by[START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF] and[START_REF] Lions | Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications[END_REF]. Analogously, estimates for the terms d 1 y 1 and d 1 v are of the same order. Let us estimate the terms appearing in α t (σ). Using[START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF], = O E sup |δy(t)| 3β = O( v 3β 3βt)| 2 |v(t)| 4 dt β O E sup |δy(t)| β T 0 |v(t)| 4 dt β O( v β 2β,2 v 2β 4β,4 ).Also, we have thatv 3 β β,1 = v 3β 3β,3 and v 3 β β,2 = v 3β 3β,6. By the Cauchy Schwarz inequality, Using the Cauchy Schwarz inequality again, we getv 3β 3β,3 = O( v β 2β,2 v 2β 4β,4). Therefore, estimate[START_REF] Zhou | A unified treatment of maximum principle and dynamic programming in stochastic controls[END_REF] follows from corollary 2.Proof of lemma 12As in the proof or proposition 8 we denote δJ := J(ū + v) -J(ū). By definition,δJ = E T 0 [ (y ū+v , ū + v) -(ȳ, ū)] dt + φ(y ū+v (T )) -φ(ȳ(T )) = I 1 + I 2 ,where, omitting the time argument in the integral,I 1 := E T 0 y δy + u v + 1 2 (y,u) 2 (δy, v) 2 + r (δy, v) 2 dt , I 2 := E φ y (ȳ(T ))δy(T ) + 1 2 φ yy (ȳ(T ))(δy(T )) 2 + r φ (ȳ(T ))(δy(T )) 2 . (108) Recalling that δy = y 1 + d 1 = y 1 + 1 2 y 2 + d 2 , assumption(8)in (H2) yieldsI 1 = E T 0 y (t)(y 1 + 1 2 y 2 ) + u (t)v + 1 2 D 2 (t)(y 1 , v) 2 dt + E T 0 y d 2 dt +O(z 1 (v)),where, omitting time from function arguments,z 1 (v) := E sup |d 1 | 2 + |d 1 (t)||y 1 | + |δy| 3 + v 1 E (sup |d 1 |) + v 3On the other hand, E φ y (ȳ(T )) y 1 (T ) + 1 2 y 2 (T ) + 1 2 φ yy (ȳ(T )) (y 1 (T )) 2 +E [φ y (ȳ(T ))d 2 (T )] + O(z 2 (v)), where z 2 (v) := E |δy(T )| 3 + |y 1 (T )||d 1 (T )| + |d 1 (T )| 2 .

2 F 2 F

 22 t ∈ [0, T ], we have that {ω ∈ Ω ; dist(v, U (t, ω)) ≤ α} = U -1 (v + α B).(113)Since U is F-adapted, it follows that so is dist(v, U (t)). Therefore, from the definition (109), for any (t, d)∈ [0, T ]× ∈ R m we have that U -1 k (d) ∈ B[0, T ] × F T and U -1 k (t, •)(d) ∈ F t . Using (110)-(111) we finally obtain that P a has the desired properties.Lemma 34. Let U be defined by (51) with U being a (B[0, T ] × F T )/B(R m )measurable, F-adapted closed-convex-valued multifunction. For u ∈ L 2 F we have that w = P U (u) iff w(t, ω) = P U (t,ω) (u(t, ω)) for a.a. (t, ω) ∈ [0, T ] × Ω.Proof. By the definition of a projection, w is characterized as the solution of the minimization problem Minw∈L , ω) -u(t, ω)| 2 dt ; w ∈ U.(114)Let u k (k ≥ 1) be a F-adapted Castaing representation of U and let the sequence u k ∈ L be defined as follows: u 0 is an arbitrary element of U, and for k ≥ 0 setu k+1 (t, ω) = u k+1 (t,ω) if |u k+1 (t, ω) -u(t, ω)| < |u k (t, ω) -u(t, ω)|, u k (t, ω) otherwise.

  Conditions d'optimalité du premier et second ordre en commande optimale stochastique Résumé : Nous considèrons un problème de commande optimale stochastique avec soit des contraintes convexes sur la commande soit un nombre fini de contraintes d'égalité et d'inégalité sur l'état final. L'approche dite variationnelle nous permet d'obtenir un dèveloppement au premier et au second ordre pour l'état et la fonction de coût, autour d'un minimum local. Avec ces dèveloppements on peut montrer des conditions générales d'optimalité de premier ordre et, sous une hypothèse géométrique sur l'ensemble des contraintes, des conditions nécessaires du second ordre sont aussi établies. On finit l'article en fournissant des conditions d'optimalité du second ordre pour de problèmes avec de contraintes en espérance sur l'état final Mots-clés : Commande optimale stochastique, approche variationnelle, conditions d'optimalité de premier et second ordre, contraintes polyédriques, contraintes sur l'état final.

