Bernt Øksendal
email: oksendal@math.uio.

Agnès Sulem
email: agnes.sulem@inria.fr

PARIS -ROCQUENCOURT

Keywords: Forward-backward SDEs, stochastic differential games, maximum principle, model uncertainty, optimal portfolio, jump diffusions Equations différentielles stochastiques forward et rétrogrades, jeux différentiels stochastiques, principes du maximum stochastique, incertitude de modèle, portefeuille optimal, diffusions avec sauts

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as (zero-sum) stochastic differential games of forward-backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the non-zero sum games (finding conditions for Nash equilibria). We then apply these results to study optimal portfolio and consumption problems under model uncertainty. We combine the optimality conditions given by the stochastic maximum principles with Malliavin calculus to obtain a set of equations which determine the optimal strategies.

Introduction

One of the aftereffects of the financial crisis is the increased awareness of the need for more advanced modeling in mathematical finance, and a focus of attention is on the problem of model uncertainty. This paper is motivated by a topic of this type. We consider a stochastic system described by a general Itô-Lévy process controlled by an agent. The performance functional is expressed as the Q-expectation of an integrated profit rate plus a terminal payoff, where Q is a probability measure absolutely continuous with respect to the original probability measure P . We may regard Q as a scenario measure controlled by the market or the environment. If Q = P the problem becomes a classical stochastic control problem of the type studied in [START_REF] Øksendal | Applied Stochastic Control of Jump Diffusions[END_REF]. If Q is uncertain, however, the agent might seek the strategy which maximizes the performance in the worst possible choice of Q. This leads to a stochastic differential game between the agent and the market. Our approach is the following: We write the performance functional as the value at time t = 0 of the solution of an associated backward stochastic differential equation (BSDE). Thus we arrive at a (zero sum) stochastic differential game of a system of forward-backward SDEs (FBSDEs) that we study by the maximum principle approach.

There are several papers of related content. Stochastic control of forwardbackward SDEs (FBSDEs) has been studied in [START_REF] Øksendal | Maximum principles for optimal control of forward-backward stochastic differential equations with jumps[END_REF] and in [START_REF] An | A maximum principle for stochastic differential games with g-expectation and partial information[END_REF] a maximum principle for stochastic differential g-expectation games of SDEs is developed. The papers [START_REF] Lim | Exponential utility maximization and indifference price in an incomplete market with defaults[END_REF], [START_REF] Øksendal | Robust stochastic control and equivalent martingale measures[END_REF] and [START_REF] Øksendal | Portfolio optimization under model uncertainty and BSDE games[END_REF] also study optimal portfolio under model uncertainty by means of BSDEs, but the approaches there are strongly linked to the exponential utility case. A key feature of the current paper is that it applies to general utility functions and also general dynamics for the state process.

Our paper is organised as follows: in Section 2, we state general stochastic maximum principles for stochastic differential games, both in the zerosum case (finding conditions for saddle points) and for the non-zero sum games (finding conditions for Nash equilibria). The proofs are given in Appendix A. In Section 3 we consider stochastic control problems under uncertainty. We formulate these problems as (zero sum) stochastic differential games of forward-backward SDEs (FBSDEs) and we study them by the maximum principle approach of Section 2. In Section 4 we apply these techniques to study an optimal portfolio and consumption problem under model uncertainty. Using the solution for linear Malliavin-differential type equations given in [START_REF] Øksendal | Maximum principles for optimal control of forward-backward stochastic differential equations with jumps[END_REF] we arrive at a set of equations which determine the optimal portfolio and consumption of the agent and the corresponding optimal portfolio scenario measure of the market.

2 Maximum principles for stochastic differential games of forward-backward stochastic differential equations

In this section, we formulate and prove a sufficient and a necessary maximum principle for general stochastic differential games (not necessarily zero-sum games) of forward-backward SDEs. Let (Ω, {F t } t≥0 , P ) be a filtered probability space. Consider a controlled forward SDE of the form dX(t) = dX (u) (t) = b(t, X(t), u(t))dt + σ(t, X(t), u(t))dB(t)

+ R γ(t, X(t), u(t), ζ) Ñ (dt, dζ) ; X(0) = x ∈ R. (2.1)
where B is a Brownian motion, and Ñ (dt, dζ) = N (dt, dζ)ν(dζ)dt is an independent compensated Poisson random measure where ν is the Lévy measure of N such that R ζ 2 ν(dζ) < ∞. We assume that IF = {F t , t ≥ 0} is the natural filtration associated with B and N . Here u = (u 1 , u 2 ), where u i (t) is the control of player i ; i = 1, 2. We assume that we are given two subfiltrations

E (i) t ⊆ F t ; t ∈ [0, T ], (2.2)
representing the information available to player i at time t ; i = 1, 2. We let A i denote a given set of admissible control processes for player i, contained in the set of

E (i) t -predictable processes ; i = 1, 2, with values in A i ⊂ R d , d ≥ 1. Denote U = A 1 × A 2 .
We consider the associated backward SDE's (i.e. BSDEs) in the unknowns

Y i (t), Z i (t), K i (t, ζ) of the form dY i (t) = -g i (t, X(t), Y i (t), Z i (t), K i (t, •), u(t))dt + Z i (t)dB(t) + R K i (t, ζ) Ñ (dt, dζ) ; 0 ≤ t ≤ T Y i (T ) = h i (X(T )) ; i = 1, 2.
(2.3)

Here g i (t, y, z, k, u) : [0, T ] × R × R × R × U → R and h i : R → R are given functions such that the BSDEs (2.3) have unique solutions. Let f i (t, x, u) : [0, T ] × R × U → R, ϕ i (x)
: R → R and ψ i (x) : R → R be given profit rates, bequest functions and "risk evaluations" respectively, of player i ; i = 1, 2. Define

J i (u) = E T 0 f i (t, X (u) (t), u(t))dt + ϕ i (X (u) (T )) + ψ i (Y i (0)) ; i = 1, 2, (2.4) RR n°7776
provided the integrals and expectations exist. We call J i (u) the performance functional of player i ; i = 1, 2.

A Nash equilibrium for the FBSDE game (2.1)-(2.4) is a pair

(û 1 , û2 ) ∈ A 1 × A 2 such that J 1 (u 1 , û2 ) ≤ J 1 (û 1 , û2 ) for all u 1 ∈ A 1 (2.5) and J 2 (û 1 , u 2 ) ≤ J 2 (û 1 , û2 ) for all u 2 ∈ A 2 . (2.6)
Heuristically this means that player i has no incentive to deviate from the control ûi , as long as player j (j = i) does not deviate from ûj ; i = 1, 2. Therefore a Nash equilibrium is in some cases a likely outcome of a game. We now present a method to find it, based on the maximum principle for stochastic control. Our result may be regarded as an extension of the maximum principles for FBSDEs in [START_REF] Øksendal | Maximum principles for optimal control of forward-backward stochastic differential equations with jumps[END_REF] and for (forward) SDE games in [START_REF] An | A maximum principle for stochastic differential games with g-expectation and partial information[END_REF].

Define the Hamiltonians

H i (t, x, y, z, k, u 1 , u 2 , λ, p, q, r) : [0, T ]×R×R×R×R×A 1 ×A 2 ×R×R×R×R → R of this game by H i (t, x, y, z, k,u 1 , u 2 , λ, p, q, r) = f i (t, x, u 1 , u 2 ) + λg i (t, x, y, z, k, u 1 , u 2 ) + pb(t, x, u 1 , u 2 ) + qσ(t, x, u 1 , u 2 ) + R r(ζ)γ(t, x, u 1 , u 2 , ζ)ν(dζ) ; i = 1, 2, (2.7)
where R is the set of functions from R 0 into R such that the integral in (2.7) converges.

We assume that H i is Fréchet differentiable (C 1 ) in the variables x, y, z, k, u i=1,2.

In the following, we are using the shorthand notation

∂H i ∂y (t) = ∂H i ∂y (t, X(t), Y i (t), Z i (t), K i (t, •), u 1 (t), u 2 (t), λ i (t), p i (t), q i (t), r i (t, •))
and similarly for the other partial derivatives of H i .

