Homogenization and kinetic models in extended phase-space

Keywords: ] Linear Boltzmann equation, Periodic homogenization, Extended phase-space, Renewal equation. Mathematics Subject Classification (2000): Primary 82C70, 35B27; Secondary 82C40, 60K05 1 -Introduction

This paper reviews recent results obtained in collaboration with E. Bernard and E. Caglioti on the homogenization problem for the linear Boltzmann equation for a monokinetic population of particles set in a periodically perforated domain, assuming that particles are absorbed by the holes. We distinguish a critical scale for the hole radius in terms of the distance between neighboring holes, derive the homogenized equation under this scaling assumption, and study the asymptotic mass loss rate in the long time limit. The homogenized equation so obtained is set on an extended phase space as it involves an extra time variable, which is the time since the last jump in the stochastic process driving the linear Boltzmann equation. The present paper proposes a new proof of exponential decay for the mass which is based on a priori estimates on the homogenized equation instead of the renewal theorem used in [Bernard-Caglioti-Golse, SIAM J. Math. Anal.

The present paper will discuss the simplest problem leading to such phenomena, i.e. the evolution of a gas of monokinetic point particles governed by a linear Boltzmann equation in a periodic distribution of holes. Particles falling into the holes are removed from the particle system, which therefore loses mass, and this suggests the following question:

To find the appropriate scaling for which the mass loss per unit of time is of order unity, and to determine the mass loss rate in that case.

As we shall see below, the solution of this seemingly simple homogenization problem is surprisingly intricate, at variance with the analogous periodic homogenization problem for a diffusion process, which leads to a conceptually simpler result.

-The diffusion case

As a warm-up, we briefly recall the answer to this question when the linear Boltzmann equation is replaced with a diffusion equation.

Let ǫZ N designate the cubic lattice with mesh size ǫ, and consider the Euclidian space with balls of radius r ǫ ∈ (0, ǫ/2) centered at lattice points removed:

Ω[ǫ, r ǫ ] = {x ∈ R N | dist (x, ǫZ N ) > r ǫ } .
Consider next the Dirichlet problem with unknown u ǫ ≡ u ǫ (x) ∈ R:

λu ǫ -∆u ǫ = f , x ∈ Ω[ǫ, r ǫ ] , u ǫ ∂Ω[ǫ,rǫ] = 0 ,
where f ∈ L 2 (R N ) and λ > 0 are given.

For this problem, there is a natural notion of critical size of the obstacles:

r ǫ = rǫ γc , with γ c = N N -2 whenever N > 2 .
The exponent γ c is critical in the following sense:

if r ǫ ≪ ǫ γc , one expects that u ǫ → (λ -∆) -1 f , if r ǫ ≫ ǫ γc , one expects that u ǫ → 0.

In the critical case, L'vov-Khruslov [START_REF] Khruslov | Perturbation of a viscous incompressible fluid by small particles[END_REF] and Cioranescu-Murat [START_REF] Cioranescu | Un terme étrange venu d'ailleurs[END_REF] proved that T h e o r e m 2.1. If r ǫ ∼ rǫ γc as ǫ → 0 + with γ c = N N -2 for N > 2, for each f ∈ L 2 (R N ), the solution u ǫ of the Dirichlet problem

λu ǫ -∆u ǫ = f , x ∈ Ω[ǫ, r ǫ ] , u ǫ ∂Ω[ǫ,rǫ] = 0 ,
extended by 0 in the holes, converges in L 2 (R N ) to the solution u of

-∆u + (λ + c)u = f , x ∈ R N ,
where c > 0 is a positive number (limiting absorption rate) independent of the data f . In other words, the effect of the periodic system of holes at the critical size on the limiting equation is given by the enhanced damping coefficient λ + cinstead of λ as in the original problem. Cioranescu and Murat coined the phrase "un terme étrange venu d'ailleurs" (a strange term coming from elsewhere) to designate this additional damping term.

The 2-dimensional case is somewhat special, as there is no critical exponent in that case. Instead, the critical size of the holes is r ǫ = Ce -1/ǫ 2 . With this modification, the result above holds verbatim. R e m a r k 2.1. The two following remarks are in order: 1) In the theorem above, the limiting problem has the same structure as the original problem.

2) The coefficient c is found to be c = rCap(S N -1 ), where Cap(S N -1 ) designates the 2-capacity of the unit sphere, defined by the usual variational problem:

Cap(S N -1 ) = inf φ∈H 1 (R 3 ) |x|≤1⇒φ(x)≥1 |x|≥1 |∇φ(x)| 2 dx .
In the context of porous media, the above result can be extended to the Stokes or Navier-Stokes equations (see Allaire [START_REF] Allaire | Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes[END_REF]): the super-critical scaling r ǫ ≫ ǫ γc leads to the d'Arcy law, whereas the critical case r ǫ ∼ rǫ γc leads to Brinkman's friction force in the Stokes equation.

Allaire's result was further extended to the case of randomly distributed holes at critical size by Desvillettes, Ricci and the author [START_REF] Desvillettes | The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow[END_REF]: the result is similar to the purely periodic case, assuming that the obstacles are well separated -we refer the interested reader to [START_REF] Desvillettes | The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow[END_REF] for a precise description of this assumptionexcept that it involves the empirical distribution of holes.

-The distribution of free path lengths

In the sequel, we shall consider a linear Boltzmann equation in space dimension N ≥ 2 set on the Euclidian space with a periodic system of balls removed:

Ω ǫ = {x ∈ R N | dist(x, ǫZ 2 ) > ǫ N/(N -1) } .
The holes in the domain Ω ǫ are at the critical size -with respect to the period ǫ, i.e. the smallest distance between neighboring ball centers. See Figure 3 for the domain Ω ǫ in space dimension 2.

Observe that, for the transport (or linear Boltzmann) equation -which a 1st order PDE instead of a 2nd order PDE like the Laplace equation -the critical exponent is γ c = N N -1 instead of N N -2 as recalled in the previous section. Henceforth, we only consider the 2-dimensional problem. Most of our results carry over to all space dimensions, except for the explicit computation of the Figure 1: The perforated domain Ω ǫ with holes at critical size in dimension 2 Figure 2: The free path length distribution of free path lengths which is not known for N > 2 (although recent progress by Marklof and Strömbergsson [START_REF] Marklof | The periodic Lorentz gas in the Boltzmann-Grad limit: asymptotic estimates[END_REF] provide an asymptotic equivalent for the tail of that distribution in all space dimensions.)

Before going further, we recall the definition of free path length. D e f i n i t i o n 3.1. The free path length for a particle moving at speed 1 in the domain Ω ǫ , starting from x ∈ Ω ǫ in the direction v ∈ S 1 is

τ ǫ (x, v) = min{t > 0 | x + tv ∈ ∂Ω ǫ } .
See Figure 3 for the geometric meaning of the free path length. Dahlqvist conjectured in [START_REF] Dahlqvist | The Lyapunov exponent in the Sinai billiard in the small scatterer limit[END_REF], and Boca and Zaharescu proved in [START_REF] Boca | The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatter Limit[END_REF] an explicit formula for the distribution of free path lengths

Φ(t) := lim ǫ→0 + Prob(τ ǫ > t) ,
assuming that x and ω are independent and uniformly distributed in Ω ǫ /ǫZ 2 (equivalently, in any period of Ω ǫ ) and in S 1 respectively. T h e o r e m 3.1 [START_REF] Boca | The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatter Limit[END_REF]. For each t > 0

Φ(t) = 24 π 2 ∞ t (s -t)g(s)ds , where
g(s) = 1 s ∈ [0, 1 2 ] , 1 2s + 2 1 -1 2s 2 ln(1 -1 2s ) -1 2 1 -1 s 2 ln |1 -1 s | s ∈ ( 1 2 , ∞)
. The proof of this result is based on Farey fractions, and on a particular 3-term partition of the 2-torus found in [START_REF] Blank | Thom's problem on irrational flows[END_REF] and [START_REF] Caglioti | On the distribution of free path lengths for the periodic Lorentz gas III[END_REF] that is reminiscent of the 3-length theorem conjectured by Steinhaus and eventually established by Sós [START_REF] Sós | On the distribution mod. 1 of the sequence nα[END_REF] and Suranyi [START_REF] Suranyi | Über die Anordnung der Vielfahren einer reelen Zahl mod[END_REF]

-Homogenization of the linear Boltzmann equation

Consider the linear Boltzmann equation in Ω ǫ with unknown f ǫ (t, x, v), with absorbing boundary condition on the edge of the holes

     ∂ t f ǫ + v • ∇ x f ǫ + σ(f ǫ -Kf ǫ ) = 0 , x ∈ Ω ǫ , |v| = 1 f ǫ (t, x, v) = 0 , x ∈ ∂Ω ǫ , v • n x > 0 f ǫ t=0 = f in ,
where σ > 0 and K is an integral operator acting on the v-variable only, i.e.