RR n°7454

 Lemma 22. The set of local constraints U defined in (51), with U (t, ω) given by (69), is polyhedric and satisfies (64).

Proof. Let ū ∈ U and v * ∈ N U (ū). For v ∈ T U (ū) ∩ (v * ) ⊥ and k ≥ 0 set

F , hence U is polyhedric and (64) holds.

The following corollary is a consequence of theorem 20 and lemmas 21, 22.

Corollary 23. Assume that (H1) -(H3) hold and suppose that ū is a local solution of (SP) where U is defined in (51), with U (t, ω) given by (69). Then, the following second order necessary conditions holds:

On the second order sufficient condition

In this section we give a second order sufficient condition for the unconstrained case and we briefly discuss the difficulties arising in the constrained case.

When U = L 2 F , (H4) is trivially satisfied and for every ū ∈ U it holds that T U (ū) = L 2 F . The following proposition is a consequence of corollary 14.

Proposition 24. Assume that (H1)-(H3) hold and that U = L 2 F . Suppose that there exist α > 0 such that ū ∈ L 2 F satisfies:

Then, there exists δ > 0 such that for all v ∈ L ∞ F with v ∞ ≤ δ, we have

Only very partial results are obtained when U = L 2 F . Let us recall (see [START_REF] Hestenes | Applications of the theory of quadratic forms in Hilbert space to the calculus of variations[END_REF]) that a quadratic form Q : H → R, where H is a Hilbert space, is a Legendre form if it is weakly lower semi-continuous (w.l.s.c.) quadratic form over H, such that, if

We have the following proposition, whose proof follows the lines of the parallel deterministic result (see [START_REF] Bonnans | Perturbation analysis of optimization problems[END_REF]Section 3.3]):

Proposition 25. Assume that (H1)-(H3) hold. In addition, assume that at ū ∈ U the quadratic form Υ 2 is a Legendre form and there exist α > 0 such that

Then, there exists δ > 0 such that for all u ∈ U with u -ū ∞ ≤ δ, we have

Therefore, using ( 25) and (33), we get (34). Now, we proceed to estimate z(v).

By [START_REF] Lions | Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications[END_REF] we have that

Estimates ( 20), ( 21) and the Cauchy Schwarz inequality yield

Analogously, using ( 21), we have

Estimate ( 20) yields

and

Adapted projections

This subsection discusses projections in L 2 F in the case of local constraints. In order to give the expression of the tangent cone to U, when U is defined by (51), we need a characterization of measurable multifunctions with closed values. We call Castaing representation of a (B[0, T ] × F T )-measurable multifunction

We say that the Castaing representation is adapted if each process (w k (t)) t∈[0,T ] is adapted. By a result due to C. Castaing (see e.g. [28, Thm 1B, p. 161]), any multifunction with closed values that is measurable in L 2 F T has a Castaing representation in the same space. We next extend the result to the adapted case.

Proposition 33. Let U : [0, T ] × Ω → P(R m ) be an (B[0, T ] × F T )-measurable, F-adapted closed-convex-valued multifunction. Then: (i) For any a ∈ R m the map (t, ω) ∈ [0, T ] × Ω → P a (t, ω) := P U (t,ω) (a) ∈ R m is (B[0, T ] × F T )/B(R m )-measurable and F-adapted. (ii) For any countable dense subset {z k } k∈N of R m we have that {P U (t,ω) (z k ), k ∈ N} is an F-adapted Castaing representation of U .


D H K Murthy

T Xu

W H Chen

A J Houtepen

T J Savenije

L D A Siebbeles

J P Nys

C Krzeminski

B Grandidier
email: bruno.grandidier@isen.iemn.univ-lille1.fr

D Stiévenard

P Pareige

F Jomard

G Patriarche

O I Lebedev

Efficient photogeneration of charge carriers in silicon nanowires with a radial doping gradient

From electrodeless time-resolved microwave conductivity measurements, the efficiency of charge carrier generation, their mobility, and decay kinetics on photoexcitation were studied in arrays of Si nanowires grown by the vapor-liquid-solid mechanism. A large enhancement in the magnitude of the photoconductance and charge carrier lifetime are found depending on the incorporation of impurities during the growth.

