dimensional Brownian diffusion processes

Universit´ es de Paris 6 & Paris 7 - CNRS (UMR 7599)

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Functional quantization of 1-dimensional Brownian diffusion processes ` H. LUSCHGY & G. PAGES

OCTOBRE 2003

Pr´epublication no 853

H. Luschgy : FB IV-Mathematik, Universit¨at Trier, D-54286 Trier. G. Pag` es : Laboratoire de Probabilit´es et Mod`eles Al´eatoires, CNRS-UMR 7599, Universit´e Paris VI & Universit´e Paris VII, 4 place Jussieu, Case 188, F-75252 Paris Cedex 05.

Functional Quantization of 1-dimensional Brownian diffusion processes Harald Luschgy∗

`s† Gilles Page

Abstract The functional quantization problem for one-dimensional Brownian diffusions on [0, T ] is investigated. First, the existence of optimal n-quantizers for every n ≥ 1 is established, based on an existence result for random vectors taking their values in abstract Banach spaces. Then, one shows under rather general assumptions that the rate of convergence of the Lp -quantization error 1 is O((log n)− 2 ) like for the Brownian motion. Several methods to construct some quasi-optimal quantizers are proposed. Finally, a special attention is given to diffusions with a Gaussian martingale term.

Key words: Functional quantization, optimal quantizers, Brownian diffusions, Lamperti transform, Girsanov Theorem. 2001 AMS classification: 60E99, 60H10.

1

Introduction

This paper is a first attempt to solve the functional quantization problems – bounds, rates, construction of nearly optimal quantizers – for diffusion processes. In particular, it will include many non-Gaussian processes. One considers a real-valued Brownian diffusion process (E) ≡ dXt = b(t, Xt ) dt + σ(t, Xt ) dWt ,

X0 = x0 ,

(1.1)

where b, σ : [0, T ] × R → R are Borel functions with at most linear growth such that the above equation (E) admits at least one (weak) solution over [0, T ] and W is a Brownian motion defined on a probability space (Ω, A, P). This solution is pathwise continuous (and hence bi-measurable on Ω × [0, T ]). It is classical background (see e.g. [2]) that supt∈[0,T ] |Xt | has r-moments for every r ∈ (0, +∞) so that the diffusion process X can be seen in particular as an Lr (P)-integrable random


variable taking its values in (LpT ,  . p ) where LpT denotes Lp ([0, T ], dt) and  g p := 0T |g(t)|p dt stands for the usual Lp -norm when p ∈ [1, +∞) (when p = +∞, g∞ = dt-ess supt∈[0,T ] |g(t)|). ∗

1

p

Universit¨ at Trier, FB IV-Mathematik, D-54286 Trier, BR Deutschland. E-mail: [email protected] Laboratoire de Probabilit´es et Mod`eles al´eatoires, UMR 7599, Universit´e Paris 6, case 188, 4, pl. Jussieu, F-75252 Paris Cedex 5. E-mail:[email protected] †

1

So for every integer n ≥ 1, we are in position to investigate the level n Lr -quantization problem for the process X, viewed as a random variable taking its values in LpT . That means minimizing the (finite) real quantity 

en,r (α, X, L ) := E min X − αi p p

1 r

r

1≤i≤n

     =  min X − αi p   1≤i≤n

(1.2)

Lr (P)

among all subset α = {α1 , . . . , αn } ⊂ LpT , |α| ≤ n (with the convention that, when |α| < n, αi may be equal to some αj with i = j). Such a set α is called a n-codebook or a n-quantizer. We will show (see Section 2 below) that en,r (α, X, Lp ) does reach a minimum over the n-quantizers and one denotes   en,r (X, Lp ) := min en,r (α, X, Lp ), α ⊂ LpT , |α| ≤ n . in some specific situation, we will mention the L∞ n level quantization problem in which the Lr (P)-norm is replaced in (1.2) by the L∞ (P)-norm defined by ZL∞ (P) := P-ess sup|Z|. Definition 1 Let α := {α1 , . . . , αn } ⊂ LpT be a n-quantizer. Any Borel partition (Ci (α))1≤i≤n of LpT satisfying

Ci (α) ⊂ β ∈ LpT | β − αi p = min β − αj p αi ∈α

is called a Voronoi partition induced by α (although, when |α| < n, Ci (α) may be equal to Cj (α) if αi = αj for some i = j). For a given n-quantizer α there are infinitely many such Voronoi partitions (see [7] for more details in a finite dimensional setting). Then, one defines the associated closest neighbour projection πα :=

n 

αi 1Ci (α)

i=1

and the induced α-quantized version (or α-quantization) of X by α := πα (X) X

(the exponent

α

will often be dropped).

(1.3)

Then, one easily checks that, for any measurable random variable Y : (Ω, A, P) → α ⊂ LpT ,

E X − Y rp ≥ E X − X α rp = E min X − αi rp 1≤i≤n

so that, finally, 

r, X σ(X)-measurable, |X(Ω)| ≤n en,r (X, Lp ) = min EX − X p 



= min X − Y rp , Y : (Ω, A, P) → LpT , |Y (Ω)| ≤ n .