Kf (t, x, v) = S 1 k(v, w)f (t, x, w)dw .
The integral kernel of the operator K is a function k ∈ C(S 1 × S 1 ) satisfying the assumptions

k(v, w) = k(w, v) ≥ 0 for each v, w ∈ S 1 ,
and

S 1 k(v, w)dw = 1 for each v ∈ S 1 .
Our purpose is to investigate the following problems:

1) to find an equation for the limit of f ǫ as ǫ → 0 + , and 2) to find the asymptotic mass loss rate, and especially to check whether or not it is exponential.

We begin with the answer to the first question, which is an analogue for the linear Boltzmann equation of the problem studied by L'vov-Khruslov [START_REF] Khruslov | Perturbation of a viscous incompressible fluid by small particles[END_REF] and Cioranescu-Murat [START_REF] Cioranescu | Un terme étrange venu d'ailleurs[END_REF], recalled in section 2. The result below was obtained by the author in collaboration with E. Bernard and E. Caglioti [START_REF] Bernard | Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes[END_REF].

T h e o r e m 4.1 (Bernard-Caglioti-Golse, 2010). In the limit as ǫ → 0 + , denoting by f ǫ both the solution of the linear Boltzmann equation defined a.e. on R + × Ω ǫ × S 1 together with its extension by 0 in the holes (i.e. in Ω c ǫ ), one has

f ǫ → ∞ 0 F (t, s, x, v)ds in L ∞ (R + × R 2 × S 1 ) weak-*, where F ≡ F (t, s, x, v) satisfies, for all (x, v) ∈ R 2 × S 1 , t, s > 0          ∂ t F + v • ∇ x F + ∂ s F = -σF + Φ ′ Φ (s ∧ t)F , F (t, 0, x, v) = σ ∞ 0 KF (t, s, x, v)ds , F (0, s, x, v) = σe -σs f in (x, v) .
Unlike in the periodic homogenization problem for the Laplace equation, the homogenization of the linear Boltzmann equation in a periodic system of holes involves an additional variable s (together with the usual variables t, x, v of the kinetic theory of gases). In other words, the homogenized equation is set on a phase-space larger than the phase-space usual in kinetic models. Specifically,

phase-space = R 2 x × S 1 v × [0, ∞) s instead of R 2 x × S 1 v ,
as in the usual kinetic theory for particles moving at unit speed.

The extra-variable s has the following physical interpretation. As is well known, the probabilistic interpretation of the linear Boltzmann equation involves a jump process in velocity, where the jump times are exponentially distributed. The variable s is the time since the last jump in velocity -in other words, F (t, s, x, v) is the conditional expectation of the particle density knowing the time of the last jump in velocity.

The theorem above can be summarized by the following diagram

F ǫ Homogenization ----------→ F lifting of initial data         integration in s f ǫ f
Perhaps the most remarkable feature in the theorem above is that the homogenization process, which is a macroscopic limit of the linear Boltzmann equation, triggers the dependence of the particle distribution function in the extra microscopic variable s, although this variable is not present in the original equation. In other words, it is quite surprising that a macroscopic limit of the linear Boltzmann equation should involve a more microscopic level of description than the original problem.

-Asymptotic mass loss rate

Now we turn to the second question raised above, i.e. finding the asymptotic mass loss rate in the long time limit.

A first step is to condition the mass at time t by the s variable; in other words, we consider m(t, s) to be the density at time t of particles whose velocities (in the stochastic process driving the linear Boltzmann equation) have jumped at time t -s. P r o p o s i t i o n 5.1 (Bernard-Caglioti-Golse [START_REF] Bernard | Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes[END_REF]). One has

1 2π R 2 ×S 1 f ǫ (t, x, v)dxdv → M (t) = ∞ 0 m(t,
         ∂ t m(t, s) + ∂ s m(t, s) + B(t, s)m = 0 , t, s > 0 , m(t, 0) = σ ∞ 0 m(t, s)ds , t > 0 , m(0, s) = σe -σs M (0) , s > 0 ,
and where

B(t, s):= σ-Φ ′ Φ (s ∧ t) , while M (0) = 1 2π R 2 ×S 1 f in (x,v)dxdv .
A somewhat surprising consequence of the structure of the equation governing m is that, in the long time limit, the total mass (=total number of particles) decays exponentially fast.

T h e o r e m 5.1 (Bernard-Caglioti-Golse [START_REF] Bernard | Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes[END_REF]). For each σ > 0, there exists a unique ξ σ ∈ (-σ, 0) such that M (t) ∼ C σ e ξσ t as t → +∞ where

C σ = 1 2π R 2 ×S 1 f in (x, v)dxdv σ ∞ 0 sΦ(s)e -(σ+ξσ )s ds .
Besides ξ σ ∼ -σ as σ → 0 + , and ξ σ → -2 as σ → +∞ .

Several remarks are in order.

First this asymptotic behavior is non uniform as σ → 0. In the case σ = 0 (i.e. in the case of the free transport equation in a periodic system of absorbing holes), the mass loss rate is given by the asymptotic behavior of the function Φ:

M (t) = Φ(t) 1 2π R 2 ×S 1 f in (x, v)dxdv , so that M (t) ∼ M (0) π 2 t as t → +∞ .
Secondly, the asymptotic equivalent of the distribution of free path lengths Φ(s) as s → +∞, i.e. Φ(s) ∼ 1 π 2 s as s → +∞ was known before the explicit computation by Boca-Zaharescu [START_REF] Boca | The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatter Limit[END_REF] recalled in Theorem 3.1 above: see [START_REF] Caglioti | On the distribution of free path lengths for the periodic Lorentz gas III[END_REF]. In fact, it was proved in [START_REF] Bourgain | Distribution of Free Path Lengths for the Periodic Lorentz Gas[END_REF] and [START_REF] Golse | On the distribution of free path lengths for the periodic Lorentz gas II. M2AN Modél[END_REF] that, in any space dimension, the distribution of free path lengths satisfies bounds of the form

C ′ s ≤ Prob(τ ǫ > s) ≤ C s , s > 1
for positive constants C > C ′ uniformly as ǫ → 0. A recent result by Marklof-Strömbergsson [START_REF] Marklof | The periodic Lorentz gas in the Boltzmann-Grad limit: asymptotic estimates[END_REF] establishes that

Φ(s) := lim ǫ→0 Prob(τ ǫ > s) ∼ π N -1 2 2 N N Γ( N +3 2 )ζ(N ) 1 s as s → +∞.
Finally, it should be mentioned that E. Bernard [3] proved exponential decay of the total mass

1 2π Ωǫ×S 1 f ǫ (t, x, v)dxdv
uniformly in ǫ > 0, before the structure of the limiting, homogenized equation was fully understood. The collisionless case σ = 0 was considered by E. Caglioti and the author in [START_REF] Caglioti | On the distribution of free path lengths for the periodic Lorentz gas III[END_REF]. The main result obtained there is as follows:

T h e o r e m 5.2 [START_REF] Caglioti | On the distribution of free path lengths for the periodic Lorentz gas III[END_REF]. Let f ǫ satisfy

   ∂ t f ǫ + v • ∇ x f ǫ = 0 , x ∈ Ω ǫ , |v| = 1 , f ǫ (t, x, v) = 0 , x ∈ ∂Ω ǫ , v • n x > 0 , f ǫ t=0 = f in .
Then, for each t > 0, f ǫ converges in the sense of Cesàro

1 | ln ǫ| 1/4 ǫ f r (t, x, v) dr r → f (t, x, v) a.e. in (x, v) ∈ R 2 × S 1
in the limit as ǫ → 0 + , where

(∂ t + v • ∇ x )f (t, x, v) = Φ ′ Φ (t)f (t, x, v) , x ∈ R 2 , |v| = 1 , f t=0 = f in .
In the limit as t → +∞ one has Φ ′ (t) Φ(t) ∼ -1 t .