To these Hamiltonians we associate a system of FBSDEs in the adjoint processes λ i (t), p i (t), q i (t) and r i (t, ζ) as follows:

(i) Forward SDE in λ i (t):        dλ i (t) = ∂H i ∂y (t)dt + ∂H i ∂z (t)dB(t) + R ∇ k H i (t, ζ) Ñ (dt, dζ) ; 0 ≤ t ≤ T λ i (0) = ψ ′ i (Y i (0)) = dψ i dy (Y i (0)) .
(2.8)

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Forward-backward SDE games and stochastic control under model uncertainty 6

(ii) Backward SDE in p i (t), q i (t), r i (t, ζ):    dp i (t) = - ∂H i ∂x (t)dt + q i (t)dB(t) + R r i (t, ζ) Ñ (dt, dζ) ; 0 ≤ t ≤ T p i (T ) = ϕ ′ i (X(T )) + h ′ i (X(T ))λ i (T ).
(2.9)

See Appendix A for an explanation of the gradient operator

∇ k H i (t, ζ) = ∇ k H i (t, ζ)(•).
• (Concavity I) The functions x → h i (x), x → ϕ i (x), x → ψ i (x) are concave, i = 1,2.
• (The conditional maximum principle)

max v∈A 1 {E[H 1 (t,
max v∈A 2 {E[H 2 (t, X(t), Ŷ2 (t), Ẑ2 (t), K2 (t, •), u 1 (t), v, λ2 (t), p2 (t), q2 (t), r2 (t, •)) | E (2) t ] ; = E[H 2 (t, X(t), Ŷ2 (t), Ẑ2 (t), K2 (t, •), û1 (t), û2 (t), λ2 (t), p2 (t), q2 (t), r2 (t, •)) | E (2) t ] (2.11)
• (Concavity II) (The Arrow conditions). The functions ĥ1 (x, y, z, k) := max

v 1 ∈A 1 E[H 1 (t, x, y, z, k, v 1 , û2 (t), λ1 (t), p1 (t), q1 (t), r1 (t, •)) | E (1) t ], and
ĥ2 (x, y, z, k) := max v 2 ∈A 2 E[H 2 (t, x, y, z, k, û1 (t), v 2 , λ2 (t), p2 (t), q2 (t), r2 (t, •)) | E (2) t ]
are concave for all t, a.s.

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Forward-backward SDE games and stochastic control under model uncertainty 7

• Moreover, assume the following growth conditions hold:

E T 0 p2 i (t) (σ(t) -σ(t)) 2 + R (r(t, ζ) -r(t, ζ)) 2 ν(dζ) + (X(t) -X(t)) 2 q2 i (t) + R r2 i (t, ζ)ν(dζ) + (Y i (t) -Ŷi (t)) 2   ∂ Ĥi ∂z 2 (t) + R ∇ k Ĥi (t, ζ) 2 ν(dζ)   + λ2 1 (t) (Z i (t) -Ẑi (t)) 2 + R (K i (t, ζ) -Ki (t, ζ)) 2 ν(dζ) dt < ∞ for i = 1, 2.
(2.12)

Then û(t) = (û 1 (t), û2 (t)) is a Nash equilibrium for (2.1)-(2.4).

Remark 2.2 Above we have used the following shorthand notation:

If i = 1, then X(t) = X (u 1 ,û 2 ) (t) and Y 1 (t) = Y (u 1 ,û 2 ) 1
(t) are the processes corresponding to the control u(t) = (u 1 (t), û2 (t)), while X(t) = X (û) (t) and Ŷ1 (t) = Y (û) 1 (t) are those corresponding to the control û(t) = (û 1 (t), û2 (t)). An analogue notation is used for i = 2.

Moreover, we put Proof. See Appendix A.

∂ Ĥi ∂x (t) = ∂H i ∂x (t
It is also of interest to prove a version of the maximum principle which does not require the concavity conditions. One such version is the following necessary maximum principle (Theorem 2.3) which requires the following assumptions:

• For all t 0 ∈ [0, T ] and all bounded, E (i) t -measurable random variables α i (ω), the control β i (t) := χ (t 0 ,T ) (t)α i (ω) belongs to A i ; i = 1, 2

(2.13) • For all u i , β i ∈ A i with β i bounded there exists δ i > 0 such that the control ũi (t) := u i (t) + sβ i (t) ; t ∈ [0, T ] belongs to A i for all s ∈ (-δ i , δ i ) ; i = 1, 2.

(2.14)

• The following derivative processes exist and belong to L 2 ([0, T ] × Ω) :

(2.15)

x 1 (t) = d ds X (u 1 +sβ 1 ,u 2 ) (t) | s=0 ; y 1 (t) = d ds Y (u 1 +sβ 1 ,u 2 ) 1 (t) | s=0 z 1 (t) = d ds Z (u 1 +sβ 1 ,u 2 ) 1 (t) | s=0 ; k 1 (t, ζ) = d ds K (u 1 +sβ 1 ,u 2 ) 1 (t) | s=0 and, similarly x 2 (t) = d ds X (u 1 ,u 2 +sβ 2 ) (t) | s=0 etc.
Note that since X (u) (0) = x for all u we have x i (0) = 0 for i = 1, 2.

In the following we write ∂b ∂x (t) for ∂b ∂x (t, X(t), u(t)) etc.

By (2.1) and (2.3) we have

dx 1 (t) = ∂b ∂x (t)x 1 (t) + ∂b ∂u 1 (t)β 1 (t) dt + ∂σ ∂x (t)x 1 (t) + ∂σ ∂u 1 (t)β 1 (t) dB(t) + R ∂γ ∂x (t, ζ)x 1 (t) + ∂γ ∂u 1 (t, ζ)β 1 (t) Ñ (dt, dζ), (2.16
)

dy 1 (t) = - ∂g 1 ∂x (t)x 1 (t) + ∂g 1 ∂y (t)y 1 (t) + ∂g 1 ∂z (t)z 1 (t) + R ∇ k g 1 (t, ζ)k 1 (t, ζ)ν(dζ) + ∂g 1 ∂u 1 (t)β 1 (t) dt + z i (t)dB(t) + R k 1 (t, ζ) Ñ (dt, dζ) ; 0 ≤ t ≤ T, y 1 (T ) = h ′ 1 (X (u 1 ,u 2 ) (T ))x 1 (T ), (2.17)
and similarly for dx 2 (t), dy 2 (t).

We are now ready to state a necessary maximum principle, which is an extension of Theorem 3.1 in [START_REF] An | A maximum principle for stochastic differential games with g-expectation and partial information[END_REF] and Theorem 3.1 in [START_REF] Øksendal | Maximum principles for optimal control of forward-backward stochastic differential equations with jumps[END_REF]. In the sequel, ∂H ∂v means ∇ v H.

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Forward-backward SDE games and stochastic control under model uncertainty 9

Theorem 2.3 (Necessary maximum principle) Suppose u ∈ A with cor- responding solutions X(t), Y i (t), Z i (t), K i (t, ζ), λ i (t), p i (t), q i (t), r i (t, ζ) of equa- tions (2.1), (2.
3), (2.8) and (2.9). Suppose (2.13), (2.14) and (2.15) hold.

Moreover, assume that

E T 0 p 2 i (t) ∂σ ∂x (t)x i (t) + ∂σ ∂u i (t)β i (t) 2 + R ∂γ ∂x (t, ζ)x i (t) + ∂γ ∂u i (t, ζ)β i (t) 2 ν(dζ) + x 2 i (t)(q 2 i (t) + R r 2 i (t, ζ)ν(dζ)) + λ 2 i (t)(z 2 i (t) + R k 2 i (t, ζ)ν(dζ)) +y 2 i (t) ( ∂H i ∂z ) 2 (t) + R ∇ k H i (t, ζ) 2 ν(dζ) dt < ∞ for i = 1, 2.
(2.18)

Then the following are equivalent:

(i) d ds J 1 (u 1 + sβ 1 , u 2 ) | s=0 = d ds J 2 (u 1 , u 2 + sβ 2 ) | s=0 = 0 for all bounded β 1 ∈ A 1 , β 2 ∈ A 2 .
(ii)

E ∂ ∂v 1 H 1 (t, X(t), Y 1 (t), Z 1 (t), K 1 (t, •), v 1 , u 2 (t), λ 1 (t), p 1 (t)q 1 (t), r 1 (t, •)) | E (1) t v 1 =u 1 (t) = E ∂ ∂v 2 H 2 (t, X(t), Y 2 (t), Z 2 (t), K 2 (t, •), u 1 (t), v 2 , λ 2 (t), p 2 (t), q 2 (t), r 2 (t, •)) | E (2) t v 2 =u 2 (t) = 0.
Proof. See Appendix A.

The zero-sum game case. In the zero-sum case we have

J 1 (u 1 , u 2 ) + J 2 (u 1 , u 2 ) = 0. (2.

19)

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Forward-backward SDE games and stochastic control under model uncertainty 10

Then the Nash equilibrium

(û 1 , û2 ) ∈ A 1 × A 2 satisfying (2.5)-(2.6) becomes a saddle point for J(u 1 , u 2 ) := J 1 (u 1 , u 2 ).
To see this, note that (2.5)-(2.6) imply that

J 1 (u 1 , û2 ) ≤ J 1 (û 1 , û2 ) = -J 2 (û 1 , û2 ) ≤ -J 2 (û 1 , u 2 )
and hence

J(u 1 , û2 ) ≤ J(û 1 , û2 ) ≤ J(û 1 , u 2 ) for all u 1 , u 2 .
From this we deduce that

inf u 2 ∈A 2 sup u 1 ∈A 1 J(u 1 , u 2 ) ≤ sup u 1 ∈A 1 J(u 1 , û2 ) ≤ J(û 1 , û2 ) ≤ inf u 2 ∈A 2 J(û 1 , u 2 ) ≤ sup u 1 ∈A 1 inf u 2 ∈A 2 J(u 1 , u 2 ).
(2.20)

Since we always have inf sup ≥ sup inf, we conclude that

inf u 2 ∈A 2 sup u 1 ∈A 1 J(u 1 , u 2 ) = sup u 1 ∈A 1 J(u 1 , û2 ) = J(û 1 , û2 ) = inf u 2 ∈A 2 J(û 1 , u 2 ) = sup u 1 ∈A 1 inf u 2 ∈A 2 J(u 1 , u 2 ). (2.21)
i.e. (û 1 , û2 ) ∈ A 1 × A 2 is a saddle point for J(u 1 , u 2 ).