(1.4)

The n level Lr -quantization problem can clearly be extended to stochastic processes which are not Brownian diffusions. Indeed, the natural extension (see e.g. [9]), is to consider an abstract

2

Banach space (E,  . ), a Radon(1 ) random variable X : (Ω, A, P) → E with E Xr < +∞ and to define for every subset α ⊂ E, |α| ≤ n, 

en,r (α, X, E) := E min X − αi r

1 r

(1.5)

1≤i≤n

in order to solve the same optimization problem en,r (X, E) := inf {en,r (α, X, E), α ⊂ E, |α| ≤ n} .

(1.6)

The main aim of Section 2 below is to provide a general condition which grants the existence of an optimal n-quantizer for this problem. This condition will always be satisfied by the Lp spaces (this result is new for p = 1 and +∞). The above preliminary facts exposed for diffusions straightforwardly extend to this abstract Banach setting. The fact that en,r (X, E) is nonincreasing and goes to 0 as n goes to infinity is easy: one considers considering a sequence of n-quantizers α(n) := {α1 , . . . , αn } where (αn )n≥1 is an everywhere dense sequence in the support of the distribution PX . On the other hand, the question of the rate of convergence is a less straightforward question. This is the second aim of this paper is to elucidate this rate of convergence for a wide class of Brownian diffusions. In finite dimension, it is a very old story which starts in the late 1940’s: if E = Rd the answer is provided to a large extend by the so-called Zador Theorem. It says (see [7]) that if E|X|r+ε < +∞ for some ε > 0, then en,r (X) ∼ c(r, d,  . ) × c(r, PX ) × n− d 1

where c(r, PX ) = g

d d+r

and g =

dPX dλd

as

n → ∞.

is the Radon-Nikodym density of the (absolutely continuous

part of) PX with respect to the Lebesgue measure λd on Rd . When PX is singular (g = 0), 1 then en,r (X) = o(n− d ). (Finding the right asymptotics for some singular measure like the uniform measure on the Cantor space or more general fractal sets has been investigated as well, see [7, 8, 3]). In infinite dimension, the case of Gaussian processes was the first to have been extensively investigated, initially in the quadratic case (i.e. p = r = 2, see [14, 15]), then for more general norms and powers (1 ≤ p ≤ +∞, 0 < r < +∞ in [9]). In the quadratic case, the upper-bounds for the rates are derived by some Hilbertian optimization methods and the lower bounds using a connection with Shannon entropy. Under some regularly varying assumptions on the reordered eigenvalues of the covariance operator of the process, the asymptotic rates of these lower and upper bounds coincide and hence provide and exact rate. This applies (see [14]) to many usual classes of Gaussian processes (Fractional Brownian motions, Gaussian stationary processes like the fractional Ornstein-Uhlenbeck processes, Gaussian diffusions, their multi-parameters random fields counterparts,. . . ). It turns out that for many (one-parameter) processes, these rates are O ((log n)−µ ) where µ is the H¨older regularity of the application t → Xt from [0, T ] into L2 (Ω, P). For many of the above examples (see [15]), this result can be refined into a sharp rate en,2 (X, L2 ) = cX (log n)−µ + o((log n)−µ )

as

n → +∞,

An E-valued random vector X is a Radon vector if its distribution PX is supported by countably many compacts. This is always the case if E is separable like the LpT spaces, 1 ≤ p < +∞. The L∞ space is not separable but the T subspace (C([0, T ]),  . sup ) is; hence, any pathwise continuous process is an L∞ Radon random vector. T 1

3

with a known constant cX , using some elements from Shannon’s rate-distortion theory. More generally, in [15] are established some sharp rate results for a wide class of Gaussian processes with an explicit real constant cX . They are based on the regular variation exponent of the covariance operator spectrum of the process. For general Lp -norms and exponents r > 0, the exact rates results still holds under similar assumptions (whereas the existence of sharp asymptotics as above remains an open question). They appear then as a consequence of the connection between functional quantization and small Lp -ball probability problem successively established in [5, 4] and [9]. The special interest of the quadratic case p = r = 2 is not limited to the existence of sharp asymptotics. The Hilbertian techniques that are developed to establish the upper-bounds are also constructive: they rely on the construction of some product quantizers of d-dimensional marginals which can be computed as soon as one has an access to optimal n-quantizers of the d-dimensional Normal distribution on Rd . Although the sharp bounds often need to consider large values of the marginal dimension d, the “scalar” product quantizers built from 1-dimensional optimal quantizers of the scalar Normal distribution provide the exact rate with a “good” constant (compared to the sharp one, see [15]). So such an approach yields some sequences (α(n) )n≥1 of L2 -quasi-optimal quantizers which can be computed (along with the mass of their Voronoi cells). By “quasi-optimal” we mean that en,2 (α(n) , X, L2 ) = O(en,2 (X, L2 )) (1.7) (with an obvious extension to general integrability parameters r and p). Let us be more specific now on the structure of these scalar product quantizers of a (centered) Gaussian process X: let (ek )k≥1 be an orthogonal basis of L2T . For every n ≥ 1, set ck := (X|ek ) √ and ξk := (X|ek )/ ck . The ξk ’s all have a N (0; 1) distribution. Then, X admits the following expansion L2 (dP⊗dt)  √ = ck ξk ek (t). (1.8) X k≥1