(In fact, the Cesàro average in r is unnecessary: see Lemma 6.1 below, which is itself a consequence of the main result in [START_REF] Boca | The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatter Limit[END_REF] in space dimension 2, and of [START_REF] Marklof | The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems[END_REF] in higher space dimensions.)

In other words, in the non collisional case, the limiting density f satisfies itself an equation of the same type as the original free transport equationexcept that the limiting equation is non autonomous. Therefore, the need for an extended phase-space, with the additional variable s, arises only for the collisional case σ > 0 -so that the result in Theorem 5.2 fails to provide any intuition about the structure of the homogenized equation in the collisional case σ > 0.

-Sketch of the proof of Theorem 4.1

How the extra variable s appear in the limiting equation obtained in Theorem 4.1 may seem somewhat mysterious. The present section gives the main ideas used in the proof of that result.

-Step 1: lifting the Boltzmann equation.

The key idea is to introduce the variable s already at the level of the linear Boltzmann equation (before passing to the limit as ǫ → 0 + ). Thus, the problem

     ∂ t f ǫ + v • ∇ x f ǫ + σ(f ǫ -Kf ǫ ) = 0 , x ∈ Ω ǫ , |v| = 1 , f ǫ (t, x, v) = 0 , x ∈ ∂Ω ǫ , v • n x > 0 , f ǫ t=0 = f in , is equivalent to f ǫ (t, x, v) = ∞ 0 F ǫ (t, s, x, v)ds ,
where F ǫ is the solution of the lifted Boltzmann equation

             ∂ t F ǫ + v • ∇ x F ǫ + ∂ s F ǫ + σF ǫ = 0 , x ∈ Ω ǫ , |v| = 1 , t, s > 0 , F ǫ (t, s, x, v) = 0 , x ∈ ∂Ω ǫ , v•n x > 0 , t, s > 0 , F ǫ (t, 0, x, v) = σ ∞ 0 KF ǫ (t, s, x, v)ds , x ∈ Ω ǫ , |v| = 1 , t > 0 , F ǫ (0, s, x, v) = f in (x, v)Π(s) , x ∈ Ω ǫ , |v| = 1 , s > 0 .
Here, Π ≡ Π(s) is any probability density on the half-line R + (for instance Π(s) = σe -σs .) There is no hidden difficulty in this statement, which can be proved by inspection. R e m a r k 6.1. Given Π, the function F ǫ is unique. Even though the function F ǫ itself depends on the specific choice of Π, one easily checks that f ǫ is independent of the choice of Π.

-

Step 2: finding an explicit formula for F ǫ .

Solving for F ǫ by the method of characteristics leads to the explicit formula

F ǫ (t, s, x, v) = F 1,ǫ (t, s, x, v) + F 2,ǫ (t, s, x, v) , where F 1,ǫ (t, s, x, v) = 1 s<t σe -σs Kf ǫ (t -s, x -sv, v)1 ǫτǫ(x/ǫ,v)>s , F 2,ǫ (t, s, x, v) = 1 t<s e -σt Π(s)f in (x -tv, v)1 ǫτǫ(x/ǫ,v)>t .

-

Step 3: passing to the limit in 1 ǫτǫ(x/ǫ,v)>t .

We begin with lemma that is an amplification of the convergence Prob(τ ǫ > t) → Φ(t) as ǫ → 0 + , established in [START_REF] Boca | The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatter Limit[END_REF].

L e m m a 6.1 (Bernard-Caglioti-Golse, 2010). In the limit as ǫ → 0 + ,

1 τǫ(x,v)≥t → Φ(t) in L ∞ (R 2 × S 1
) weak-* for each t > 0 .

See [START_REF] Bernard | Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes[END_REF] for a proof of this result.

At this point, one extends F ǫ and f ǫ by 0 inside the holes, and calls {F ǫ } and {f ǫ } the densities so extended.

Clearly, as a straightforward consequence of Lemma 6.1,

{F 2,ǫ }(t, s, x, v) → 1 t<s e -σt Π(s)f in (x -tv, v)Φ(t) in L ∞ (R + × R 2 × S 1 ) weak-* as ǫ → 0.
The main difficulty in the problem is therefore to pass to the limit in the product Kf ǫ (t -s, x -sv, v)1 ǫτǫ(x/ǫ,v)>s .

-Step 4: compactness by velocity averaging.

On the other hand

(∂ t + v • ∇ x ){f ǫ } = (v • n x )f ǫ ∂Ωǫ×S 1 δ ∂Ωǫ + O(1) L ∞ , so that    f ǫ L ∞ ≤ C (by the maximum principle), (∂ t + v • ∇ x ){f ǫ } = O(1) M([0,T ]×B(0,R)×S 1 ) ,
where M(X) designates the space of bounded Radon measures on X. That the holes are at the critical size in the problem considered here is reflected precisely in the second bound above.

The key is to use these estimates together with compactness by velocity averaging. We recall that, since the transport equation is hyperbolic, L p bounds on (∂ t + v • ∇ x ){f ǫ } do not entail strong relative compactness on {f ǫ } itself in L p loc . However, such bounds do imply strong relative compactness in L p loc on moments (averages of {f ǫ } in velocity) of the form

S 1 {f ǫ }(t, x, v)φ(v)dv
for each φ ∈ L ∞ (S 1 ). See [START_REF] Agoshkov | Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation[END_REF], [START_REF] Golse | Un résultat de compacité pour les équations du transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport[END_REF] and [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF] for the first results in that direction.

In the specific situation considered in the present paper, we apply Theorem 1.8 on p. 29 of [START_REF] Bouchut | Kinetic Equations and Asymptotic Theory[END_REF], and finds that, for each 1

≤ p < ∞ {Kf ǫ } = K{f ǫ } is strongly relatively compact in L p loc (R + × R 2 × S 1 ) .
Therefore, by weak+strong convergence

F 1,ǫ → 1 s<t σe -σs Kf (t -s, x -sv, v)Φ(s) in L p loc (R + × R 2 × S 1 ) for each p ∈ [1, ∞).
The remaining part of the proof is routine.

-Proof of exponential decay of the total mass

The equation for m is obtained by integrating the equation for F in x and v, since

m(t, s) := 1 2π R 2 ×S 1 F (t, s, x, v)dxdv .
At this point, the key observation is that, by applying the method of characteristics to the transport equation satisfied by m, the total mass at time t, i.e.

M (t) = ∞ 0 m(t, s)ds
satisfies the integral equation of renewal type

M (t) = σe -σt Φ(t) + t 0 σe -σ(t-s) Φ(t -s)M (s)ds , t ≥ 0 ,
with defective kernel e -σs Φ(s). In other words, one has 0 ≤ e -σs Φ(s) for each s > 0 and ∞ 0 e -σs Φ(s)ds < 1 .

Applying the classical results on the asymptotic behavior of integral equations of renewal type (see for instance Theorem 2 on p. 349 in Feller's classical treatise [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF]), one finds that M (t) ∼ C σ e ξσt as t → +∞ where ξ σ is defined as the only solution of

∞ 0 e -(σ+ξ)t Φ(t)dt = 1 σ , -σ < ξ σ < 0 , while C σ := c σ 2π R 2 ×S 1 f in (x, v)dxdv with 1 c σ = ∞ 0 σe -(σ+ξσ )t tΦ(t)dt .
If one is not interested in obtaining the exact asymptotic equivalent, but only in the exponential decay of M , one can avoid using the renewal theorem in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF] and proceed with elementary a priori estimates, as follows.

L e m m a 7.1.

For C ≥ | Φ(2)| -1 , one has - Φ(τ ) Φ(τ ) ≥ 1 C + τ ,
for each τ ≥ 0 .