We now state the necessary maximum principle for the zero sum game problem:

Choose

g i = g, h i = h, f 1 = f = -f 2 , ϕ 1 = ϕ = -ϕ 2 and ψ 1 = ψ = -ψ 2 ; i = 1, 2. For u = (u 1 , u 2 ) ∈ A 1 × A 2 define J(u 1 , u 2 ) = E T 0 f (t, X (u) (t), u(t))dt + ϕ(X (u) (T )) + ψ(Y (0)) , (2.22) where X (u) (t), Y (t) = Y i (t), Z(t) = Z i (t) and K(t, ζ) = K i (t,
H(t, x, y, z, k, u 1 , u 2 , λ, p, q, r) = f (t, x, u 1 , u 2 ) + λg(t, x, y, z, k, u 1 , u 2 ) + pb(t, x, u 1 , u 2 ) + qσ(t, x, u 1 , u 2 ) + R r(ζ)γ(t, x, u 1 , u 2 , ζ)ν(dζ).
(2.23) Let λ = λ i , p = p i , q = q i and r = r i i = 1, 2 be as in (2.8)-(2.9). Theorem 2.4 (Necessary maximum principle for zero-sum forward-backward games) Assume the conditions of Theorem 2.3 hold. Then the following are equivalent:

(i) d ds J(u 1 + sβ 1 , u 2 ) | s=0 = d ds J(u 1 , u 2 + sβ 2 ) | s=0 = 0 (2.24) for all bounded β 1 ∈ A 1 , β 2 ∈ A 2 .
(ii)

E ∂ ∂v 1 H(t, X(t), Y (t), Z(t), K(t, •), v 1 , u 2 (t), λ(t), p(t), q(t), r(t, •)) | E (1) t v 1 =u 1 (t) = E ∂ ∂v 2 H(t, X(t), Y (t), Z(t), K(t, •), u 1 (t), v 2 , λ(t), p(t), q(t), r(t, •)) | E (2) t v 2 =u 2 (t) = 0. (2.25)
Proof. The proof is similar to the proof of Theorem 2.3 and is omitted. Let H(t, x, y, z, k, u 1 , u 2 , λ, p, q, r) be as in (2.25). Let (û 1 , û2 ) ∈ A 1 × A 2 , with corresponding solutions X(t), Ŷ (t), Ẑ(t), K(t), λ(t), p(t), q(t), r(t, ζ) of equation (2.1), (2.2), (2.8) and (2.9) for g i = g, h i = h, f i = f, ϕ i = ϕ and ψ i = ψ. Suppose the following holds

Corollary 2.5 Let u = (u 1 , u 2 ) ∈ A 1 × A 2 be
• The functions x → h(x), x → ϕ(x) and x → ψ(x) (2.26)
are affine.

• (The conditional maximum principle)

max v 1 ∈A 1 E[H(t, X(t), Ŷ (t), Ẑ(t), K(t, •), v 1 , û2 (t), λ(t), p(t), q(t), r(t, •)) | E (1) t ] = E[H(t, X(t), Ŷ (t), Ẑ(t), K(t, •), û1 (t), û2 (t), λ(t), p(t), q(t), r(t, •)) | E (1)
t ] (2.27)

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Forward-backward SDE games and stochastic control under model uncertainty 12 and

min v 2 ∈A 2 E[H(t, X(t), Ŷ (t), Ẑ(t), K(t, •), û1 (t), v 2 , λ(t), p(t), q(t), r(t, •)) | E (2) t ] = E[H(t, X(t), Ŷ (t), Ẑ(t), K(t, •), û1 (t), û2 (t), λ(t), p(t), q(t), r(t, •)) | E (2) t ].
(2.28)

• (The Arrow conditions) The function ĥ(x, y, z, k) := max

v 1 ∈A 1 E[H(t, x, y, z, k, v 1 , û2 (t), λ(t), p(t), q(t), r(t, •)) | E (1) t ]
is concave, and the function

∨ h (x, y, z, k) := min v 2 ∈A 2 E[H(t, x, y, z, k, û1 (t), v 2 , λ(t), p(t), q(t), r(t, •)) | E (2) t ]
is convex, for all t ∈ [0, T ], a.s.

• The growth condition (2.14) holds with

p i = p etc. Then û(t) = (û 1 (t), û2 (t)) is a saddle point for J(u 1 , u 2 ).
Proof. The proof is similar to the proof of Theorem 2.1 and is omitted.

3 Stochastic control under model uncertainty

Let X(t) = X v
x (t) be a controlled Itô-Lévy process of the form

dX(t) = b(t, X(t), v(t))dt + σ(t, X(t), v(t))dB(t) + R γ(t, X(t), v(t), ζ) Ñ (dt, dζ) ; 0 ≤ t ≤ T X(0) = x ∈ R (3.1)
where v(•) is the control process. We consider a model uncertainty setup, represented by a probability measure Q = Q θ which is equivalent to P , with the Radon-Nikodym derivative on F t given by

d(Q | F t ) d(P | F t ) = G θ (t) (3.2) RR n°7776
Forward-backward SDE games and stochastic control under model uncertainty 13

where, for 0 ≤ t ≤ T , G θ (t) is a martingale of the form

dG θ (t) = G θ (t -)[θ 0 (t)dB(t) + R θ 1 (t, ζ) Ñ (dt, dζ)] G θ (0) = 1. (3.3)
Here θ = (θ 0 , θ 1 ) may be regarded as a scenario control. Let A 1 denote a given family of admissible controls v and A 2 denote a given set of admissible scenario controls θ such that E[

T 0 {|θ 2 0 (t)| + R θ 2 1 (t, ζ)ν(dζ)}dt] < ∞ and θ 1 (t, ζ) ≥ -1+ǫ for some ǫ > 0. Let E (1)
0≤t≤T and E

(2) 0≤t≤T be given subfiltrations of F 0≤t≤T , representing the information available to the controllers at time t. It is required that v ∈ A 1 be E 1 t -predictable, and θ ∈ A 2 be E 2 t -predictable. We consider the stochastic differential game to find

(v, θ) ∈ A 1 × A 2 such that sup v∈A 1 inf θ∈A 2 E Q θ [W (v, θ)] = E Q θ [W (v, θ)] = inf θ∈A 2 sup v∈A 1 E Q θ [W (v, θ)], (3.4)
where

W (v, θ) = U 2 (X v (T )) + T 0 U 1 (s, X v (s), v(s))ds + T 0 ρ(θ(t))dt. (3.5)
Here, U 1 : [0, T ]×R×V → R and U 2 : R → R are given functions, concave and increasing with a strictly decreasing derivative, and ρ is a convex function.

The term Λ(θ

) := E Q θ [ T 0 ρ(θ(t)
)dt] can be seen as a penalty term, penalizing the difference between Q θ and the original probability measure P . Put

F (t, x, u) = U 1 (t, x, v) + ρ(θ); u = (v, θ) = (c, π, θ 0 , θ 1 ). (3.6)
Then

E Q θ [W (v, θ)] = E[G θ (T )U 2 (X v (T )) + T 0 G θ (s)F (s, X v (s), u(s))ds]. (3.7)
We now define

Y (t) = Y v,θ (t) by
Y (t) = E[ G θ (T ) G θ (t) U 2 (X v (T )) + T t G θ (s) G θ (t) F (s, X v (s), u(s))ds | F t ]; t ∈ [0, T ].
(3.8) Then we recognize Y (t) as the solution of the linear BSDE (see Lemma B.1)

dY (t) = -[F (t, X v (t), u(t)) + θ 0 (t)Z(t) + R θ 1 (t, ζ)K(t, ζ)ν(dζ)]dt + Z(t)dB(t) + R K(t, ζ) Ñ (dt, dζ); 0 ≤ t ≤ T (3.9) Y (T ) = U 2 (X v (T )).

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Forward-backward SDE games and stochastic control under model uncertainty 14

Note that

Y (0) = Y v,θ (0) = E Q θ [W (v, θ)]. (3.10)
Therefore the problem (3.4) can be written

sup v∈A 1 inf θ∈A 2 Y v,θ (0) = Y v, θ(0) = inf θ∈A 2 sup v∈A 1 Y v,θ (0), (3.11)
where Y v,θ (t) is given by the forward-backward system (3.1) & (3.9). This is a zero-sum stochastic differential game (SDG) of forward-backward SDEs of the form (2.22) with f = ϕ = 0 and ψ = Id.