When this basis is the eigenbasis of the covariance operator of X, the above expansion on (ek )k≥1 is known as the Karhunen-Lo`eve expansion: the ck are the eigenvalues of the covariance operator and the random variables ξk in (1.8) are then i.i.d. so that (1.8) also holds (at least) P-a.s. for almost every t ∈ [0, T ]. A scalar product n-quantizer is then defined as follows: let (nk )k≥1 be a sequence of integers (n ) (n ) such that n = n1 n2 · · · nk · · · and let x(nk ) := {x1 k , . . . , xnkk } be an optimal nk -quantizer for the (1) (scalar) Normal distribution N (0; 1). Note that nk = 1 for k large enough and that x1 = {0}. Then one sets √ (n ) αi := ck xik k ek , i := (i1 , . . . , ik , . . . , 1, 1, . . .). (1.9) k≥1

One further interesting feature is that when the eigenbasis has no closed form, the upper-bound with the asymptotic exact rate can sometimes be obtained using another explicit orthonormal basis of L2 : This is the case e.g. of the Haar basis for the fractional Brownian motion. Such product quantizers are easy to compute since they only require the access to some 1-dimensional optimal quadratic n-quantizers of the standard Normal distribution. For a fixed n, this optimal quantizer is unique (for the standard normal distribution) and can be obtained as the solution of a simple optimization problem by the classical Newton-Raphson procedure (see [16]). This can be extended – at least partially – to product quantizers with “medium”-dimensional marginal blocks 4

since some optimal quantizers of distributions N (0; Σ) can be efficiently computed using stochastic optimization techniques as long as 2 ≤ d ≤ 10 (see [16] again). When the eigenbasis is explicit, what was mentioned in the former paragraph is even easier to apply. Thus, the Karhunen-Lo`eve expansion of the standard Brownian motion reads (in L2T and a.s.) √    2 2T (2n − 1)t , t ∈ [0, T ], Wt = ξn sin π π(2n − 1) 2T n≥1 so that we have access to some sequences of n-quantizers that provide a rate of convergence 1 c(log n)− 2 with an explicit real constant c close to the optimal one. The same remark holds true for the Brownian bridge. We can make explicit now the three main problems that we address in this paper: – Existence of (Lr ,  .p )-optimal quantizers for a wide class of processes (Xt )t∈[0,T ] . – Rate of convergence O((log n)− 2 ) of the Lp -quantization error is generic for a wide class of 1-dimensional Brownian diffusions. 1

– Exhibition of some sequences of quasi-optimal n-quantizers in the sense of (1.7) for Brownian diffusions built from (quasi-)optimal quadratic quantizers for the Brownian motion, using some transforms can be numerically approximated. We will state and prove the existence results for optimal quantizers in full generality since its range of application is much wider than Brownian diffusions. On the other hand, for the sake of simplicity, we will mostly investigate the rate problem for Brownian diffusions in the case r = p ∈ [1, +∞). In fact this case is the natural extension of the quadratic case r = p = 2. Furthermore, there will be no real loss of generality: the extension to the most general case is essentially straightforward up to some tedious computations. To this end we introduce the following notations for every p ∈ [1, +∞), en (α, X, Lp ) := en,p (α, X, Lp )

and

en (X, Lp ) := en,p (X, Lp ).

Some results will also be presented in the case p = +∞ and r = 1 (for gaussian martingales). Our results are mainly focused on 1-dimensional Brownian diffusions because we make an extensive use of the Lamperti transform. However, when the diffusion coefficient σ only depends on time t, most of what we did extends to d-dimensional diffusions. The paper is organized as follows: in Section 2, a theorem about the existence of optimal n-quantizers for Banach-valued (tight) random vectors is established under some very general assumptions. This existence problem is connected with its (bi-)dual counterpart and enlightened by a counterexample. Then the stationarity property is deeply investigated. This turns out to be an essential key for the next section. Section 3 is entirely devoted to the Lp -quantization rate for Brownian diffusions. Two settings are developed: the Lipschitz continuous setting is based on the Lamperti transform and the bounded one based on the Girsanov Theorem. Section 4 is devoted to the special case where the Brownian diffusion has a Gaussian martingale term. In that case some assumptions are relaxed and some specific constructive procedures are proposed to get quantizers achieving the optimal rate in the purely martingale case. Notations: • Let (E,  . E ) denote a Banach space and r > 0. We will denote by LrE (Ω, A, P) or LrE (P) the vector space of E-valued Radon random variables such that E(XrE ) < +∞. Then 5

1

 XLr (P) := (EXrE ) r will be called the “Lr -norm” or “Lr (P)-norm” on LrE (P) (although it is a true norm only when r ≥ 1). When E = R we will write Lr (P) instead of LrR (Ω, A, P).

• C([0, T ], R) will denote the set of real-valued continuous functions on [0, T ]. • an ∼ bn if an = bn + o(bn ) and an ≈ bn if an = O(bn ) and bn = O(an ).

• The letter C denotes a positive real constant that may vary from line to line. When it depends on some exogenous parameters (b, σ, T , etc) they may appear as subscripts. • The supremum process of a process (Xt )t∈[0,T ] will be denoted X T := sups∈[0,t] |Xs |. ◦

• A and A will stand for the closure and the interior of the set A respectively.

2

Existence and stationarity of optimal n-quantizers

Let X be a Radon (E,  . )-valued random vector with distribution PX . We will assume throughout this section that X satisfies the integrability condition

E Xr < +∞. Then lim en,r (X, E) = 0.

n→∞

(2.1)

As a matter of fact, the support of PX being separable there exists a countable subset {ξn , n ≥ 1} everywhere dense in supp(PX ). It is clear that 0 ≤ ern,r (X, E) ≤ E min X − ξi r → 0 as n → ∞ 1≤i≤n

by the Lebesgue dominated convergence Theorem. On the other hand, the existence of optimal quantizers, i.e. the fact that en,r (X, E) actually stands as a minimum needs much more care.