P r o o f. According to the first formula on p. 427 in [START_REF] Boca | The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatter Limit[END_REF], one has

Φ(τ ) = n≥1 φ n τ -n , τ ≥ 2 ,
with φ n > 0 for each n ≥ 1 and φ n+1 /φ n → 2 as n → +∞, so that

-Φ(τ ) = n≥1 nφ n τ -n-1 ≥ Φ(τ ) τ , τ > 2 .
On the other hand, Φ is decreasing and convex, so that

-Φ(τ ) ≥ -Φ(2) = -Φ(2)Φ(0) ≥ -Φ(2)Φ(τ ) whenever τ ∈ [0, 2] .
Putting these two inequalities together leads to

(| Φ(2)| -1 + τ )| Φ(τ )| ≥ Φ(τ ) for each τ ∈ R + ,
which is the desired inequality.

With the inequality in Lemma 7.1, we proceed as follows. Obviously m ≥ 0, so that 0 = ∂ t m(t, s) + ∂ s m(t, s) + B(t, s)m(t, s)

≥ ∂ t m(t, s) + ∂ s m(t, s) + σ + 1 C + t ∧ s m(t, s) ≥ ∂ t m(t, s) + ∂ s m(t, s) + σ + 1 C + s m(t, s) , with C = | Φ(2)| -1 .
Another key computation is the following one: for each ψ ∈ C 1 (R + ) decaying exponentially fast as s → +∞, one has

d dt ∞ 0 m(t, s)ψ(s)ds = ∞ 0 ψ(s)(-∂ s m(t, s) -B(t, s)m(t, s))ds = m(t, 0)ψ(0) + ∞ 0 ψ(s)(-∂ s m(t, s) -B(t, s)m(t, s))ds = ∞ 0 m(t, s)(ψ ′ (s) -B(t, s)ψ(s) + σψ(0))ds .
Therefore, choose ψ so that

     ψ ′ (s) -σ + 1 C + s ψ(s) + σψ(0) = -αψ(s) , s > 0 , ψ(0) = 1 ,
where α ∈ (0, σ) is chosen so that

J := ∞ 0 σe (α-σ)τ Cdτ C + τ < 1 .
Solving for ψ the differential equation above leads to

ψ(s) = e (σ-α)s C + s C 1 - s 0 σe (α-σ)τ Cdτ C + τ ≥ e (σ-α)s C + s C J ≥ J > 0 .
With this choice of ψ, one has

d dt ∞ 0 m(t, s)ψ(s)ds ≤ -α ∞ 0 m(t, s)ψ(s)ds since B(t, s) ≥ σ + 1 C + s ,
and therefore

JM (t) ≤ ∞ 0 m(t, s)ψ(s)ds ≤ e -αt ∞ 0 m(0, s)ψ(s)ds = e -αt ∞ 0 σe -αt C + s C ds .
Summarizing the argument above, which is a simplified variant of the discussion of the asymptotic behavior of solutions of the renewal equation in [START_REF] Perthame | Transport equations in biology[END_REF], we have proved the following:

P r o p o s i t i o n 7.1. Set C = | Φ(2)| -1 and let α ∈ (0, σ) be such that ∞ 0 σe (α-σ)τ Cdτ C + τ < 1 .
Then the total mass of the particle system decays as follows:

M (t) = O(e -αt ) as t → +∞ .

-Some open questions

While the asymptotic behavior of the total mass is the first obvious question to be answered, it would be interesting to investigate the long time behavior of the solution F ≡ F (t, s, x, v) of the homogenized equation.

For instance, assume that the initial data is periodic with period 1 in the space variable x, and that ǫ = 1/n. Then the following question arises naturally: Does there exists α ∈ R and Ψ ≡ Ψ(s, x, v) such that e -αt F ǫ (t, s, x, v) ∼ Ψ(s, x, v) as t → +∞?

If Ψ ≡ Ψ(s, v) is sought independent of x, then it should satisfy

     -αΨ + ∂ s Ψ = -σΨ + Φ ′ Φ (s)Ψ , v ∈ S 1 , s > 0 , Ψ(0, v) = σ ∞ 0 KΨ(s, v)ds , v ∈ S 1 . It is found that Ψ(s, v) = ψ(v)e -(σ-α)s Φ(s) , with ψ(v) = σKψ(v) ∞ 0 e -(σ-α)s Φ(s)ds .
An obvious solution is ψ = Kψ and 1 = σ ∞ 0 e -(σ-α)s Φ(s)ds

In that case ψ is a constant in v, while the second equation above has α = ξ σ < 0 as unique solution.

This suggests the following line of investigation: a) Is that solution Ψ the only possible one? b) Does the solution F of the homogenized problem satisfy F (t, s, x, v) → σe -(σ-α)s Φ(s)

T 2 ×S 1 f in (x, v)dxdv
in some sense to be made precise as t → +∞? c) If so, what is the convergence rate? d) Is there a Lyapunov functional for the homogenized problem?

In the case of a random, Poisson distribution of holes, all the results established above hold verbatim with Φ(s) = e -λs , for some λ > 0 .

In that case, the additional coefficient

∂ t f + v • ∇ x f + σ(f -Kf ) + λf = 0 .
In other words, in the random, Poisson case, the presence of the holes leads to an additional exponential damping term in the linear Boltzmann equationexactly as in the diffusion case recalled in section 2. In that case, the questions a)-d) above are answered by the spectral gap property for the linear Boltzmann operator in the torus.

Therefore, the questions a)-d) above corresponds with a generalization of that spectral gap property for the kinetic theory in extended phase space obtained in Theorem 4.1. These questions are left for future investigations. The general issue of analyzing renewal equations with energy estimates based on some Lyapunov functional follows the strategy proposed by Perthame in his monograph [START_REF] Perthame | Transport equations in biology[END_REF].

Finally, we feel that the general idea of using extended phase spaces to study other singular limit problems in PDEs should perhaps be given more attention.

Figure 3 :

 3 Figure 3: The graphs of 24 π 2 g (left) and Φ (right)

0 F

 0 s) = -λ in the homogenized equation obtained in Theorem 4.1 and in Proposition 5.1 is constant, so that the extra variable s can be averaged out, and the limiting distribution function f (t, x, v) = ∞ (t, s, x, v)ds satisfies the linear Boltzmann equation


Patrick Cheridito

Ying Hu

Optimal consumption and investment in incomplete markets with general constraints *

We study an optimal consumption and investment problem in a possibly incomplete market with general, not necessarily convex, stochastic constraints. We provide explicit solutions for investors with exponential, logarithmic as well as power utility and show that they are unique if the constraints are convex. Our approach is based on martingale methods that rely on results on the existence and uniqueness of solutions to BSDEs with drivers of quadratic growth.

Introduction

We consider an investor receiving stochastic income who can invest in a financial market. The question is how to optimally consume and invest if utility is derived from intermediate consumption and the level of remaining wealth at some final time T . More specifically, we assume our investor receives income at rate e t and a lump sum payment E at the final time. The investor chooses a rate of consumption c t and an investment policy so as to maximize the expectation E T 0 αe -βt u(c t )dt + e -βT u(X T + E) , where α and β are constants, u : R → R ∪ {-∞} is a concave utility function and X T is his/her wealth immediately before receiving the lump sum payment E. There exists an extensive literature on problems of this form; see for instance, Karatzas and Shreve [START_REF] Karatzas | Methods of mathematical finance[END_REF] for an overview.

The novelty of this paper is that we put general, not necessarily convex, stochastic constraints on consumption and investment. We provide explicit solutions for investors with exponential, logarithmic and power utility in terms of solutions to BSDEs with drivers of quadratic growth. Our approach is based on an extension of the arguments of Hu et al. [START_REF] Hu | Utility maximization in incomplete markets[END_REF], where investment problems without intermediate consumption are studied. To every admissible strategy we associate a utility process, which we show to always be a supermartingale and a martingale if and only if the strategy is optimal. This method relies on results from Kobylanski [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF] and Morlais [START_REF] Morlais | Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem[END_REF] on the existence and properties of solutions to BSDEs with drivers of quadratic growth. We formulate constraints on consumption and investment in terms of subsets of predictable processes and use conditional analysis results from Cheridito et al. [START_REF] Cheridito | Conditional analysis on R d[END_REF] to obtain the existence of optimal strategies. For related results in a slightly different setup, see Nutz [START_REF] Nutz | The Bellman equation for power utility maximization with semimartingales[END_REF], where dynamic programming is used to derive the Bellman equation for power utility maximization in general semimartingale models with stochastic constraints on investment.