Proceeding as in Section 2, define the Hamiltonian

H : [0, T ] × R × R × R 0 × R × A 1 × A 2 × R × R × R × R → R by H(t, x, y, z, k, v, θ, λ, p, q, r) = [F (t, x, u) + θ 0 z + R θ 1 (ζ)k(ζ)ν(dζ)]λ +b(t, x, v)p + σ(t, x, v)q + R γ(t, x, v, ζ)r(ζ)ν(dζ).
(3.12)

where R is the set of functions r : R 0 → R such that (3.12) converge. Define a pair of FBSDEs in the adjoint processes λ(t), p(t), q(t), r(t, ζ) as follows: Forward SDE for λ(t):

dλ(t) = ∂H ∂y (t)dt + ∂H ∂z (t)dB(t) + R ∇ k H(t, ζ) Ñ (dt, dζ) = λ(t)θ 0 (t)dB(t) + λ(t) R θ 1 (t, ζ)(•) Ñ (dt, dζ); t ∈ [0, T ] λ(0) = 1 (3.13)
Backward SDE for p(t), q(t), r(t, ζ):

dp(t) = - ∂H ∂x (t)dt + q(t)dB(t) + R r(t, ζ) Ñ (dt, dζ) = -{ ∂F ∂x (t) + p(t) ∂b ∂x (t) + q(t) ∂σ ∂x (t) + R r(t, ζ) ∂γ ∂x (t, ζ)ν(dζ)}dt + q(t)dB(t) + R r(t, ζ) Ñ (dt, dζ); t ∈ [0, T ] p(T ) = λ(T )U ′ 2 (X(T )). ( 3
+q(t) ∂σ ∂v (t, X(t), v(t)) + R r(t, ζ) ∂γ ∂v (t, X(t), v(t), ζ)ν(dζ) | E (1)
t ] = 0 E[ λ(t)( ∂ρ ∂θ 0 ( θ(t)) + Ẑ(t)) | E (2)
t ] = 0 E[ λ(t)(∇ θ 1 F (t, X(t), û(t)) + R (•) K(t, ζ)ν(dζ)) | E (2)
t ] = 0.

Note that both ∇ θ 1 F and R (•) K(t, ζ)ν(dζ) are linear functionals, the latter being defined by the action

ϕ → R ϕ(ζ) K(t, ζ)ν(dζ)
for all bounded continuous functions ϕ : R 0 → R.

Portfolio and consumption problem under model uncertainty

We now apply this to the following portfolio and consumption problem under model uncertainty. Consider a financial market consisting of a bond with unit price S 0 (t) = 1 ; 0 ≤ t ≤ T, and a stock, with unit price S(t) given by

dS(t) = S(t -)[b 0 (t)dt + σ 0 (t)dB(t) + R γ 0 (t, ζ) Ñ (dt, dζ)], (4.1)

RR n°7776

where b 0 (t) = b 0 (t, ω), σ 0 (t) = σ 0 (t, ω) and γ 0 (t, ζ) = γ 0 (t, ζ, ω) are given {F t }-predictable processes such that γ 0 ≥ -1 + ǫ for some ǫ > 0 and

E[ T 0 {|b 0 (t)| + σ 2 0 (t) + R γ 2 0 (t, ζ)ν(dζ)}dt] < ∞.
Note that this system is non-Markovian since the coefficients are random processes.

We introduce the state price density Γ(t) defined by

Γ(t) := exp( t 0 - b 0 (s) σ 0 (s) dB(s) - 1 2 t 0 ( b 0 (s) σ 0 (s) ) 2 ds). (4.2)
Let X(t) = X v (t) be the wealth process corresponding to a portfolio π(t) and a consumption rate c(t), i.e.

dX(t) = π(t)[b 0 (t)dt + σ 0 (t)dB(t) + R γ 0 (t, ζ) Ñ (dt, dζ)] -c(t)dt, t ∈ [0, T ] X(0) = x ∈ R, (4.3
) and put v = (π, c). We consider the stochastic differential game (3.4)-(3.5). For i = 1, 2, I i will denote the inverse of U ′ i , in the sense that

I i (y) = (U ′ i ) -1 (y); 0 ≤ y ≤ y i 0 y > y i (4.4)
where y i = lim x→0 + U ′ i (x). We assume that ρ ′ (θ) has an inverse. We have seen in Section 3, that the problem (3.4)-(3.5) can be written as

sup v∈A 1 inf θ∈A 2 Y v,θ (0) = Y v, θ(0) = inf θ∈A 2 sup v∈A 1 Y v,θ (0), (4.5)
where Y (t) = Y v,θ (t) is given by equation (3.9) and (4.3). We now apply the necessary maximum principle given by Theorem 3.1. The Hamiltonian for the problem (4.5) is, by (3.12),

H(t, x, y, z, k, v, θ, λ, p, q, r) = [U 1 (t, c) + ρ(θ) + θ 0 z + R θ 1 (ζ)k(ζ)ν(dζ)]λ +(πb 0 (t) -c)p + πσ 0 (t)q + π R γ 0 (t, ζ)r(ζ)ν(dζ).

RR n°7776

The forward SDE for λ(t) = λ θ (t) and the BSDE for p(t), q(t), r(t, ζ) are (see (3.13)-(3.14))

dλ(t) = λ(t)[θ 0 (t)dB(t) + R θ 1 (t, ζ) Ñ (dt, dζ)]; t ∈ [0, T ] λ(0) = 1 (4.6) dp(t) = q(t)dB(t) + R r(t, ζ) Ñ (dt, dz); t ∈ [0, T ] p(T ) = λ(T )U ′ 2 (X(T )). (4.7)
Maximizing H with respect to (c, π) gives the following first order conditions:

E[λ(t) | E (1) t ] ∂U 1 ∂c (t, c(t)) = E[p(t) | E (1) t ] (4.8) E[b 0 (t)p(t) + σ 0 (t)q(t) + R γ 0 (t, ζ)r(t, ζ)ν(dζ) | E (1)
t ] = 0 (4.9)

Minimizing H with respect to θ = (θ 0 , θ 1 ) gives the following first order conditions:

∂ρ ∂θ 0 (θ(t)) + E[Z(t) | E (2)
t ] = 0 (4.10)

∇ θ 1 ρ(θ(t))(•) + E[ R (•)K(t, ζ)ν(dζ) | E (2)
t ] = 0 (4.11)

We now restrict ourselves to the case when there are no jumps, i.e. Ñ = ν = K = θ 1 = 0 and E (1) t

= E

(2) t = F t . For simplicity of notation, we write θ instead of θ 0 . Then equations (4.6)-(4.11) simplify to:

λ(t) = exp( t 0 θ(s)dB(s) - t 0 1 2 θ 2 (s)ds) (4.12) p(t) = E[λ(T )U ′ 2 (X(T )) | F t ] ; (4.13) λ(t) ∂U 1 ∂c (t, c(t)) = p(t) (4.14) b 0 (t)p(t) + σ 0 (t)q(t) = 0 (4.15) ρ ′ (θ(t)) + Z(t) = 0 (4.16)
and by the generalized Clark-Ocone formula [START_REF] Aase | White noise generalizations of the Clark-Haussmann-Ocone theorem, with application to mathematical finance[END_REF],

q(t) = E[D t (λ(T )U ′ 2 (X(T ))) | F t ], (4.17)

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where D t denotes the Malliavin derivative at t with respect to B(•). (See e.g. [START_REF] Di Nunno | Malliavin Calculus for Lévy Processes with Applications to Finance[END_REF]).