They are explained by the internal electric field that builds up, due to a higher doped sidewalls, as revealed by detailed analysis of the nanowire morphology and chemical composition.

Thin films consisting of arrays of semiconductor nanostructures exhibit electronic, optical and thermal properties that derive to a large extent from the periodic arrangement of their nanometer scale constituents. For example, arrays of Si wires have been shown to successfully increase light-trapping compared to planar semiconductor thin films [i,ii], while providing a better geometry to collect photoexcited carriers [iii,iv]. Periodic nanoholes etched in thin films of silicon modify the mean free paths of phonons and greatly reduce the thermal conductivity of the film [v]. As a result, arrays of semiconductor nanostructures offer the prospect to improve the energy conversion efficiency in devices such as solar cells or Peltier elements.

Whatever the foreseen application is, the conversion efficiency partly depends on the transport properties of the array. Although these properties have been investigated for single semiconductor nanowires (NWs) [vi] their electrical conductivity may vary from one nanowire to another in the same array depending on slight variations in the NW morphology such as its diameter or its surface roughness. Key transport parameters, like the carrier lifetime and mobility, are still not well known and are usually assumed to be similar to their bulk counterparts in order to determine the conversion efficiency of the NWs [vii]. In addition, difficulties in establishing Ohmic electrical contacts often complicate interpretation of the measurements [viii,ix,x]. Therefore, new experimental tools are required to investigate the transport properties of semiconductor nanostructure ensembles.

As an answer to this challenge, we show that the time-resolved microwave conductivity (TRMC) technique is a promising method to measure, without any electrode, the transient photoconductance of Si NW arrays of semiconductor nanostructures [xi]. As a prototypical example, we investigated the photoconductive properties of arrays of Si NWs grown by the vapor-liquid-solid (VLS) technique using gold as catalyst. Depending on the incorporation of impurities, the TRMC technique allows to measure strong differences in the efficiency of the charge carrier photogeneration, their mobility, and decay over time on photoexcitation. Such differences are rationalized from the detailed knowledge of the NW structural and chemical composition, obtained using transmission electron microscopy (TEM) and atom probe tomography (APT). They show how formation of an internal electric field due to differences in radial doping can be advantageously used to separate photoexcited electron hole pairs. Si NWs were prepared by the VLS method from silane on quartz substrates using the chemical vapor deposition (CVD) technique. Gold nanoparticles were used as catalysts. They were formed at 700°C in the CVD chamber from the deposition of a gold film with a thickness of 2 nm. The growth was performed with a silane partial pressure of 1 mbar, a total pressure in the chamber of 10 mbar and a temperature of 430°C. N-type doped Si NWs were also grown with a silane flow rate of 300 sccm (standard cubic centimeters per minute), whereas the flow rate of phosphine diluted in H 2 was 1 or 5 sccm. The morphology of the Si NWs was characterized by scanning (SEM) and transmission (TEM) electron microscopies.

Their chemical composition was obtained with the atom probe tomography (APT) measurements, that was conducted with a laser-assisted atom probe tomography (CAMECA LAWATAP). In contrast to the previous method that we developed for specimen preparation [xii], Si NWs were directly manipulated on their substrate and welded from the top of the NW onto a pre-prepared W supported tip in a Cross Beam (Focused Ion Beam -Gas Injection System-SEM) workstation. Typically, when NWs are taken from the substrate, they break up in a part of the shaft that is far from both ends. The analysis of the chemical composition that takes place in this region over several hundreds of nm along the growth axis is thus representative of the chemical composition of the rest of the NWs, with the exception of the extremities, where the growth conditions are different. The orientation of the NWs on the tip can be controlled by changing the W tip vertex angle and the welding position. As The specimen analyzed by APT should have good mechanical properties, (Me 3 )MeCpPt and Pt are chosen as the deposition precursor and the solder for reinforcement. The APT characterization was performed using a UV laser as the pulse source (wavelength of 343 nm, power of 4 mW).

The chamber pressure and specimen temperature were maintained at 2×10 -10 mbar and 80 K respectively. The detection limit of the phosphorus atom is 1.2x10 18 at.cm -3 .