The structure of the paper is as follows: Section 2 introduces the model. In Section 3 we discuss the case of constant absolute risk aversion, corresponding to exponential utility functions. Section 4 treats the case of constant relative risk aversion, which is covered by logarithmic and power utility functions. The specification of the constraints and the definition of admissible strategies will be slightly different from case to case. Section 5 concludes with a discussion of the assumptions and potential generalizations.

The model

Let T ∈ R + be a finite time horizon and (W t ) 0≤t≤T an n-dimensional Brownian motion on a probability space (Ω, F, P). Denote by (F t ) the augmented filtration generated by (W t ). We consider a financial market consisting of a money market and m ≤ n stocks. Money can be lent to and borrowed from the money market at a constant interest rate r ≥ 0 and the stock prices follow the dynamics

dS i t S i t = µ i t dt + σ i t dW t , S i 0 > 0, i = 1, . . . , m,
for bounded predictable processes µ i t and σ i t taking values in R and R 1×n , respectively. If m < n, the stocks do not span all uncertainty and the market is incomplete even if there are no constraints.

Consider an investor with initial wealth x ∈ R receiving income at a predictable rate e t and an F T -measurable lump sum payment E at time T who can consume at intermediate times and invest in the financial market. If the investor consumes at a predictable rate c t and invests according to a predictable trading strategy π t taking values in R 1×m , where π i t is the amount of money invested in stock i at time t, his/her wealth evolves like

X t = x + t 0 X s - m i=1 π i s rds + m i=1 t 0 π i s S i s dS i s + t 0 (e s -c s )ds.
Denote by σ t the matrix with rows σ i t , i = 1, . . . , m. Assume that σσ T is invertible ν ⊗ P-almost everywhere, where ν is the Lebesgue measure on [0, T ], and the process We assume our agent chooses c and π so as to maximize

θ = σ T (σσ T ) -1 (µ -r1) is bounded. Then for p = πσ, one can write X (c,p) t = x + t 0 X (c,p) s rds + t 0 p t [dW t + θ t dt] + t 0 (e s -c s )ds. (2.1) Note that if T 0 (|e t | + |c t | + |p t | 2 )dt
E T 0 αe -βt u(c t )dt + e -βT u X (c,p) T + E (2.2)
for given constants α > 0, β ∈ R and a concave function u : R → R ∪ {-∞}. The specific cases we will discuss are:

• u(x) = -exp(-γx) for γ > 0 • u(x) = log(x) • u(x) = x γ /γ for γ ∈ (-∞, 0) ∪ (0, 1
). As usual, for γ > 0, we understand x γ /γ to be -∞ on (-∞, 0) while log(x) and x γ /γ for γ < 0 are meant to be -∞ on (-∞, 0].

To formulate consumption and investment constraints we introduce non-empty subsets C ⊂ P and Q ⊂ P 1×m , where P denotes the set of all real-valued predictable processes (c t ) 0≤t≤T and P 1×m the set of all predictable processes (π t ) 0≤t≤T with values in R 1×m . In Section 3 we do not put restrictions on consumption and just require the investment strategy π to belong to Q. In Section 4 consumption and investment will be of the form c = cX and π = πX, respectively, and we will require c to be in C and π in Q.

Note that the expected value (2.2) does not change if (c, p) is replaced by a pair (c ′ , p ′ ) which is equal ν ⊗ P-a.e. So we identify predictable processes that agree ν ⊗ P-a.e. and use the following concepts from Cheridito et al. [START_REF] Cheridito | Conditional analysis on R d[END_REF]: We call a subset A of P 1×k sequentially closed if it contains every process a that is the ν ⊗ P-a.e. limit of a sequence (a n ) n≥1 of processes in A. We call it P-stable if it contains 1 B a + 1 B c a ′ for all a, a ′ ∈ A and every predictable set B ⊂ [0, T ] × Ω. We say A is P-convex if it contains λa + (1λ)a ′ for all a, a ′ ∈ A and every process λ ∈ P with values in [0, 1]. In the whole paper we work with the following

Standing assumption C and Q are sequentially closed and P-stable.

This will allow us to show existence of optimal strategies. If, in addition, C and Q are P-convex, the optimal strategies will be unique. Note that P = {πσ : π ∈ Q} is a P-stable subset of P 1×n , which, since we assumed σσ T to be invertible for ν ⊗ P-almost all (t, ω), is P-convex if and only if Q is. Moreover, it follows from [START_REF] Cheridito | Conditional analysis on R d[END_REF] that P is sequentially closed.

For a process q in P 1×n , we denote by dist(q, P ) the predictable process dist(q, P ) := ess inf p∈P |q -p|, where ess inf denotes the greatest lower bound with respect to the ν ⊗ P-a.e. order. It is shown in [START_REF] Cheridito | Conditional analysis on R d[END_REF] that there exists a process p ∈ P satisfying |q -p| = dist(q, P ) and that it is unique (up to ν ⊗ P-a.e. equality) if P is P-convex. We denote the set of all these processes by Π P (q). By P 1×n BMO we denote the processes Z ∈ P 1×n for which there exists a constant D ≥ 0 such that

E T τ |Z t | 2 dt | F τ ≤ D for all stopping times τ ≤ T.
For every Z ∈ P 1×n BMO , . 0 Z s dW s is a BMO-martingale and E(Z • W ) t , 0 ≤ t ≤ T , a positive martingale. Moreover, if Z, V belong to P 1×n BMO , then Z is also in P 1×n BMO with respect to the Girsanov transformed measure

Q = E(V • W ) T • P;
see for instance, Kazamaki [START_REF] Kazamaki | Continuous exponential martingales and BMO[END_REF].

CARA or exponential utility

We first assume that our investor has constant absolute risk aversion -u ′′ (x)/u ′ (x) = γ > 0.

Then, up to affine transformations, the utility function u is of the exponential form

u(x) = -exp(-γx).
Here we do not constrain consumption, that is, C = P, and we assume that the set P of possible investment strategies contains at least one bounded process p. Moreover, we assume that the rate of income e and the final payment E are both bounded. Define the bounded positive function h on [0, T ] by

h(t) = 1/(1 + T -t) if r = 0 and h(t) = r 1 -(1 -r) exp(-r(T -t)) if r > 0.
Note that in both cases h solves the quadratic ODE

h ′ (t) = h(t)(h(t) -r), h(T ) = 1. Definition 3.1 If u(x) = -exp(-γx), an admissible strategy consists of a pair (c, p) ∈ P × P such that T 0 (|c t | + |p t | 2 )dt < ∞ P-a.s., exp -γh(t)X (c,p) t 0≤t≤T
is of class (D) and

T 0 E e -γct dt < ∞.
Consider the BSDE

Y t = E + T t f (s, Y s , Z s )ds + T t Z s dW s (3.1)
with driver

f (t, y, z) = - γ 2 dist 2 t z + 1 γ θ, hP + zθ t + 1 2γ |θ t | 2 + h(t)(e t -y) + h(t) γ log h(t) α -1 + β γ .
Since θ, e, E and h are bounded and the set P contains a bounded process p, there exists a constant

K ∈ R + such that |f (t, y, z)| ≤ K(1 + |y| + |z| 2 ) and |f (t, y 1 , z 1 ) -f (t, y 2 , z 2 )| ≤ K(|y 1 -y 2 | + (1 + |z 1 | + |z 2 |)|z 1 -z 2 |).
So it follows from Kobylanski [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF] that equation (3.1) has a unique solution (Y, Z) such that Y is bounded and from Morlais [START_REF] Morlais | Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem[END_REF] that Z belongs to P 1×n BMO .

Theorem 3.2 The optimal value of the optimization problem

(2.2) for u(x) = -exp(-γx) over all admissible strategies is -exp [-γ(h(0)x + Y 0 )] , (3.2)
and (c * , p * ) is an optimal admissible strategy if and only if

c * = hX (c * ,p * ) + Y - 1 γ log h α ν ⊗ P-a.e. and p * ∈ Π P Z + θ/γ h . (3.3)
In particular, an optimal admissible strategy exists, and it is unique up to ν ⊗ P-a.e. equality if the set P is P-convex.