The FBSDEs (4.3)-(3.9) simplify to:

dX(t) = π(t)[b 0 (t)dt + σ 0 (t)dB(t)] -c(t)dt, 0 ≤ t ≤ T X(0) = x > 0 (4.18) dY (t) = -[U 1 (t, c(t)) + ρ(θ(t)) + θ(t)Z(t)]dt + Z(t)dB(t); 0 ≤ t ≤ T Y (T ) = U 2 (X(T )). (4.19) Put R = p(T ) = λ(T )U ′ 2 (X(T )). (4.20)
Then (4.15) can be written

b 0 (t)E[R | F t ] + σ 0 (t)E[D t R | F t ] = 0. (4.21)
Following [START_REF] Øksendal | Maximum principles for optimal control of forward-backward stochastic differential equations with jumps[END_REF] we call this a Malliavin-differential type equation in the unknown random variable R. By Theorem A.1 in [START_REF] Øksendal | Maximum principles for optimal control of forward-backward stochastic differential equations with jumps[END_REF], the general solution of this equation is R = R β (T ); where

R β (t) = βΓ(t); 0 ≤ t ≤ T, (4.22)
for some constant β, where Γ(t) is defined in (4.2). Note that R β (t) is a martingale. Hence since p(T ) = R β (T ), we get by (4.13) that

p(t) = R β (t); 0 ≤ t ≤ T. (4.23)
Modulo the unknown constant β we can now find the optimal terminal wealth X β (T ) by (4.20) as follows:

X β (T ) = I 2 ( βΓ(T ) λ(T ) ), (4.24)
Similarly the optimal consumption rate is, by (4.14),

c(t) = c β (t) = I 1 (t, βΓ(t) λ(t) ); 0 ≤ t ≤ T (4.25)
The optimal scenario parameter is, by (4. 16)

θ(t) = θ β (t) = (ρ ′ ) -1 (-Z β (t)); 0 ≤ t ≤ T (4.26)
where (Y β (t), Z β (t)) is the solution of the corresponding BSDE (4.19), i.e.

dY β (t) = -[U 1 (t, c β (t)) + ρ(θ(t)) + θ(t)Z β (t)]dt + Z β (t)dB(t); 0 ≤ t ≤ T Y β (T ) = U 2 (I 2 ( βΓ(T ) λ(T ) )). (4.27) RR n°7776
Forward-backward SDE games and stochastic control under model uncertainty 19

Let us consider the case when

U 1 = c = 0 (no consumption) and ρ(θ) = 1 2 θ 2 . ( 4
   dY β (t) = 1 2 θ 2 (t)dt -θ(t)dB(t) ; 0 ≤ t ≤ T Y β (T ) = U 2 (I 2 ( βΓ(T ) λ(T ) )).
(4.29)

Integrating (4.29), and using (4.12) at t = T , we get

- 1 2 T 0 θ 2 (s)ds + T 0 θ(s)dB(s) = Y β (0) -U 2 (I 2 ( β Γ(T ) λ(T ) )). (4.30)
Taking exponentials in (4.30) we obtain

λ(T ) = exp T 0 θ(s)dB(s) - 1 2 T 0 θ 2 (s)ds = exp Y β (0) exp(U 2 (I 2 ( β Γ(T ) λ(T ) )))
.

(4.31) Therefore λ(t) is given as the solution of the BSDE (or more precisely SDE with terminal condition)

dλ(t) = λ(t)θ(t)dB(t) ; 0 ≤ t ≤ T λ θ (T ) = L (4.32)
where L = L(β, Y β (0)) is the solution of the equation:

L exp(U 2 (I 2 ( β Γ(T ) L ))) = exp Y β (0). (4.33)
By the generalized Clark-Ocone formula [START_REF] Aase | White noise generalizations of the Clark-Haussmann-Ocone theorem, with application to mathematical finance[END_REF] this gives

λ(t)θ(t) = E[D t L | F t ] ; 0 ≤ t ≤ T. ( 4
dλ(t) = E[D t L | F t ]dB(t) ; 0 ≤ t ≤ T λ(0) = 1 (4.35)
and

θ(t) = E[D t L | F t ] λ(t) ; 0 ≤ t ≤ T. (4.36) RR n°7776
Forward-backward SDE games and stochastic control under model uncertainty 20

Note that E[L] = 1 by the martingale property of λ(t).

It remains to determine β and Y β (0). To this end, we consider the equation (4.18) for X(t) as a BSDE as follows:

Put

Zβ (t) = π(t)σ 0 (t).
Then

π(t) = Zβ (t) σ 0 (t) (
X(T ) = I 2 ( β Γ(T ) L ). (4.39)
The solution of this linear BSDE is

X(t) = E[I 2 ( β Γ(T ) L ) exp( T t - 1 2 ( b 0 (s) σ 0 (s) ) 2 ds - T t b 0 (s) σ 0 (s) dB(s)) | F t ] = E[I 2 ( β Γ(T ) L ) Γ(T ) Γ(t) | F t ]. (4.40)
In particular, putting t = 0, we get

x = E[I 2 ( β Γ(T ) L )Γ(T )]. (4.41)
Finally, by taking expectation in (4.30), we deduce that

Y β (0) = E U 2 (I 2 ( β Γ(T ) L )) - 1 2 T 0 θ 2 (s))ds (4.42)
which, together with (4.41) gives the value of β and the solution Y β (0) = Y π, θ(0) of (3.11).

We summarize what we have proved Theorem 4.1 Consider the problem to find (π, θ) such that

sup π∈A 1 inf θ∈A 2 E Q θ [W (v, θ)] = E Q θ [W (π, θ)] = inf θ∈A 2 sup v∈A 1 E Q θ [W (π, θ)], (4.43)
with

W (π, θ) = U 2 (X π (T )) + T 0 θ(t) 2 dt (4.44) RR n°7776
where

dX(t) = π(t)[b 0 (t)dt + σ 0 (t)dB(t)], 0 ≤ t ≤ T X(0) = x > 0. (4.45)
This problem is equivalent to

sup π∈A 1 inf θ∈A 2 Y π,θ (0) = Y π, θ(0) = inf θ∈A 2 sup π∈A 1 Y v,θ (0), (4.46)
where Y = Y π,θ is given by

dY (t) = -[ 1 2 θ(t) 2 + θ(t)Z(t)]dt + Z(t)dB(t); 0 ≤ t ≤ T Y (T ) = U 2 (X(T )). (4.47)
Then, the optimal scenario parameter is

θ = E[D t L | F t ](1 + t 0 E[D s L | F s ]dB(s)) -1 .
The optimal portfolio π is given by

π = D t X(t) σ 0 (t)
where X(t) is the optimal state process given by (4.40), with β and Y β (0) given by (4.41)-(4.42) with θ = θ, and hence L = L(β, Y β (0)) given by (4.33).

Proof. The argument above shows that, by the necessary maximum principle (Theorem 3.1), if there is an optimal pair (π, θ), then it is given as in the theorem.

Conversely, if we define (π, θ) as in the theorem, we can show that (π, θ) must be optimal, as follows:

Fix an arbitrary π ∈ A 1 in the BSDE (4.47). Then, proceeding as in [START_REF] Øksendal | Portfolio optimization under model uncertainty and BSDE games[END_REF], by the comparison theorem for BSDEs, we obtain the minimal value Y π, θ(0) and its minimizer θ simply by minimizing the driver of (4.47), i.e. by minimizing for each t and ω the function:

θ → 1 2 θ 2 + θZ(t).
This gives

θ(t) = -Z(t), (4.48)

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Forward-backward SDE games and stochastic control under model uncertainty 22 which is identical to (4.16). Substituting this into (4.47), we have reduced the original game problem to the following FBSDE control problem:

Find π ∈ A 1 such that sup π∈A 1 Y π (0) = Y π(0), (4.49)
where

dY π (t) = 1 2 Z(t) 2 dt + Z(t)dB(t); 0 ≤ t ≤ T Y π (T ) = U 2 (X π (T )) (4.50)
and X π (t) given in (4.45). This problem is of the type discussed in [START_REF] Øksendal | Maximum principles for optimal control of forward-backward stochastic differential equations with jumps[END_REF]. If we apply the sufficient maximum principle (Theorem 2.3) of that paper, we get that the optimal π is given as the maximizer π of the associated Hamiltonian:

H 0 (t, x, y, z, π, λ, p, q)
:= - 1 2 λz 2 + π(p b 0 (t) + qσ 0 (t)). (4.51)
This gives the equation

p(t) b 0 (t) + q(t)σ 0 (t) = 0, (4.52)
which is (4.15). Moreover, again by Theorem 2.3 in [START_REF] Øksendal | Maximum principles for optimal control of forward-backward stochastic differential equations with jumps[END_REF], the equation for the associated process λ(t) is

dλ(t) = -Z(t)λ(t)dB(t) = λ(t)θ(t)dB(t), (4.53) λ
(0) = 1 (4.54)
which is (4.12). We conclude that, since the pair (π, θ) of Theorem 4.1 does indeed satisfy the sufficient conditions (4.48), (4.52), and (4.53), it also satisfies all the conditions of the sufficient maximum principle of Theorem 2.3 in [START_REF] Øksendal | Maximum principles for optimal control of forward-backward stochastic differential equations with jumps[END_REF] and hence the pair is optimal.

The logarithmic utility case. In this case, substituting U 2 (x) = ln x and I 2 (x) = 1 x in the general formulas above, we get:

β = 1 x (4.55) L = Γ(T ) 1/2 E[Γ(T ) 1/2 ] (4.56) θ = E[D t L | F t ](1 + t 0 E[D s L | F s ]dB(s)) -1 (4.57) Y β (0) = ln x + E T 0 ( 1 2 ( b 0 (s) σ 0 (s) ) 2 -θ 2 (s))ds (4.58) X(t) = x E[Γ(T ) 1/2 | F t ] E[Γ(T ) 1/2 ]Γ(t) . ( 4

.59)

The case with no model uncertainty. In this case, θ = 0 and λ = 1 and the problem reduces to maximizing

Y (0) = E[ T O U 1 (t, c(t))dt + U 2 (X(T ))]
which is a classical optimal portfolio/consumption problem. Then the optimal terminal wealth X(T ) is given by :

X β (T ) = I 2 (βΓ(T ))
and by (4.25), and the optimal consumption rate c(t) is given by c β (t) = I 1 (t, βΓ(t)).