A detailed description of the TRMC technique can be found elsewhere [xiii,xiv] . In brief, Si NWs grown on quartz substrates were placed in an X-band (8.4 GHz) microwave cell and photoexcited with a linearly polarized laser pulse from an Optical Parametric Oscillator pumped at 355 nm with the third harmonic of a Q-switched Nd:YAG laser (Vibrant, Opotek).

    t G K P t P     (1)
The dimensions of the microwave cell and the geometrical and properties of the media in the microwave cell determine the sensitivity factor, K (see supporting information). For equation 1 to be applicable P/P is kept below 510 -3 since higher values might change the microwave field pattern in the cell. The absolute value of the conductance is related to the number of negative (electrons) and positive (holes) charge carriers and their mobilities.

Combining the maximum change in conductance (G max ) with the incident light leads to the following equation:

0 max eJ G       (2)
in which the parameter ηΣμ denotes the product of the charge carrier generation yield (η) per incident photon and the sum of the electron and hole mobilities (Σμ). β is the ratio between the broad and narrow inner dimensions of the waveguide used, e is the elementary charge, J 0 is the laser fluence i.e. the number of incident photons per unit area per pulse.

Figure 1 shows typical images of the Si NWs grown on quartz substrates obtained with SEM and TEM. A high density of Si NWs is observed. Their sidewalls are straight and parallel to the growth axis, as it is seen in the high resolution TEM (HRTEM) image of Fig. 1(b). When the growth takes place with the addition of phosphine, the overall morphology of the NWs does not change. For both types of NWs, the histograms of the NW diameter distribution reveals, in Fig. 1(d), that doping the NWs do not modify their diameter, the mean diameter being 65 nm for doped NWs and 66 nm for non doped NWs (the diameter is measured just below the Au seed particle). Finally, we note that the NWs are slightly tapered.

From the high-angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) image of the n-type Si NW shown in Fig. 1(c), we measured a variation of 21 nm between the base and the top of the NW, indicating that the lateral growth rate is about 140 times slower than the axial growth rate. Despite this lateral deposition, the sidewalls do not appear defective. Indeed, HRTEM images of the Si NWs, such as the one presented in Fig. 1(c), show that the cristallographic structure of the core is still preserved at the surface of the NW.

First, TRMC experiments were performed on non-doped and n-doped Si NWS. While the amplitude of the photoconductance signal was weak on photoexciting the samples through the quartz substrate, more than 10 times higher signals could be measured when the the NWs were directly photoexcited, as shown in Figure 2(a). This difference is explained as follows.

The growth of Si NWs leads to the formation of a very thin layer of amorphous Si between the Si NWs, on the top of the quartz substrate. This layer contains a lot of Au impurities that can act as traps for the charge carriers, leading to the formation of a disordered layer with numerous point defects in contrast to the top part of the sample, where the Si NWs are crystalline, as demonstrated by the TEM images. Such variation in the structural quality between the top and the bottom parts of the structure grown on the quartz substrate accounts for the significant change in the measured amplitude of the photoconductance.

Focussing on the photoconductance traces obtained for the top part of the samples only, a constant mean diameter of the Si NWs between the doped and non-doped wires (see Fig. 1(d)) ensures similar optical absorptions for both types of arrays [xvii]. As a result, the magnitude and lifetimes measured on the photoconductance traces can be compared between the different samples. On formation of mobile charges due to the nanosecond laser pulse, the TRMC signal increases during the first few nanoseconds, as shown in Fig. 2(b). Note that, since no electrodes are applied, the decay of the TRMC signal is solely due to recombination and/or trapping of charges. Clearly, n-doping leads to a substantial enhancement of G max . In addition, a longer half-lifetime of the charges ( 1/2 = 42 ± 2 ns) as compared to the charges generated in the non-doped wires ( 1/2 = 6.5 ± 2 ns) is found.

Si NWs are prone to have many intraband gap surface states and dangling bonds [xviii,xix]. N-type doping of Si NWs is therefore expected to lead to the population of these surface states with electrons [xx]. Hence the outer surface of n-type doped Si NWs carries excess negative charge that is expected to influence the band bending at the sidewalls of the NW [xxi,xxii]. In order to unravel if the beneficial effects of doping on the photoconductance are not due to this charge trapping effect, an additional etching treatment was applied. Etching with HF removes the oxide layer on the surface that ultimately minimizes the defect density and/or dangling bonds by a factor of 50 as reported earlier [Erreur ! Signet non défini.]. As a consequence, the charge trapping effect at the surface of the NW is largely reduced. Note that the TRMC experiments were performed immediately after etching in the absence of oxygen to avoid re-oxidation of these wires. As shown in Figure 2(c) the magnitude of photoconductance is increased by a factor of less than two after etching as compared to the value before etching. For the 5 sccm doped NWs a similar observation was made. Most strikingly, the photoconductance decays are hardly affected by the etching process, indicating that charge trapping by intra band-gap surface states on the Si NW sidewall has a negligible effect on the lifetime of the photoexcited carriers.