Proof. For every admissible strategy (c, p), equation (2.1) defines a continuous stochastic process X (c,p) . The process

R (c,p) t = -e -βt e -γ h(t)X (c,p) t +Yt - t 0 αe -βs e -γcs ds satisfies R (c,p) 0 = -e -γ(h(0)x+Y 0 ) , R (c,p) T = -e -βT e -γ X (c,p) T +E - T 0
αe -βs e -γcs ds and dR

(c,p) t = γe -βt e -γ h(t)X (c,p) t +Yt (h(t)p t -Z t )dW t + A (c,p) t dt ,
where

A (c,p) t = h(t)p t θ t - γ 2 |h(t)p t -Z t | 2 -f (t, Y t , Z t ) +h(t)(e t -c t ) - α γ e γ h(t)X (c,p) t +Yt e -γct + h ′ (t)X (c,p) t + h(t)rX (c,p) t + β γ .
First note that

h(t)p t θ t - γ 2 |h(t)p t -Z t | 2 = - γ 2 h(t)p t -Z t + 1 γ θ t 2 + Z t θ t + 1 2γ |θ t | 2 ≤ - γ 2 dist 2 t Z + 1 γ θ, hP + Z t θ t + 1 2γ |θ t | 2 ,
and the inequality becomes a ν ⊗ P-a.e. equality if and only if

p ∈ Π P Z + θ/γ h . Furthermore, for fixed (t, ω) ∈ [0, T ] × Ω, z → -h(t)z - α γ e γ h(t)X (c,p) t +Yt e -γz
is a strictly concave function that is equal to its maximum

h(t) γ log h(t) α -h 2 (t)X (c,p) t -h(t)Y t - h(t) γ if and only if z = h(t)X (c,p) t + Y t - 1 γ log h(t) α .
Therefore, one has

h(t)(e t -c t ) - α γ e γ h(t)X (c,p) t +Yt e -γct + h ′ (t)X (c,p) t + h(t)rX (c,p) t + β γ ≤ h(t)e t + h(t) γ log h(t) α -h 2 (t)X (c,p) t -h(t)Y t - h(t) γ + h ′ (t)X (c,p) t + h(t)rX (c,p) t + β γ = h(t)e t + h(t) γ log h(t) α -h(t)Y t - h(t) γ + β γ , (3.4)
where the inequality is attained if and only if

c = hX (c,p) + Y - 1 γ log h α (note that in (3.4) the X (c,p) t
-terms disappear due to our choice of the function h). It follows that for every admissible pair (c, p), R (c,p) is a local supermartingale, which by our definition of admissible strategies, is of class (D). Therefore, it is a supermartingale, and one obtains

R (c,p) 0 ≥ E R (c,p) T ,
where the inequality is strict if the pair (c, p) does not satisfy condition (3.3). On the other hand, if we can show that each pair (c * , p * ) satisfying (3.3) is admissible and

R * = R (c * ,p * ) is a martingale, we can conclude that R * 0 = E [R * T ] ,
+ E T 0 e -γc * t dt ≤ M E [-R * T ] < ∞,
where M is a suitable constant and the inequality E [-R * T ] < ∞ follows from Fatou's lemma. By Girsanov's theorem,

W Q t = W t + t 0 θ s ds
is an n-dimensional Brownian motion under the measure

Q = E(-θ • W ) T • P,
and one has

d(h(t)X * t ) = h ′ (t)X * t dt + h(t)p * t dW t + h(t)[X * t r + p * t θ t + e t -c * t ]dt = h ′ (t)X * t dt + h(t)p * t dW t + h(t) X * t r + p * t θ t + e t -h(t)X * t -Y t + 1 γ log h(t) α dt = h(t)p * t dW t + h(t) p * t θ t + e t -Y t + 1 γ log h(t) α dt = h(t)p * t dW Q t + h(t) e t -Y t + 1 γ log h(t) α dt. (3.5)
Since p * belongs to P 1×n BMO , the process V t = t 0 h(s)p * s dW Q s is a BMO-martingale under Q, and it can be seen from (3.5) that there exist constants

d 1 , d 2 such that e -γh(t)X * t ≤ d 1 e -γVt and e -γVt ≤ d 2 e -γh(t)X * t for all t ∈ [0, T ].
Hence, one obtains for every stopping time τ ≤ T ,

e -γh(τ )X * τ ≤ d 1 e -γVτ ≤ d 1 E Q e -γ 2 V T | F τ 2 = d 1 E e -γ 2 V T E(-θ • W ) T | F τ 2 E(-θ • W ) -2 τ ≤ d 1 E e -γV T | F τ E E(-θ • W ) 2 T | F τ E(-θ • W ) -2 τ ≤ d 1 d 2 E e -γX * T | F τ E E(-θ • W ) 2 T | F τ E(-θ • W ) -2 τ .
But since θ is bounded, there exists a constant d 3 such that

E E(-θ • W ) 2 T | F τ E(-θ • W ) -2 τ = E E(-2θ • W ) T E(-2θ • W ) τ exp T τ |θ s | 2 ds | F τ ≤ d 3 for every stopping time τ ≤ T.
So one has

e -γh(τ )X * τ ≤ d 1 d 2 d 3 E e -γX * T | F τ for every stopping time τ ≤ T.
This shows that exp (-γh(t)X * t ) 0≤t≤T is of class (D). Therefore, (c * , p * ) is admissible and R * a martingale.

It remains to show that a pair (c * , p * ) satisfying (3.3) exists and that it is unique up to ν ⊗ Pa.e. equality if the set P is P-convex. It is shown in [START_REF] Cheridito | Conditional analysis on R d[END_REF] that a process p * in Π P Z+θ/γ h exists and that it is unique up to ν ⊗ P-a.e. equality if P is P-convex. As we have seen above, every p * ∈ Π P Z+θ/γ h is also in P 1×n BMO . So there exists a unique continuous process (X t ) satisfying

X t = x + t 0 X s rds + t 0 p * t [dW t + θ t dt] + t 0 e s -h(s)X s -Y s + 1 γ log h(s) α ds. But X = X (c * ,p * ) for c * t = h(t)X t + Y t - 1 γ log h(t) α .
So (c * , p * ) satisfies condition (3.3), and it is unique up to ν ⊗ P-a.e. equality if the set P is P-convex.

CRRA utility

We now assume that the investor has constant relative risk aversion -xu ′′ (x)/u ′ (x) = δ > 0. For δ = 1, this corresponds to u(x) = log(x), and for δ = 1 to u(x) = x γ /γ, where γ = 1δ. We discuss the cases δ = 1 and δ = 1 separately. In both of them we assume E = 0. We here suppose that the initial wealth is strictly positive: x > 0. To avoid -∞ utility, the agent must keep the wealth process positive. Therefore, we can parameterize e, c and π by ẽ = e/X, c = c/X and π = π/X, respectively. If one denotes p = πσ, the corresponding wealth evolves according to

dX (c,p) t X (c,p) t = pt (dW t + θ t dt) + (r + ẽt -ct )dt, X (c,p) 0 = x,
and one can write

X (c,p) t = x E p • W Q t exp t 0 (r + ẽs -cs )ds > 0, (4.1)
where E is the stochastic exponential and W Q t = W t + t 0 θ s ds. In the whole section we assume that ẽ is bounded and the constraints are of the following form: c must be in the set C and π in Q, or equivalently, p in P = {πσ : π ∈ Q}. Additionally, c will be required to be positive or non-negative depending on the specific utility function being used. Also, c and p will have to satisfy suitable integrability conditions. For all CRRA utility functions u we make the following assumption: there exists a pair (c, p) ∈ C × P such that u(c)c and p are bounded.

(4.2)

Note that this implies that u(c) and c are both bounded.

Logarithmic utility

In the case u(x) = log(x), we introduce the positive function

h(t) = 1 + α(T -t) if β = 0 α/β + (1 -α/β)e -β(T -t) if β > 0 ,
and notice that h ′ (t) = βh(t)α with h(T ) = 1.