2.1

An abstract existence result on a Banach space

A set α ⊂ E with 1 ≤ |α| ≤ n is called an Lr -optimal n-quantizer for X if en,r (α, X, E) = en,r (X, E).

(2.2)

(where (en,r (α, X, E))r = E mina∈α X − ar ). Let Cn,r (X, E) denote the set of all Lr -optimal n-quantizers. The first results of existence for optimal quantizers are due to Cuesta-Albertos and Matr´ an (1988) and P¨ arna (1990) for uniformly convex and reflexive Banach spaces, respectively. We provide an extension to Banach spaces having the property that the closed balls form a compact system. A system K of subsets of E is called compact if each subsystem K0 of K which has the finite intersection property (i.e. the intersection of each finite subsystem of K0 is not empty) has a nonempty intersection. Let B(x, ρ) := {y ∈ E : y − x ≤ ρ} be the closed ball of radius ρ centered at x. Theorem 1 Assume that {B(x, ρ), x ∈ E, ρ > 0} is a compact system in E. Then, for every integer n ≥ 1 and every r > 0, Cn,r (X, E) = ∅. 6

Proof. Fix r > 0 and n ∈ N. Let τo denote the topology on E generated by the system {B(x, ρ)c : x ∈ E, ρ > 0} and let τ be the product topology on E n (these topologies usually do not satisfy the Hausdorff axiom). The family {B(x, ρ) : x ∈ E, ρ > 0} being a compact system, one easily checks that E is τ0 -quasi-compact(2 ). Consequently, E n is τ -quasi-compact. It is obvious that any lower semi-continuous (l.s.c.) function defined on E n then reaches a minimum. Hence, the proof X : E n → R defined by amounts to showing that the function Dn,r + X Dn,r (a) = E min X − ai r

(2.3)

1≤i≤n

X is the so-called level n Lr -distortion function). is τ -lower semi-continuous (Dn,r For every x ∈ E and a ∈ E n , set d(x, a) := min x − ai . Then 1≤i≤n

{a ∈ E n : d(x, ·)r ≤ c} =

n

{a ∈ E n : ai ∈ B(x, c1/r )}

i=1

is τ -closed for every c ≥ 0. Hence, a → d(x, a)r is τ -lower semi-continuous. In turn any convex X (and (D X )1/r ) are τ -lower combination of such functions are τ -l.s.c. as well. This implies that Dn,r n,r semi-continuous provided |supp(PX )| < ∞. X > c} is τ -open. First note that from (2.2) For general X we will show that for every c ≥ 0, {Dn,r m : Ω → E, |X m (Ω)| ≤ m, such that and (1.4), there exists a sequence of quantizations X m r lim X − X L (P) = 0. m

Consider first the case r ≥ 1. It follows from Minkowski’s inequality that, for every a ∈ E n , X (a))1/r is 1-Lipschitz on Lr (P): X → (Dn,r  

 

X Y (a)1/r − Dn,r (a)1/r | = d(X, a)Lr (P) − d(Y, a)Lr (P)  |Dn,r

≤ d(X, a) − d(Y, a)Lr (P) ≤ X − Y Lr (P) .

(2.4)

m 1/r X )1/r > c}. It follows from (2.4) that, the τ -open set {(D X m r } > c+X − X Let a ∈ {(Dn,r L ( P) n,r ) X )1/r > c}. Furthermore, it contains a for large enough m, still by (2.4). is always contained in {(Dn,r X )1/r > c} is τ -open and D X is τ -l.s.c. Hence {(Dn,r n,r When 0 < r < 1, one concludes the same way round, using now that |ur − v r | ≤ |u − v|r for every u, v ∈ R+ , one derives that for every a ∈ E n , X Y (a) − Dn,r (a)| ≤ E |d(X, a) − d(Y, a)|r ≤ X − Y rLr (P) . |Dn,r

♦

Corollary 1 (a) If there exists a linear projection Π from the bidual E ∗∗ onto E with Π ≤ 1, then Cn,r (X, E) is not empty. (b) In particular, this condition is fulfilled if E is a dual space or an abstract L-space. 2

i.e. satisfies the Borel-Lebesgue axiom – from any open covering one may extract a finite open covering – but possibly not the Hausdorff axiom.