To find the unknown β, we consider the equation (4. Forward-backward SDE games and stochastic control under model uncertainty 24

The solution of this linear BSDE is

X(t) = E[I 2 (β.Γ(T )) Γ(T ) Γ(t) + T t Γ(s) Γ(t) I 1 (s, β.Γ(s))ds | F t ].
Putting t = 0, we get

x = E[I 2 (βΓ(T ))Γ(T ) + T 0 Γ(s)I 1 (s, βΓ(s))ds]
and this equation determines β. We thus recover by a completely different method the results obtained by the classical martingale method, (see [START_REF] Karatzas | Methods of Mathematical Finance[END_REF], Chapter 3).

A Proofs of the maximum principles for FB-SDE games

We first recall some basic concepts and results from Banach space theory. Let

V be an open subset of a Banach space X with norm • and let F : V → R.

(i) We say that F has a directional derivative (or Gâtaux derivative) at x ∈ X in the direction y ∈ X if

D y F (x) := lim ε→0 1 ε (F (x + εy) -F (x))
exists.

(ii) We say that F is a Fréchet differentiable at x ∈ V if there exists a linear map

L := X → R such that lim h→0 h∈X 1 h |F (x + h) -F (x) -L(h)| = 0.
In this case we call L the gradient (or Fréchet derivative) of F at x and we write L = ∇ x F.

(iii) If F is Fréchet differentiable, then F has a directional derivative in all directions y ∈ X and

D y F (x) = ∇ x F (y).
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I 2 ≤ E[ϕ ′ 1 ( X(T ))(X(T ) -X(T ))] = E[p 1 (T )(X(T ) -X(T ))] -E[ λ1 (T )h ′ 1 ( X(T ))(X(T ) -X(T ))] = E T 0 p1 (t -)(dX(t) -d X(t)) + T 0 (X(t -) -X(t -))dp 1 (t) + T 0 q1 (t)(σ(t) -σ(t))dt + T 0 R r1 (t, ζ)(γ(t, ζ) -γ(t, ζ))ν(dζ)dt -E[ λ1 (T )h ′ 1 ( X(T ))(X(T ) -X(T ))] = E T 0 p1 (t)(b(t) -b(t))dt + T 0 (X(t) -X(t)) - ∂ Ĥ1 ∂x (t) dt + T 0 q1 (t)(σ(t) -σ(t))dt + T 0 R r1 (t, ζ)(γ(t, ζ) -γ(t, ζ))ν(dζ)dt -E[ λ1 (T )h ′ 1 ( X(T ))(X(T ) -X(T ))]. (A.6)

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Forward-backward SDE games and stochastic control under model uncertainty 27

By concavity of ψ 1 , (2.8), and concavity of ϕ 1 :

I 3 = E[ψ 1 (Y 1 (0)) -ψ 1 ( Ŷ1 (0))] ≤ E[ψ ′ 1 ( Ŷ1 (0))(Y 1 (0) -Ŷ1 (0))] = E[ λ1 (0)(Y 1 (0) -Ŷ1 (0))] = E[(Y 1 (T ) -Ŷ1 (T )) λ1 (T )] -E T 0 (Y 1 (t -) -Ŷ1 (t -))d λ1 (t) + T 0 λ1 (t -)(dY 1 (t) -d Ŷ1 (t)) + T 0 ∂ Ĥ1 ∂z (t)(Z 1 (t) -Ẑ1 (t))dt + T 0 R ∇ k Ĥ1 (t, ζ)(K 1 (t, ζ) -K1 (t, ζ))ν(dζ)dt = E[(h 1 (X(T )) -h 1 ( X(T ))) λ1 (T )] -E T 0 ∂ Ĥ1 ∂y (t)(Y 1 (t) -Ŷ1 (t))dt + T 0 λ1 (t)(-g 1 (t) + ĝ1 (t))dt + T 0 ∂ Ĥ1 ∂z (t)(Z 1 (t) -Ẑ1 (t))dt + T 0 R ∇ k Ĥ1 (t, ζ)(K 1 (t, ζ) -K1 (t, ζ))ν(dζ)dt ≤ E[ λ1 (T )h ′ 1 ( X(T ))(X(T ) -X(T ))] -E T 0 ∂ Ĥ1 ∂y (t)(Y 1 (t) -Ŷ1 (t))dt + T 0 λ1 (t)(-g 1 (t) + ĝ1 (t))dt + T 0 ∂ Ĥ1 ∂z (t)(Z 1 (t) -Ẑ1 (t))dt + T 0 R ∇ k Ĥ1 (t, ζ)(K 1 (t, ζ) -K1 (t, ζ))ν(dζ)dt . (A.7)
Adding (A.5), (A.6) and (A.7) we get

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Forward-backward SDE games and stochastic control under model uncertainty 28

∆ = I 1 + I 2 + I 3 ≤ E T 0 H 1 (t) -Ĥ1 (t) - ∂ Ĥ1 ∂x (t)(X(t) -X(t)) - ∂H 1 ∂y (t)(Y 1 (t) -Ŷ1 (t)) - ∂ Ĥ1 ∂z (t)(Z 1 (t) -Ẑ1 (t)) - R ∇ k Ĥ1 (t, ζ)(K 1 (t, ζ) -K1 (t, ζ))ν(dζ) dt (A.8)
Since ĥ1 (x, y, z, k) is concave, it follows by a standard separating hyperplane argument (see e.g. [START_REF] Rockafellar | Convex analysis[END_REF], Chap.5, Sect. 23) that there exists a supergradient a = (a 0 , a 1 , a 2 , a 3

(•)) ∈ R 3 × R for ĥ1 (x, y, z, k) at x = X(t), y = Ŷ1 (t), z = Ẑ1 (t -) and k = K1 (t -, •) such that if we define ϕ 1 (x, y, z, k) := ĥ1 (x, y, z, k) -ĥ1 ( X(t -), Ŷ1 (t -), Ẑ1 (t -), K1 (t, •)) -[a 0 (x -X(t)) + a 1 (y -Ŷ1 (t)) + a 2 (z -Ẑ1 (t)) + R a 3 (ζ)(k(ζ) -K(t, ζ))ν(dζ)
then ϕ 1 (x, y, z, k) ≤ 0 for all x, y, z, k.

On the other hand we clearly have

ϕ 1 ( X(t), Ŷ (t), Ẑ(t), K1 (t, •)) = 0. It follows that ∂ Ĥ1 ∂x (t) = ∂ ĥ1 ∂x ( X(t), Ŷ1 (t), Ẑ1 (t), K1 (t, •)) = a 0 ∂ Ĥ1 ∂y (t) = ∂ ĥ1 ∂y ( X(t), Ŷ1 (t), Ẑ1 (t), K1 (t, •)) = a 1 ∂ Ĥ1 ∂z (t) = ∂ ĥ1 ∂z ( X(t), Ŷ1 (t), Ẑ1 (t), K1 (t, •)) = a 2 ∇ k Ĥ1 (t, ζ) = ∇ k ĥ1 ( X(t), Ŷ1 (t), Ẑ1 (t), K1 (t, •)) = a 3 .
Combining this with (A.8) we get

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Forward-backward SDE games and stochastic control under model uncertainty 29

∆ ≤ ĥ1 (X(t), Y 1 (t), Z 1 (t), K 1 (t, •)) -ĥ1 ( X(t), Ŷ1 (t), Ẑ1 (t), K1 (t, •)) - ∂ ĥ1 ∂x ( X(t), Ŷ1 (t), Ẑ1 (t), K1 (t, •))(X(t) -X(t)) - ∂ ĥ1 ∂y ( X(t), Ŷ1 (t), Ẑ1 (t), K1 (t, •))(Y 1 (t) -Ŷ1 (t)) - ∂ ĥ1 ∂z ( X(t), Ŷ1 (t), Ẑ1 (t), K1 (t, •))(Z 1 (t) -Ẑ1 (t)) - R ∇ k ĥ1 ( X(t), Ŷ1 (t), Ẑ1 (t), K1 (t, •))(K 1 (t, ζ) -K1 (t, ζ))ν(dζ) ≤ 0 since ĥ1 is concave. Hence J 1 (u 1 , û2 ) ≤ J 1 (û 1 , û2 ) for all u 1 ∈ A 1 .
The inequality J 2 (û 1 , u 2 ) ≤ J 2 (û 1 , û2 ) for all u 2 ∈ A 2 is proved similarly. This completes the proof of Theorem 2.1.