Definition 4.1 For u(x) = log(x), an admissible strategy is a pair (c, p) ∈ C × P satisfying

E T 0 | log(c t )|dt + T 0 ct dt + T 0 |p t | 2 dt < ∞. (4.3)
Remember that we understand log(x) to be -∞ for x ≤ 0. Therefore, (4.3) implies c > 0 ν ⊗P-a.e.

Let us set max

c∈C α h log(c) -c := ess sup c∈C α h log(c) -c , (4.4)
where ess sup is the smallest upper bound with respect to ν ⊗ P-a.e. inequality. By

arg max c∈C α h log(c) -c (4.5)
we denote the set of all processes in C which attain the ess sup. It follows from Cheridito et al.

[5] that (4.5) is not empty and, up to ν ⊗ P-a.e. equality, contains exactly one process if

C is P-convex. Note that α h log(c) -c ≤ max c∈C α h log(c) -c ≤ α h log α h -1 ,
where c is the process of assumption (4. with driver In particular, an optimal admissible strategy exists, and it is unique up to ν ⊗ P-a.e. equality if the sets C and P are P-convex.

f (t, y) = 1 2 dist 2 t (θ, P ) - 1 2 |θ t | 2 - αy h(t) -max c∈C α h log(c) -c t -r -ẽt . ( 4
Proof. For every admissible strategy (c, p), define the process where

R (c,p) t = h(t)e -βt log X (c,p) t -Y t + t 0 αe -βs log(c s )ds. Then R (c,p) 0 = h(0)(log(x) -Y 0 ), R (c,p) T = e -
A (c,p) t = pt θ t - 1 2 |p t | 2 + αY t h(t) + f (t, Y t ) + α h(t) log(c t ) + r + ẽt -ct . First note that pt θ t - 1 2 |p t | 2 + αY t h(t) = - 1 2 |p t -θ t | 2 + 1 2 |θ t | 2 + αY t h(t) ≤ - 1 2 dist 2 t (θ, P ) + 1 2 |θ t | 2 + αY t h(t) ,
and the inequality becomes a ν ⊗ P-a.e. equality if and only if

p ∈ Π P (θ) . Furthermore, α h log(c) + r + ẽ -c ≤ max z∈C α h log(z) -z + r + ẽ,
where ν ⊗ P-a.e. equality is attained if and only if

c ∈ arg max z∈C α h log(z) -z .
It follows that for every admissible pair (c, p), the process R (c,p) is a local supermartingale. But it can be seen from (4.10) that the local martingale part of R (c,p) is a true martingale and its finite variation part is of integrable total variation. So R (c,p) is a supermartingale and one obtains

R (c,p) 0 ≥ E R (c,p) T ,
where the inequality is strict if (c, p) does not satisfy condition (4.9). If (c * , p * ) satisfies (4.9), then the pair is in C ×P . Moreover, we have seen above that it follows from assumption (4.2) that the process log(c * ) is bounded. The same is true for p * because θ is bounded and, by assumption (4.2), P contains a bounded process p. In particular, (c * , p * ) is admissible and the corresponding process R * a martingale. One concludes

R * 0 = E [R * T ] ,
which shows that (c * , p * ) is optimal. That a strategy satisfying (4.9) exists follows from [START_REF] Cheridito | Conditional analysis on R d[END_REF] as well as its uniqueness (up to ν ⊗ P-a.e. equality) in case C and P are P-convex.

Example 4.3 If consumption is unconstrained, that is C = P, then c * = α h , max c∈C α h log(c) -c = α h log α h -1 ,
and the driver (4.7) becomes

f (t, y) = 1 2 dist 2 t (θ, P ) - 1 2 |θ t | 2 - αy h(t) - α h(t) log α h(t) -1 -r -ẽt .

Power utility

Let us now turn to the case u(x) = x γ /γ for γ ∈ (-∞, 0) ∪ (0, 1). The definition of admissible strategies is slightly different for γ > 0 and γ < 0. But the optimal value of the optimization problem (2.2) as well as the optimal strategies will in both cases be of the same form. For γ < 0, we additionally require the process (X (c,p) ) γ to be of class (D) and

E T 0 c γ t dt < ∞.
Note that for γ < 0, since we assume x γ to be ∞ if x ≤ 0, the condition

E T 0 c γ t dt < ∞ implies c > 0 ν ⊗ P-a.e.
For every continuous bounded process Y , define

max c∈C α γ cγ e Y -c := ess sup c∈C α γ cγ e Y -c , (4.11)
where ess sup denotes the smallest upper bound with respect to ν ⊗ P-a.e. ordering, and denote by

arg max c∈C α γ cγ e Y -c (4.12)
the set of all processes in C which attain the ess sup. It follows from Cheridito et al. [START_REF] Cheridito | Conditional analysis on R d[END_REF] that (4.12) is not empty and, up to ν ⊗ P-a.e. equality, contains exactly one process if C is P-convex.

For the process c of assumption (4.2), one has

α γ cγ e Y -c ≤ max c∈C α γ cγ e Y -c ≤ 1 -γ γ α 1/(1-γ) e Y /(1-γ) .
This implies that max c∈C with driver

f (t, y, z) = γ 1 -γ 2 dist 2 t z + θ 1 -γ , P - |z + θ t | 2 2(1 -γ) - 1 2γ |z| 2 -max c∈C α γ cγ e y -c t -r -ẽt + β γ .
(4.14) Note that f (t, y, z) grows exponentially in y. But it satisfies Assumption (A.1) in Briand and Hu [START_REF] Briand | Quadratic BSDEs with convex generators and unbounded terminal conditions[END_REF]. So it can be deduced from Proposition 3 in [START_REF] Briand | Quadratic BSDEs with convex generators and unbounded terminal conditions[END_REF] that (4.13) has a solution (Y, Z) such that Y is bounded. That Z is in P 1×n BMO and the uniqueness of such a solution then follow from [START_REF] Morlais | Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem[END_REF].

Theorem 4.5 If u(x) = x γ /γ for γ ∈ (-∞, 0) ∪ (0, 1), the optimal value of the optimization problem (2.2) over all admissible strategies is In particular, an optimal admissible strategy exists, and it is unique up to ν ⊗ P-a.e. equality if the sets C and P are P-convex.

1 γ x γ e -Y 0 , ( 4
Proof. For every admissible strategy (c, p) define the process

R (c,p) t = e -βt 1 γ X (c,p) t γ e -Yt + t 0 αe -βs 1 γ c γ s ds. Then R (c,p) 0 = 1 γ x γ e -Y 0 , R (c,p) T = e -
|Z t | 2 = γ -1 2 pt - Z t + θ t 1 -γ 2 + 1 2(1 -γ) |Z t + θ t | 2 + 1 2γ |Z t | 2 ≤ γ -1 2 dist 2 t Z + θ 1 -γ , P + 1 2(1 -γ) |Z t + θ t | 2 + 1 2γ |Z t | 2 ,
and the inequality becomes a ν ⊗ P-a.e. equality if and only if

p ∈ Π P Z + θ 1 -γ . Furthermore, α γ cγ e Y + ẽ -c + r - β γ ≤ max z∈C α γ zγ e Y -z + ẽ + r - β γ ,
where

Z + θ 1 -γ , p * = dist Z + θ 1 -γ , P ≤ dist Z + θ 1 -γ , p ,
that dist Z+θ 1-γ , p is in P BMO and p * in P 1×n BMO . This shows that (c * , p * ) is admissible. Furthermore, the corresponding process X * satisfies

X * t = x E p * • W Q t exp t 0 (r + ẽs -c * s )ds ≤ M E p * • W Q t (4.18) for some constant M ∈ R + . Choose γ < γ ′ < 1 and let ε = 1 -γ ′ . Since θ is bounded, one has E Q E θ • W Q 1/ε T < ∞,
and by Hölder's inequality, one obtains for every stopping time τ ≤ T ,

E (X * τ ) γ ′ = E Q (X * τ ) γ ′ E(θ • W Q ) T ≤ E Q [(X * τ )] γ ′ E Q E θ • W Q 1/ε T ε ≤ M γ ′ E Q E θ • W Q 1/ε T ε .
It follows that (X * ) γ is of class (D) and R * a martingale. In particular,