7

Proof.(a) It is enough to show that the system K := {B(x, ρ) : x ∈ E, ρ > 0} is compact.   Compactness of the system {B(x, ρ) : x ∈ E, ρ > 0}, where B(x, ρ) is the closed ball in E ∗∗ ∗∗ ∗ with radius ρ centered at x, follows from the σ(E , E )-compactness of closed balls in E ∗∗ . The existence of a linear projection onto E of norm ≤ 1 easily implies that K is compact. (b) Dual spaces clearly satisfy (a) and all abstract L-spaces satisfy condition (a) as well (cf. [19], chap.II. 8.3). ♦ It has to be noticed that Cn,r (X, E) = ∅ for the spaces of our main interest i.e. E = LpT , 1 ≤ p ≤ +∞. In the non-quantization setting n = 1, Theorem 1 and Corollary 1 are due to Herrndorf (see [10]). Note that in general, Cn,r (X, E) may be empty even for n = 1 (see [10]). Remarks. • Concerning the Banach spaces E = LpT , the above theorem provides new existence results for the Lr -optimal quantizers in the cases p = 1 and p = ∞. • Any pathwise continuous process (Xt )t∈[0,T ] is an L∞ ([0, T ], dt)-Radon random variable since (C([0, T ],  . ∞ ) is a Polish subspace of E = L∞ ([0, T ], dt) (any probability on a Polish space is tight i.e. Radon). The above existence theorem shows that if X∞ ∈ Lr (P) for some r > 0, then, for every n ≥ 1, X has at least one Lr -optimal n-quantizer for the  . ∞ -norm. However, nothing is known about the pathwise regularity of these optimal quantizers. Surprisingly, we will see below that, for the same process, (Lr ,  . p )-optimal n-quantizers have much more regular paths (i.e. considering E = LpT and r ≥ p). The example below shows that the existence of optimal n-quantizer may fail when the assumption of Theorem 1 is not fulfilled, even for n = 1. Optimal 1-quantizers may not exist Let (E,  . ) = (c0 (N),  . sup ) where c0 (N) denotes the set of real valued sequences ξ = (ξk )k≥0 such that limk ξk = 0 and ξsup = supk |ξk |. Let (u(n) )n≥0 (n)

denote the canonical basis of c0 (N) defined by uk = δn,k where δi,j is for the Kronecker symbol. One considers an E-valued random vector X supported by {u(n) , n ≥ 0} with a distribution pn = P(X = u(n) ), n ≥ 0 satisfying pn ∈ (0, 1/2) for every n ≥ 0. Now E ∗ = l1 (N) so that E ∗∗ = ∞ (N). Consequently Assumption (a) of Corollary 1 is obviously not fulfilled. One checks in fact that the assumption of Theorem 1 is not fulfilled either since the family {B(u(n) , 1/2), n ≥ 0} has an empty intersection whereas any finite sub-family has a non empty intersection. So let n = 1 and r = 1. We will show that e1,1 (X, c0 (N)) =

inf

ξ∈c0 (N)

EX − ξsup = 1/2

and

C1,1 (X, c0 (N)) = ∅.

More precisely we will show that corresponding level 1 distortion problem extended to the Banach space ∞ (N) does have a unique solution ξmin in ∞ (N) given by (ξmin )k = 1/2, k ≥ 0, that is C1,1 (X, ∞ (N)) = {ξmin } which in turn implies that it admits no solution in c0 (N). In fact,

E X − ξmin sup =

∞ 

pn u(n) − ξmin sup = 1/2.

n=0

For an arbitrary ξ ∈ for every n = n0 ,

∞ (N)

one gets the following: if u(n0 ) − ξsup < 1/2 for some n0 ≥ 0, then,

u(n) − ξsup Hence

≥ u(n) − u(n0 ) sup − un0 − ξsup = 1 − u(n0 ) − ξsup .

E X − ξsup =



pn u(n) − ξsup

n≥0

8

(2.5)

≥



pn (1 − u(n0 ) − ξsup ) + pn0 u(n0 ) − ξsup

n =n0

= 1 − pn0 − (1 − 2pn0 )u(n0 ) − ξsup 1 > 1 − pn0 − (1 − 2pn0 ) 2 = 1/2. In case u(n) − ξsup ≥ 1/2 for every n ≥ 0, one clearly obtains

E X − ξsup =



pn u(n) − ξsup ≥ 1/2.

n≥0

According to the above reasoning, any ξ ∈ ∞ (N) that achieves the infimum must satisfy u(n) − ξsup = 1/2 for every n ≥ 0 which clearly implies ξ = ξmin . Finally e1,1 (X, ∞ (N)) = EX − ξmin sup = 1/2 and EX − ξsup > 1/2, ξ = ξmin , ξ ∈ ∞ (N). / c0 (N), it follows that C1,1 (X, c0 (N)) is empty (as defined above with respect Now since ξmin ∈ to c0 (N)). On the other hand, as a minimizing sequence from c0 (N) one may choose a(m) = 1 2

m 

n=0

u(n) , m ≥ 0. Then

EX − a

(m)

sup =



pn u

(n)

−a

(m)

sup

n≥0

m ∞  1 m→+∞ = pn + pn −→ 1/2. 2 n=0 n=m+1

This example is enlightened by the general Theorem 2. This theorem solves the correspondence between the quantization problem in E and in E ∗∗ . It shows that the quantization error does not decrease when X is seen as random vector in the bidual E ∗∗ of E and that the set of its optimal n-quantizers as an E-valued random vector is made up with those of its optimal n-quantizers as an E ∗∗ -valued random vector that lie in E. In particular, Cn,r (X, E) = ∅ corresponds to the phenomenon that any optimal n-quantizer of Cn,r (X, E ∗∗ ) has at least one element in E ∗∗ \ E : this is precisely what happens in the above example. Theorem 2 (a) We have In particular,

en,r (X, E) = en,r (X, E ∗∗ ). Cn,r (X, E) = {α ∈ Cn,r (X, E ∗∗ ) : α ⊂ E}.