In order to explain the measured recombination lifetimes between non-doped and ntype doped Si NWs, an analysis of the NW chemical composition was performed. The chemical composition of the n-type doped Si NWs was investigated with the APT and secondary ion mass spectrometry (SIMS) techniques. Figure 3(a) shows a selected volume reconstruction of the core of a doped Si NW, where phosphorus impurities are clearly detected. The concentration of phosphorus impurities establishes at 2x10 18 at.cm -3 in the center of the Si NWs and gradually increases towards the surface, in agreement with previous reports [xxiii,xxiv]. At the surface of the Si NWs, the doping level may be different since the incorporation rate of impurities on the NW sidewall has been previously found to be different when lateral growth occurs [xxv]. A comparison with the SIMS experiments performed on a film grown simultaneously with the NWs reveals that the film thickness is similar to the overgrown layer on the sidewall and that the phosphorus impurity concentration in the film amounts to 5x10 19 at.cm -3 . Therefore we expect the shell of the n-type doped Si NWs to have a similar concentration of phosphorus impurities, which is an order of magnitude higher than the one found in the core of the NWs. Such an inhomogeneous radial distribution of the dopants suggests a variation of the position of the Fermi level with respect to the edge of the bands in the radial direction.

SEM and TEM images also show bright protrusions at the top part of the Si NWs. The protrusions are attributed to Au-rich clusters, because Au is known to diffuse from the seed particle at the end of the growth due to the reduction of the silane partial pressure, while the temperature is still high in the CVD chamber [xxvi,xxvii,xxviii]. As this region only extends over 200-300 nm below the seed particles as pointed out by the brackets in Fig. 1(a), it is quite small in comparison with the rest of the shaft (see Fig. 1(c)) and its contribution in the photoconductance can thus be neglected. From the knowledge of the chemical composition of the NWs, the following model arises: the radial variation of the phosphorus impurities in the Si NWs leads to the formation of a n-n+ core-shell junction. Therefore, the bands are not flat across the NW diameter and an internal electric field is built up.

The profile of the radial phosphorus distribution in the doped Si NWs was used to obtain the internal electric field in the NWs by solving the 2D Poisson equation (DESSIS -Release Z.2007.03, Synopsys Co., Ltd). We consider that they are covered with a thin oxide layer, with the Fermi level pinned midgap at the surface of the NWs. We also presume that all the impurities are electrically active. Figure 4 shows the variation of the electric field across the diameter of a n-type doped Si NW. For comparison, the dopant concentration is also drawn in the inset of Fig. 4. An abrupt variation of the electrical field is found at the interface between the n + doped shell and the n doped core. Then, there is almost no electrical field in the rest part of the shell, due to the high doping level in this region, whereas the electric field varies smoothly in the core to reach zero at the center of the NW. The existence of an internal electric field allows separating the photoexcited carriers. The electrons get preferentially trapped in the highly doped shell, whereas the holes drift towards the core. The recombination lifetime thus becomes longer in comparison with the non doped Si NWs, because recombination is hindered by the spatial separation of the carriers.

The presence of the electric field also accounts for the negligible effect of the chemical composition of the NW sidewall on the charge carrier lifetime as demonstrated in the inset of Figure 2(c). Indeed, holes are repelled from the surface avoiding recombination with electrons through surface states. Such a result is confirmed by studying the product of the quantum yield (η) and the total mobility (Σμ) as a function of the laser fluence. Figure 5 depicts the variation of this product before and after etching the sidewalls of the Si NWs. While the maximum photoconductance increases with laser pump fluence, J 0 , according to G max  J 0  , the product ηΣμ is proportional to J 0 -1 . For the doped Si NWs the value of α amounts to 0.82 and 0.80 before and after etching, respectively. The fact that the slopes do not change substantially on etching (α values are very similar) implies that recombination on fast time scales (< 1 ns) is not mediated by the intraband-gap surface states. This observation renders support to our previous conclusion that recombination takes place in the core of the NWs.