Proof of Theorem 2.3 (Necessary maximum principle) Consider

D 1 := d ds J 1 (u 1 + sβ 1 , u 2 ) | s=0 = E T 0 ∂f 1 ∂x (t)x 1 (t) + ∂f 1 ∂u 1 (t)β 1 (t) dt + ϕ ′ 1 (X (u 1 ,u 2 ) (T ))x 1 (T ) + ψ ′ 1 (Y 1 (0))y 1 (0) . (A.9)
By (2.9), (2.12) and the Itô formula,

RR n°7776

Forward-backward SDE games and stochastic control under model uncertainty 30

E[ϕ ′ 1 (X (u 1 ,u 2 ) (T ))x 1 (T )] = E[p 1 (T )x 1 (T )] -E[h ′ 1 (X (u 1 ,u 2 ) (T ))λ 1 (T )] = E T 0 p 1 (t -)dx 1 (t) + x 1 (t -)dp 1 (t) + q 1 (t) ∂σ ∂x (t)x 1 (t) + ∂σ ∂u 1 (t)β 1 (t) dt + R r 1 (t, ζ) ∂γ ∂x (t, ζ)x 1 (t) + ∂γ ∂u 1 (t, ζ)β 1 (t, ζ) ν(dζ)dt -E[h ′ 1 (X (u 1 ,u 2 ) (T ))λ 1 (T )] = E T 0 p 1 (t) ∂b ∂x (t)x 1 (t) + ∂b ∂u 1 (t)β 1 (t) + x 1 (t) - ∂H 1 ∂x (t) + q 1 (t) ∂σ ∂x (t)x 1 (t) + ∂σ ∂u 1 (t)β 1 (t) + R r 1 (t, ζ) ∂γ ∂x (t, ζ)x 1 (t) + ∂γ ∂u 1 (t, ζ)β 1 (t, ζ) ν(dζ) dt -E[h ′ 1 (X (u 1 ,u 2 ) (T ))λ 1 ((T )]. (A.10)
By (2.8), (2.12) and the Itô formula

E[ψ ′ 1 (Y 1 (0))y 1 (0)] = E[λ 1 (0)y 1 (0)] = E[λ 1 (T )y 1 (T )] -E T 0 {λ 1 (t -)dy 1 (t) + y 1 (t -)dλ 1 (t) + ∂H 1 ∂z (t)z 1 (t)dt + R ∇ k H 1 (t, ζ)k 1 (t, ζ)ν(dζ)dt = E[λ 1 (T )h ′ 1 (X (u 1 ,u 2 ) (T ))] -E T 0 λ 1 (t) - ∂g 1 ∂x (t)x 1 (t) - ∂g 1 ∂y (t)y 1 (t) - ∂g 1 ∂z (t)z 1 (t) - R ∇ k g 1 (t, ζ)k 1 (t, ζ)ν(dζ) - ∂g 1 ∂u 1 (t)β 1 (t) + ∂H 1 ∂y (t)y 1 (t) + ∂H 1 ∂z (t)z 1 (t) + R ∇ k H 1 (t, ζ)k 1 (t, ζ)ν(dζ) dt . (A.11)
Adding (A.10) and (A.11) we get, by (A.9),

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Forward-backward SDE games and stochastic control under model uncertainty 31 (B.4)

D 1 = E T 0 ∂f 1 ∂x (t) + p 1 (t) ∂b ∂x (t) + q 1 (t) ∂σ ∂x (t) + R r 1 (t,

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Forward-backward SDE games and stochastic control under model uncertainty 33

Hence Υ(t, s) = Υ(0, s) Υ(0, t) , Υ(t, T ) = Υ(0, T ) Υ(0, t) .

Proof. For completeness we give the proof, but it is also given in [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF]. Existence and uniqueness follow by general theorems for BSDEs with Lipschitz coefficients. See e.g. [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF]. Hence it only remains to prove that if we define Y (t) to be the solution of (B.1), then (B.2) holds. To this end, define Υ(s) = Υ(0, s).

Then by the Itô formula (see e.g. [START_REF] Øksendal | Applied Stochastic Control of Jump Diffusions[END_REF], Ch.

ln( 1 +

 1 ξ 1 (u)) Ñ (du, dζ)).

  ζ) are defined by (2.1) and (2.3). In this case only one Hamiltonian is needed, namely (see (2.7)):

  .14) Suppose that the conditions of Theorem 2.3 hold. Let (v, θ) ∈ A

	Forward-backward SDE games and stochastic control under model uncertainty 15
	Here we have used the abbreviated notation
	∂H ∂y	(t) =	∂H ∂y	(t, X(t), Y (t), Z(t), K(t, •), v(t), θ(t), λ(t), p(t), q(t), r(t, •))
	and similarly for the other partial derivatives. We now present a necessary
	maximum principle for the forward-backward stochastic differential game
	(3.1), (3.9), (3.11) by adapting Theorem 2.4 to this case.
	Theorem 3.1 E[ λ(t)	∂U 1 ∂v	(t, X(t), v(t)) + p(t)	∂b ∂v	(t, X(t), v(t))
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1 ×A 2 , with corresponding solutions X(t), Ŷ (t), Ẑ(t), K(t, •), λ(t), p(t), q(t), r(t, •) of equations (3.1), (3.9), (3.14) and (3.13). Suppose (3.11) holds, together with (2.12). Then the following holds:

  If D 1 = 0 for all bounded β 1 ∈ A 1 , then this holds in particular for β 1 of the form in (a1), i.e.β 1 (t) = χ (t 0 ,T ] (t)α 1 (ω),where α 1 (ω) is bounded andE )α 1 = 0 for a.a. t 0 . (u 1 , u 2 + sβ 2 ) | s=0 = 0 for all bounded β 2 ∈ A 2 .This shows that (i) ⇒ (ii). The argument above can be reversed, to give that (ii) ⇒ (i). We omit the details.B Linear BSDEs with jumpsLemma B.1 [Linear BSDEs with jumps]. Let F be a F T -measurable and square-integrable random variable. Let β and ξ 0 be bounded predictable processes and ξ 1 a predictable process such that ξ 1 (t, ζ) ≥ C 1 with C 1 > -1 and |ξ 1 (t, ζ)| ≤ C 2 (1 ∧ |ζ|) for a constant C 2 ≥ 0. Let ϕ be a predictable process such that E[

	Forward-backward SDE games and stochastic control under model uncertainty 32
	A similar argument gives that
						E	∂H 2 ∂u 2	(t) | E t (2)	= 0
	ζ) (t) + λ 1 (t) ∂γ ∂x (t, ζ)ν(dζ) -∂g 1 ∂y (t) y 1 (t) ∂H 1 ∂x (t) + λ 1 (t) ∂g 1 ∂z (t) z 1 (t) (t) + p 1 (t) ∂H 1 ∂y ∂H 1 ∂z ∂f 1 + -provided that + -+ + ∂u 1 ∂b ∂u 1 (t) + q 1 (t) ∂σ (t) + λ 1 (t) ∂u 1 (t) + R r 1 (t, ζ) ∂γ ∂u 1 (t, ζ)ν(dζ) + ∂g 1 ∂u 1 (t) β 1 (t) dt ∂g 1 ∂x = E T 0 ∂H 1 ∂u 1 (t)β 1 (t)dt = E T 0 E ∂H 1 ∂u 1 (t)β 1 (t) | E (1) t dt . (1) t 0 -measurable. Hence E T t 0 E ∂H 1 ∂u 1 (t) | E (1) t α 1 dt = 0. Differentiating with respect to t 0 we get E ∂H 1 ∂u 1 D 2 := d ds J 2 Υ(t, t) = 1 i.e. (t 0 Since this holds for all bounded E (1) t 0 -measurable random variables α 1 we (t) x 1 (t) (A.12) (B.3) conclude that Υ(t, s) = exp( s t {β(u) -s 1 2 ξ 2 ξ 0 (u)dB(u) 0 (u)}du + t s
	+	t	R	E	∂H 1 ∂u 1	(t) | E t (1)
	RR n°7776

R [-∇ k H 1 (t, ζ) + λ 1 (t)∇ k g 1 (t, ζ)] k 1 (t, ζ)ν(dζ) = 0 for a.a. t ∈ [0, T ]. T 0 ϕ 2 (t)dt] < ∞. Then the linear BSDE dY (t) = -[ϕ(t) + β(t)Y (t) + ξ 0 (t)Z(t) + R ξ 1 (t, ζ)K(t, ζ)ν(dζ)]dt + Z(t)dB(t) + R K(t, ζ) Ñ (dt, dζ); 0 ≤ t ≤ T Y (T ) = F (B.1)

has the unique solution

Y (t) = E[F Υ(t, T ) + T t Υ(t, s)ϕ(s)ds | F t ]; 0 ≤ t ≤ T,

(B.2)

where Υ(t, s); 0 ≤ t ≤ s ≤ T ; is defined by

dΥ(t, s) = Υ(t, s -)[β(s)ds + ξ 0 (s)dB(s) + R ξ 1 (s, ζ) Ñ (ds, dζ)]; t ≤ s ≤ T

{ln(1 + ξ 1 (u))ξ 1 (u)}ν(dζ)du +

RR n°7776

Éditeur Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

Acknowledgments. We thank Olivier Menoukeu Pamen and Marie-Claire Quenez for helpful comments.

no and Norwegian School of Economics and Business Administration (NHH)The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no [228087].