If |supp(PX )| ≥ n, then e1,r (X, E) > · · · > en,r (X, E). (b) Assume that situation (a) of Corollary 1 is satisfied. Further assume supp(PX ) = E. Then Cn,r (X, E) = Cn,r (X, E ∗∗ ). The proof of this theorem is deferred in the Annex 1 at the end of the paper. Remark. It is to be noticed that the situation C1,r (X, E) = ∅ never occurs for Gaussian (Radon) random vectors X. In view of Lemma 3 (in the Annex below), we may assume without loss of 9

generality that X is centered. Let r > 0. It follows from the Anderson inequality (see [12]) that, for every ξ ∈ E,

EX − ξr =

 +∞ 0

P(X − ξr ≥ ρ)dρ ≥

 +∞ 0

P(Xr ≥ ρ)dρ = EXr

so that {0} ∈ C1,r (X, E) = ∅. However, it remains an open question whether Cn,r (X, E) may be empty for n ≥ 2 or not. We provide some further interesting properties of n-optimal quantizers (they can be seen as necessary conditions for n-optimality). Their proofs are literally the same as those (established in finite-dimension) of Theorem 4.1 and Theorem 4.2 in [7] respectively. They are related with the Voronoi partitions induced by a n-quantizer α := {α1 , . . . , αn } ∈ E: these are the Borel partitions (Ci (α))αi ∈α of E which satisfy



Ci (α) ⊂ Vi (α) := ξ ∈ E | ξ − αi  = min ξ − αj  . αj ∈α

(2.6)

Let us note that for every αi ∈ α, 



ξ ∈ E | ξ − αi  < min ξ − αj  αj =αi

◦

◦

⊂ C i (α) ⊂ V i (α) ⊂ Ci (α) ⊂ Vi (α).

Furthermore, as soon as the norm (E,  . ) is strictly convex (3 ), any Voronoi partition satisfies for every αi ∈ α Ci (α) = Vi (α) (2.7) 

and ◦

◦

C i (α) =V i (α) =



ξ ∈ E | ξ − αi  < min ξ − αj  . αj =αi

Proposition 1 Assume that |supp(PX )| ≥ n. (a) Let α ∈ Cn,r (X, E). Then |α| = n and for every αi ∈ α,

PX (Ci (α)) > 0

{αi } ∈ C1,r (PX ( · |Ci (α)), E), i = 1, . . . , n.

and

(b) Assume that E is smooth (4 ) and strictly convex. If α ∈ Cn,r (X, E) and (r > 1)

or

(r = 1 and P(X ∈ α) = 0),

then

PX (Vi (α) ∩ Vj (α)) = 0

for every i, j ∈ {1, . . . , n}, i = j.

Note that, under the strict convexity assumption, (2.8) is then equivalent to both (∀ i ∈ {1, . . . , n}, 3 4

PX (∂Ci (α)) = 0) and (∀ i ∈ {1, . . . , n},

PX (∂Vi (α)) = 0) .

i.e. B .  (0; 1) is a strictly convex set: ∀x, y ∈ S .  (0; 1), x = y, ∀λ∈ (0, 1), λx + (1 − λ)y < 1. i.e. the norm is Gateaux-differentiable at every x = 0.

10

(2.8)

2.2

The Lr -stationary quantizers

We will introduce a notion of Lr -stationary quantizer as the critical points of level n Lr -distortion X formerly defined by Equation (2.3). function Dn,r Definition 2 A n-quantizer α := {α1 , . . . , αn } ⊂ E of size n is called admissible for X if 

(i)

PX (Vi (α)) > 0,

i = 1, . . . , n,

(ii) PX (Vi (α) ∩ Vj (α)) = 0,

i, j = 1, . . . , n, i = j.

A n-tuple (α1 , . . . , αn ) ∈ E n is admissible if its associated n-quantizer is. X is Proposition 2 Assume that E is smooth. Let r > 1. Then the Lr -distortion function Dn,r Gateaux-differentiable at every admissible n-tuple (α1 , . . . , αn ) with a Gateaux differential given by







X (α1 , . . . , αn ) = r E 1Ci (α)\{αi } (X)X − αi r−1 ∇ . (αi − X) ∇Dn,r

1≤i≤n

∈ (E ∗ )n

where (Ci (α))1≤i≤n denotes any Voronoi partition induced by α. If the norm is Fr´echet-differentiable X (α , . . . , α ) is the Fr´ at every x = 0, then ∇Dn,r echet derivative. Furthermore, if E is uniformly 1 n X (α , . . . , α ) is continuous on the set of admissible n-tuples smooth, then (α1 , . . . , αn ) → ∇Dn,r 1 n (where E ∗ is endowed with its norm). When r = 1, the above results extend to admissible n-tuples with PX ({α1 , . . . , αn }) = 0. Remark. In case E = L1T , the above theorem does not apply since the  . 1 -norm is neither smooth nor strictly convex. Proof. A straightforward adaptation of Lemma 4.10 in [7] yields both differentiability properties. Then, if E is uniformly smooth, the mapping x → ∇ . (x) is continuous (see [1]). One derives X by the Lebesgue dominated convergence theorem using that ∇ .  takes the continuity of ∇Dn,r its values in the unit ball of E ∗ . ♦ Definition 3 Let E be a Banach space and let r ≥ 1. A n-quantizer α for X is called Lr -stationary for X if
 E 1Ci (α)\{αi } (X)X − αi r−1 ∇ . (αi − X) = 0, i = 1, . . . , n. (2.9) where (Ci (α))1≤i≤n denotes any Voronoi partition induced by α. (This requires that the Gateauxdifferential ∇ . (αi −ξ) is defined PX (dξ)-a.e. on Ci (α)\{αi } and, furthermore, that P(X ∈ α) = 0 when r = 1). This finally leads to the following proposition which makes the (expected) connection between optimality and stationarity. Proposition 3 (a) Assume that E is smooth and strictly convex. Let r > 1. Assume that X ∈ LrE (P) and |supp PX | ≥ n. Then any Lr -optimal n-quantizer α is Lr -stationary (and admissible) for X. This extends to r = 1 if PX (α) = 0. (b) If E = L1T , assertion (a) (except for admissibility) remains true if X satisfies furthermore

PXt

is continuous on R

dt-a.e.