Interestingly, for all fluences studied the n-doped Si NWs show an order of magnitude higher  values as compared to non-doped Si NWs. If only first order decay processes occur,  is constant and α = 1 [Erreur ! Signet non défini.,xxix]. If also higher order decay processes take place such as bimoleculer recombination, α < 1. For the non-doped Si

NWs the slope α amounts to 0.90 and 0.91 before and after etching. The difference in slopes between the non doped (α = 0.90) and n-type doped Si NWs (α = 0.82) indicates that recombination processes in both types of NWs are different. On assuming  to be 1, the minimum mobility valuefor 1 sccm NWs at the lowest incident intensity is then found to be 8 and 16 cm 2 /Vs before and after etching, respectively. For NWs grown with a phosphine flow rate of 5 sccm, similar values are found. Although no discrimination can be made between the individual contributions from positive or negative charge carriers to the total mobility in using the TRMC technique, we can compare the measured AC mobility to the DC hole and electron mobilities in bulk crystalline silicon for similar dopant concentrations. These mobilities are expected to be around 100 cm 2 /Vs for electrons in the shell and 5 cm 2 /Vs for holes in the core [xxx,xxxi,xxxii]. The measured mobility is thus consistent with the DC hole mobility in ntype Si bulk. Although the total mobility could be higher if the actual yield of charge carriers on the time scale of the TRMC experiment is less than 1, we rather suspect that the electron mobility in the Si NWs is small or even negligible. Indeed, due to the internal electric field, electrons drift to the shell. As the NWs stand predominantly in an upright position, electrons oscillate in the shell along the radial direction due to the high frequency electric field component of the microwaves. But, because of a limited shell thickness, that does not exceed 10 nm at the base of the NWs and is smaller towards the top, electrons are likely to be constantly scattered at the boundaries of the shell. As a result their mobility is significantly lowered, suggesting that only holes contribute to the measured mobility.

In transients observed for non-doped and n-doped Si NWs photoexcited at 420 nm using an incident intensity of 55×10 -9 J/cm 2 .pulse for different flow rates of phosphine diluted in H 2 .

The flow rate is expressed in units of standard cubic centimeters per minute (sccm). (c)

Dependence of the photoconductance transients on the surface chemistry for n-doped Si NWs grown with 1 sccm of phosphine diluted in H 2 . The excitation was set at 420 nm using an incident intensity of 55×10 -9 J/cm 2 .pulse. All the insets show corresponding normalized photoconductance transients.

  Photogeneration of mobile charge carriers in the sample leads to an increase of the conductance, ΔG(t), and consequently to the enhanced absorption of microwave power by the sample. The time-dependent change of the conductance is obtained from the normalized change in reflected microwave power (ΔP(t)/P) from the cell, according to [Ref. Erreur ! Signet non défini.,Erreur ! Signet non défini.,xv,xvi]:

FIGURE 1 .

 1 photoconductive properties of the Si NWs requires their analysis with sophisticated complimentary characterization techniques such as TEM and APT, key parameters can be routinely measured using the non-invasive TRMC technique. Due to its simplicity, this technique can be applied to a wide range of nanostructure ensembles and offers the prospect to perform in-situ analyses.

FIGURE 2 .

 2 FIGURE 2. (a) Photoconductance transients observed for doped Si NWs when the excitation is performed from the front (FS) or the back (BS) side of the sample. The photoexcitation was set at 420 nm using an incident intensity of 0.65 J/cm 2 .pulse. (b) Photoconductance

FIGURE 3 .

 3 FIGURE 3. (a) End-on view and side view of the atomically resolved three-dimensional phosphorus distribution in the core of a 70 nm-diameter Si NW. Phosphorus atoms appear as pink dots, whereas the black matrix represents Si atoms. (b) Average phosphorus concentration along the radial axis starting from the middle of the NW towards the edge.Inset: SIMS analysis of the phosphorus concentration in the 25 nm-thick amorphous Si film grown on a SiO 2 surface during the growth of the P-doped Si NWs.

FIGURE 4 . 4 FIGURE 5 .

 445 FIGURE 4. Calculated radial distribution of the electrical field in a n-type doped Si NWs grown, based on the phosphorus doping profile shown in the inset.

  Figure 2

Acknowledgment. The work of DHKM forms the research program of the Dutch

Polymer Institute (DPI, project #681). We acknowledge Daniela Ullien and Louis de Smet from the section Nano-Organic Chemistry of Chemical Engineering department at TU Delft for assistance in etching with HF. A.J.H. acknowledges the 3TU Centre for Sustainable