Proof of Theorem 2.1 (Sufficient maximum principle). We first prove that

To this end, fix u 1 ∈ A 1 and consider

where

2)

3)

By (2.7) we have


Micky Rakotondrabe

Ioan Alexandru Ivan
email: ivan@valahia.ro

Valentin Stihi
email: vstihi@yahoo.com

Simona Noveanu
email: simona.noveanu@mmfm.utcluj.ro

Eugénia Minca
email: minca@valahia.ro

Eugenia Minca

DESIGN AND MODELING OF A PIEZOELECTRICALLY ACTUATED MICROVALVE *

Keywords: piezoelectric, actuator, microvalve, dynamic modeling

teaching and research institutions in France or abroad, or from public or private research centers.

Design and modeling of a piezoelectrically actuated microvalve.

INTRODUCTION

Electrovalves are usually actuated by solenoids. Their response time is often too big for fast switching applications and they offer discrete output only [START_REF] Nesbitt | Guide to European valves: for control, isolation and safety[END_REF]. Using geared motors a continuous flow control is achieved but with the inconvenient of an even bigger idle time. Miniaturization of these valves is also technically challenging. Pneumatic valves [START_REF] Beater | Drives: System Design, Modelling and Control[END_REF] present similar challenges. Piezoelectrically driven valves began to develop as a viable alternative. Piezoceramic materials such as PZT (lead zirconate titanate) [START_REF] Moulson | Electroceramics: Materials, Properties, Applications 2 nd Edition[END_REF] provide high energy density and electromechanical coupling factor. Their bandwidth allows response time in millisecond order and extremely low power consumption under one milliwatt. Finally, the high resolution that piezoelectric materials provide is very compatible to the required accuracy in microvalves.

In this paper, we propose a new principle of microvalve based on piezoelectric PZT actuator. The objective is to obtain a very accurate and with a very high settling time. In order to understand the behavior of the whole system, we developed a model. While the developed model is strongly nonlinear, it has been simulated for different input conditioning.

The paper is organized as follows. In Section-2, we present the piezoelectrically actuated microvalve. Section-3 is dedicated to the modeling of the while system.

PRINCIPLE OF THE PIEZOELECTRICALLY ACTUATED MICROVALVE

Let us consider a tube whose fluid outlets to an end-use vat. The objective is to control the flow 3 / o qms   that comes into the vat. The natural way to adjust it is to vary the opening at the extremity of the tube. To perform that, we propose to use an end-effector (steel) actuated by a piezoelectric cantilever (piezocantilever) (Fig. 1-a). Initially, while the piezocantilever is not electrically excited, the endeffector obstructs the tube and no flow goes out from it, i.e.

MODELING OF THE SYSTEM

This section is dedicated to provide the behavior equations of the system. The considered input control is the voltage applied to the piezocantilever, while the output is the flow o q . The modeling is separated into three parts. First we model the piezocantilever. Afterwards, we give the equation that the end-effector plays. Then, we give the equation of the fluid flow part. Finally, we give the governing equations of the whole system.

EQUATION OF THE PIEZOCANTILEVER

A piezocantilever is made up of one or more piezoelectric layers and one or more passive layers. Unimorph piezocantilevers, which are composed of one piezolayer and one passive layer, are appreciated because of their ease of mass production. In this study, we particularly use a PZT-layer and Copper-layer. When an external voltage is applied to the PZT-layer, it expands (or contracts) that leads to a bending [ ] µm δ of the piezocantilever (Fig. 3). 1) represent the static model of the piezocantilever. It has been shown that a second-order model is well adapted to approximate the dynamic part of such an actuator [START_REF] Rakotondrabe | Complete open-loop control of hysteretic, creeped and oscillating piezoelectric cantilever[END_REF]. Introducing the dynamic part, we finally have: While δ and α can be considered as internal states of the actuator, the momentum M is unknown. The next subsection is focused on the expression of M by using the geometrical modeling of the end-effector. The expression of the momentum is:

+ = + ( ) ( ) ( ) () ( ) ()( ) ()
( ) na t a M FL F L w ≈-+ ( 3 )
where the tangential and normal components of the force are:

( )

sin t FF α = and
( )

cos n FF α = respectively.
On the other hand, based on the Fig. 4-b, the distance a L can be expressed as follows:

() () ( )

1 sin cos aa o a LL w δ α α =+ + ( 4 )
Inserting Eq. ( 4) and the components of the force into Eq. ( 3), we derive the expression of the momentum:

( ) (
)

sin ao M LL F δα ≈+ - ( 5 )
The force F is unknown, it can be expressed by calculating the one applied by the fluid on the end-effector which is called jet force. This is the aim of the next subsection.

EQUATION OF THE FORCE BY MODELING THE JET FORCE

Consider the Fig. 5 where the interaction between the end-effector and the fluid from the tube is zoomed. The diameter of the tube is Using dynamic fluid motion law (see for example [START_REF] Durst | Fluid Mechanics, an introduction to the theory of fluid flows[END_REF]), we have:

( ) ( ) cos AL a A A A qv v p S F α - =-∞ ( 6 )
where

2 4 A A d S π =
is the section of the tube at its extremity. Replacing La v by its expression and a L by Eq. ( 4), we deduce the force:

() ( ) sin A AA a o a A d Fp S q L w v dt α δα  =- + + -   ( 7 )

EXPRESSION OF THE EFFECTIVE OUTPUT FLOW o q

The effective output flow o q is considered as the fluid that comes out from the tube. It depends on the opening gap s d between the tube and the end-effector.

We have:

oo A qS v = ( 8 )
tan sin oA a o a qd L w π αδ α =+ + have: ( ) ( ) ( )
4. SIMULATION

THE FINAL MODEL

Based on the previous section, the final model that governs the piezoelectrically actuated microvalve is given by the following strongly nonlinear dynamic model to be introduced into Simulink: ( The state matrix X is available either by sensors or by using a state observer [START_REF] Luenberger | Observing the state of a linear system[END_REF]. 148

SIMULATION RESULTS

Actuator geometry insures that displacement and flow values are negative for positive applied voltages. At null voltage but for a given non-zero fluid pressure the model correctly provide a flow offset value. This can be avoided by a reverse voltage or by pre-compressing the end-effector. A control system would better work in closed-loop, incorporating at least one external sensor for hydrostatic pressure or directly for flow rate. Based on above geometry and input parameters, second-order simulation model provided that settling time is 20 ms. Static flow rate is -0.0228 mm 3 /V at 1600 Pa and related mechanical opening is 0.7mm/V. At null input voltage there is a constant flow of 15.6mm 3 /bar; a reverse voltage of 2.994 mV/Pa is required to compensate the hydrostatic pressure.

CONCLUSION

We proposed a fast piezoelectrically driven valve for controlling small fluid flows in a range of up to several cubic millimeter per second. Expected resolution may reach several dozens picoliters per second, depending on outlet diameter.

However we must admit that we used several simplifying hypothesis that should be further refined in the model, such as: ideal fluid (laminar flow, uncompressible, no capillarity etc.), infinite end-effector stiffness, and linear piezoelectric material. Mechanical structure may be optimized with an appropriate damping to avoid actual significant flow overshoot and further improve response time.

0 oq

 0 = . When a voltage is applied to the piezocantilever, it bends and the end-effector effectuates an angle [ ] rad α allowing the output flow as in Fig. 1-b. Fig. 2 pictures the design of the piezoelectrically actuated microvalve.

Fig. 1 .

 1 Fig. 1. Principle of the system.

Fig. 2 .

 2 Fig. 2. Design of the piezoelectrically-actuated microvalve.

Fig. 3 .

 3 Fig. 3. A unimorph piezocantilever subjected to a voltage excitation. Smits and Choi [4] provided static equations for the unimorph piezocantilever tip deflection δ and the angle α for a given voltage U and a mechanical momentum M : 11 12 21 22 mM mU mM mU α δ =+   =+ 

  inertial and damping coefficients respectively.

3. 2 .

 2 Fig. 4 pictures a simplified scheme of the piezocantilever and related endeffector. Initially, the outlet of the tube is at the distance ao L of the tube axis. When

Fig. 4 .

 4 Fig. 4. Simplified scheme of the system.

  a d . At its extremity, the flow, the pressure and the speed of the fluid are denoted by A q , A p and A v respectively. They are supposed as known. When the end-effector turns, the tangential speed at distance a distance between the tube and the end-effector is denoted by s d .

Fig. 5 .

 5 Fig. 5. Zoom of interaction between the end-effector and the fluid motion from the tube.

  area of the fluid between the tube and the end-

Fig. 6 Fig. 6 .

 66 Fig. 6 pictures the systemic scheme of Eq. (8). It describes how the different coupling and effects are connected. In the figure, the term d dt α can be derived either from the derivative of α or by multiplication the state vector X with a matrix d C α such as:

Fig. 7 .

 7 Fig. 7. Flow rate of a step response of an input signal of -40V.

Piezoelectrically actuated microvalve