(Note that assumptions |supp PX | ≥ n and PX (α) = 0 are then always fulfilled). 11

Proof. (a) Any Lr -optimal n-quantizer α is admissible by Proposition 1(b), hence the GateauxX (α) does exist and is 0 which exactly means stationarity. differential ∇Dn,r (b) The space E is not smooth. However,  . 1 is Gateaux-differentiable at every ξ such that ξ(t) = 0 dt-a.e.. Now, by the Fubini Theorem, one has for every Borel integrable function a : [0, T ] → R, 

Ω

λ(t : Xt (ω) = a(t)) P(dω) =

 T 0

P(Xt = a(t)) dt = 0

i.e. (Xt − a(t) = 0 dt-a.e.) P-a.s.. Let αi ∈ α and Pi := P( . |{X ∈ Ci (α)}). This definition is consistent since P({X ∈ Ci (α)}) > 0 by Proposition 1(a). It follows easily that Ψi : a →  X − ar1 dPi is Gateaux differentiable with a Gateaux-differential given by ∇Ψi (a) = r



X − ar−1 sign(a − X)dPi . 1

Now, still following Proposition 1(a), αi is a minimum for Ψi so that its Gateaux differential is zero. Hence, for every αi ∈ α, 

2.3

1Ci (α) (X)X − ar−1 sign(a − X)dP = 0. 1

♦

Stationarity for stochastic processes and applications

Let X = (Xt )t∈[0,T ] be a bi-measurable process defined on a probability space (Ω, A, P) and let p, r ∈ [1, +∞). Assume that Xp ∈ Lr (Ω, P) i.e.

E

 T 0

r/p

|Xt | dt p

< +∞.

(2.10)

Then, the process X can be seen as a random vector taking its values in the Banach space (E,  . ) = (LpT ,  . p ) satisfying an Lr -integrability property, that is X ∈ LrLp (Ω, P). When p = 1, T the LpT spaces are uniformly smooth and strictly convex, so the abstract results obtained in the former section apply. Furthermore, if q denotes the conjugate H¨ older exponent of p, for every f ∈ LpT , f ≡ 0, 

∇ . p (f ) =

|f | f p

p−1

signf ∈ LqT .

so that the (Lr ,  . p )-stationarity condition reads for any Voronoi partition (Ci (α))1≤i≤n ,




E 1Ci (α) (X)X − αi r−p |αi − X|p−1 sign(αi − X) = 0, p with the convention

0 |0|

i = 1, . . . , n.

(2.11)

= 0. When p = 1, the condition is formally the same. This may be written

:= X α of X defined by (1.3), namely: in a more synthetic way by introducing the α-quantization X




r−p |X − X| p−1 sign(X − X) | X = 0. E X − X p

(2.12)

When p = 2, r ≥ 2 (and P(X = x) = 0 for every x ∈ L2T if r > 2), Equation (2.11) looks simpler and reads r−2 L2 E(X1Ci (α) X − αi 2 ) αi = 1 ≤ i ≤ n. (2.13) E(1Ci (α) X − αi r−2 ) 2 One derives from Proposition 3 the following corollary. 12

Corollary 2 Let p, r ∈ [1, +∞), let n ≥ 1. If  p, r > 1      

and |suppPX | ≥ n,

p > 1, r = 1 and PX is continuous,

(2.14)

and PXt is dt-a.e. continuous,

p = 1, r ≥ 1

then, any (Lr ,  . p )-optimal n-quantizer is (Lr ,  . p )-stationary in the sense of (2.11). 2.3.1

Pathwise regularity of stationary quantizers (1 ≤ p ≤ r < +∞)

We will derive from Equations (2.11) (and (2.13)) some pathwise continuity result for the (Lr ,  . p )stationary quantizers (which extends a result established in [14] in the purely quadratic case p = r = 2). Theorem 3 Let p, r ∈ [1, +∞), r ≥ p. Let X be a bi-measurable process satisfying (2.10) and ∀ t ∈ [0, T ],

Xt ∈ Lr−1 (Ω, P).