R * 0 = E [R * T ] ,
which shows that (c * , p * ) is optimal. If γ < 0, R (c,p) is for every admissible pair (c, p) a supermartingale due to our assumption that (X (c,p) ) γ is of class (D) and E

[-R * T ] < ∞. Hence, E (X * T ) γ + T 0 (c * t ) γ dt < ∞, (4.19)
and it follows that (c * , p * ) is admissible. It can be seen from (4.1) that (4.19) implies

E E(p * • W Q ) γ T < ∞,
where W Q t = W t + t 0 θ s ds. So one obtains from Jensen's inequality that for every stopping time τ ≤ T ,

E(p * • W Q ) γ τ ≤ E Q E(p * • W Q ) γ/2 T |F τ 2 = E E(p * • W Q ) γ/2 T E(-θ • W ) T E(-θ • W ) τ |F τ 2 ≤ E E(p * • W Q ) γ T |F τ E E(-θ • W ) 2 T E(-θ • W ) 2 τ |F τ ≤ M E E(p * • W Q ) γ T |F τ
for some constant M ∈ R + . This shows that E(p * • W Q ) γ and (X * ) γ are of class (D). Hence, (c * , p * ) is admissible and R * is a martingale. In particular, R * 0 = E [R * T ], and it follows that (c * , p * ) is optimal.

In both cases, γ > 0 and γ < 0, existence of an optimal strategy follows from [START_REF] Cheridito | Conditional analysis on R d[END_REF] as does uniqueness (up to ν ⊗ P-a.e. equality) if the sets C and P are P-convex.

Conclusion

We gave solutions to optimal consumption and investment problems for expected utility optimizers in the three cases of exponential, logarithmic and power utility. In the exponential case we assumed that the income rate e as well as the final payment E were bounded and consumption c was unconstrained. The proof of Theorem 3.2 relies on the fact that the BSDE (3.1) has a unique solution (Y, Z) such that Y is bounded and Z is in P 1×n BMO . There exist extensions of the result of Kobylanski [START_REF] Kobylanski | Backward stochastic differential equations and partial differential equations with quadratic growth[END_REF] showing that equation (3.1) also has a unique solution for certain unbounded random variables E; see Briand and Hu [START_REF] Briand | BSDE with quadratic growth and unbounded terminal value[END_REF][START_REF] Briand | Quadratic BSDEs with convex generators and unbounded terminal conditions[END_REF], Ankirchner et al. [START_REF] Ankirchner | On measure solutions of backward stochastic differential equations[END_REF], Delbaen et al. [START_REF] Delbaen | On the uniqueness of solutions to quadratic BS-DEs with convex generators and unbounded terminal conditions[END_REF]. However, if E is not bounded, Y is not bounded and Z not necessarily in P 1×n BMO . It is still possible to show that there exist strategies satisfying condition (3.3). But one would need a new argument to show that and in which sense they are optimal. If one introduces restrictions on consumption, the proof of Theorem 3.2 does not go through. If one can show existence of an optimal strategy when consumption is constrained, it obviously has to be different from (3.3). In the cases of logarithmic and power utility we assumed ẽ = e/X to be bounded and E = 0. The first assumption is technical and ensures that the BSDEs (4.6) and (4.13) both have unique solutions (Y, Z) such that Y is bounded and Z in P 1×n BMO . Again, if (4.6) or (4.13) can be solved for unbounded ẽ, one can still show that there exist strategies satisfying (4.9) or (4.16), respectively. But then again, one would have to find a new explanation why and in which sense they are optimal. For E = 0 the proofs of Theorems 4.2 and 4.5 do not work because the process R (c,p) does not have the correct terminal value. One would have to find a new way to construct R (c,p) to cover this case. We point out that also in Sections 3 and 4 of Hu et al. [START_REF] Hu | Utility maximization in incomplete markets[END_REF] as well as in Nutz [START_REF] Nutz | The Bellman equation for power utility maximization with semimartingales[END_REF] it is assumed that E = 0.

  2). It follows that max c∈C α h log(c)c as well as every process c ∈ arg max z∈C α h log(z)z is bounded. In particular, log(c) is bounded for every c ∈ arg max z∈C α h log(z)z . Consider the BSDE Y t = T t f (s, Y s )ds + T t Z s dW s (4.6)

Definition 4 . 4 0 ct dt + T 0 |p t | 2

 4402 In the case γ > 0, an admissible strategy is a pair (c, p) ∈ C × P such that c ≥ 0 ν ⊗ P-a.e. and T dt < ∞ P-a.s.

  α γ cγ e Yc as well as u(c) and c for every c ∈ arg max z∈C α γ zγ e Yz , are bounded processes. Now consider the BSDE Y t = T t f (s, Y s , Z s )ds + T t Z s dW s (4.13)

  if (c, p) does not fulfill condition (4.16). If (c * , p * ) satisfies (4.16), it follows as in the case γ > 0, that u(c * ) and c * are bounded and p * belongs to P 1×n BMO . In particular, c * > 0 ν ⊗ P-a.e. Moreover, -R * is a positive local martingale. So -R * is a supermartingale and E

Example 4 . 6

 46 If consumption is unconstrained, that is C = P, thenc * = α 1/(1-γ) e Yt/(1-γ) , max c∈C α γ cγ e yc = 1γ γ α 1/(1-γ) e y/(1-γ) ,and the driver (4.14) becomesf (t, y, z) = γ 1γ) e y/(1-γ)rẽt + β γ .

  and it follows that (c * , p * ) is optimal. But if (c * , p * ) satisfies (3.3), c * is continuous in t. In particular, it belongs to P and Moreover, since θ as well as h are bounded and P contains a bounded process p, there exists a constant L such that |p * | ≤ L(1 + |Z|). It follows that p * ∈ P 1×n BMO , and hence,

	∞ P-a.s.	T 0 |c * t |dt <

T 0 |p * t | 2 dt < ∞ P-a.s. Since A * := A (c * ,p * ) = 0, -

R * is a positive local martingale, and one obtains E e -γX * T

  .7) f (t, y) is of linear growth in y, and all the other terms are bounded. It is known from Pardoux and Peng[START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF] that (4.6) has a unique solution (Y, Z) such that Y is square-integrable, and it follows from Morlais[START_REF] Morlais | Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem[END_REF] that Y is bounded and Z ∈ P 1×n BMO .

	Theorem 4.2 For u(x) = log(x), the optimal value of the optimization problem (2.2) over all
	admissible strategies is
		h(0)(log(x) -Y 0 ),	(4.8)
	and (c * , p * ) is an optimal admissible strategy if and only if
	c * ∈ arg max c∈C	α h	log(c) -c	and p * ∈ Π P (θ) .	(4.9)

  βT log X

			(c,p) T	+	0	T	αe -βs log(c s )ds
	and	dR t (c,p)	= h(t)e -βt (p t + Z t )dW t + A (c,p) t	dt ,	(4.10)

  .15) and (c * , p * ) is an optimal admissible strategy if and only if

	c * ∈ arg max c∈C	α γ	cγ e Y -c	and p * ∈ Π P	Z + θ 1 -γ	.	(4.16)

  ν ⊗ P-a.e. equality is attained if and only if , p * ) be a strategy satisfying(4.16). Then, as we have seen above, u(c * ) and c * are bounded processes. In particular c * ≥ 0 ν ⊗ P-a.e. Moreover, since θ and p are bounded and Z is in P 1×n BMO , it follows from dist

	with strict inequality if condition (4.16) is violated. Now let (c
	c ∈ arg max z∈C	α γ	zγ e Y -z .
	The next step of the proof is slightly different for the two cases γ > 0 and γ < 0. Let us
	first assume γ > 0. Then for every admissible pair (c, p), the process R (c,p) is a positive local
	supermartingale, and hence, a supermartingale. In particular,
	R	(c,p) 0	≥ E R T (c,p)	(4.17)

*

The paper was written during periods when the authors could work together at the Université Rennes 1 or Princeton University. The hospitality of both institutions is greatly appreciated. We also thank an anonymous referee for valuable comments.

† Partially supported by NSF Grant DMS-0642361 ‡ Partially supported by funds from the Marie Curie ITN Grant, "Controlled Systems", GA no.213841/2008.