When r > p, assume furthermore that its distribution PX is continuous. Let α := (α1 , . . . , αn ) be an (Lr ,  . p )-stationary n-quantizer (in the sense of (2.11)). (a) If p = 1, if X is pathwise continuous with supp PX = C([0, T ], R) (in case X is viewed as a C([0, T ], R) random vector) and if the distribution PXt of Xt is continuous on the real line for every t ∈ [0, T ] (or simply on (0, T ] if X0 = x0 ∈ R), then α has a version consisting of continuous functions (such that αi (0) = x0 , i = 1, . . . , n, if X0 = x0 ). (b) If p ∈ (1, +∞) and if t → Xt is continuous from [0, T ] into Lr−1 (Ω, P), then α has a version consisting of continuous functions. Furthermore, if X0 = x0 ∈ R, then αi (0) = x0 , i = 1, . . . , n. The continuity assumption is fulfilled e.g. if X is pathwise continuous and X T ∈ Lr−1 (Ω, P). (c) If p = 2, then α has a version such that ∀ s, t ∈ [0, T ],

max |αi (t) − αi (s)| ≤ CX,α Xt − Xs Lr−1 (Ω,P)

1≤i≤n

for some real constant CX,α > 0. Proof: For every i ∈ {1, . . . , n}, set Qi,r = 1Ci (α) (X)X − αi r−p .P. The measure Qi,r is finite: if p r = p, this is obvious, otherwise,


Qi,r (Ω) ≤ E X − αi r−p ≤ EX − αi rp p

1− p r

< +∞.

On the other hand, Qi,r is a nonzero measure since P(X ∈ Ci (α)) > 0 and P(X = αi ) = 0. Now, define on R × [0, T ] the function Φi by 

Φi (a, t) := Ω

ϕp−1 (a − Xt )dQi,r

where

ϕq (x) = sign(x)|x|q .

First note that the function Φi is real valued. If r > p > 1, the Young inequality with p = r−1 and q  = r−p implies ≤ C(|a − Xt |r−1 + X − αi r−1 ) |a − Xt |p−1 αi − Xr−p p p + |Xt |r−1 + Xr−1 ). ≤ C(|a|r−1 + αi r−1 p p 13

r−1 p−1

so that |a − Xt |p−1 αi − Xr−p ∈ L1 (Ω, P). When r = p (or p = 1), the result is obvious. p (b) For every fixed t ∈ [0, T ] and p > 1, a → ϕp−1 (a − Xt ) is (strictly) increasing, hence a → Φi (a, t) is strictly increasing too. The continuity of a → Φi (a, t) on R follows from the Lebesgue dominated convergence Theorem. Furthermore, for every a ≥ 0, Φi (a, t) ≥

 {Xt ≤a}

ϕp−1 (a − Xt )dQi,r −



|Xt |p−1 dQi,r

so that lim Φi (a, t) = +∞ by Fatou’s Lemma. Similarly, lim Φi (a, t) = −∞. a→+∞

a→−∞

The proof reduces to providing an argument for the continuity of t → Φi (a, t). – If 1 < p ≤ 2, one starts from the inequality |ϕp−1 (u) − ϕp−1 (v)| ≤ 22−p |u − v|p−1

u, v ∈ R.

When r > p, the Holder inequality applied with the conjugate exponents

r−1 p−1

r−1 r−p

and

yields

r−p |Φi (a, t) − Φi (a, s)| ≤ 22−p Xt − Xs p−1 Lr−1 (P)  X − αi p Lr−1 (P) .

This still holds if r = p. – if p > 2, one starts from |ϕp−1 (u) − ϕp−1 (v)| ≤ (p − 1)(|u| ∨ |v|)p−2 |u − v|, Since r > 2 the Holder Inequality applied with r − 1 and

r−1 r−2

u, v ∈ R.

yields






1Ci (α) (X) |Φi (a, t)−Φi (a, s)| ≤ (p−1)E |Xt − Xs | (|a − Xt |∨|a − Xs |)p−2 X − αi r−p p  

≤ (p−1)Xt −Xs Lr−1 (P) E (|a−Xt |∨|a−Xs |)

(p−2)(r−1) r−2

(r−p)(r−1) r−2 p

 r−2

X −αi 

r−1

.

A new application of the Holder Inequality to the expectation in the right hand side of the above inequality yields 

r−p

  |Φi (a, t) − Φi (a, s)| ≤ (p − 1)Xt − Xs Lr−1 (P) |a − Xt | ∨ |a − Xs |p−2 Lr−1 (P) X − αi p Lr−1 (P)




p−2 ≤ Cp,αi Xt − Xs Lr−1 (P) |a|p−2 + Xs p−2 Lr−1 (P) + Xt Lr−1 (P) .

Owing to these properties, one easily checks that for every t ∈ [0, T ], the equation Φi (ai (t), t) = 0 admits a unique solution and that the implicitly defined function t → ai (t) is continuous. On the other hand the function αi satisfies dt-a.e. Φi (αi (t), t) = 0 so that αi (t) = ai (t) dt-a.e.. If X0 = x0 ∈ R, then Φi (a, 0) = ϕp−1 (a − x0 )Qi,r (Ω) so that ai (0) = x0 . (a) When p = 1,



Φi (a, t) = Ω

sign(a − Xt ) dQi,r .

The continuity on R × [0, T ] (on R × (0, T ] respectively) follows from the pathwise continuity of X and from the continuity of PXt by the Lebesgue dominated convergence Theorem: the sign function is bounded and Qi,r  P. Similarly one shows that lim Φi (a, t) = ±Qi,r (Ω). It is also obvious a→±∞

14

that a → Φi (a, t) is nondecreasing. To establish strict monotonicity, one proceeds as follows: let us consider the subset of Ci (α) defined by Ui (α) := {ξ ∈ C([0, T ], R) : ξ − αi 1 < min ξ − αj 1 }. j =i

It is a nonempty open subset of (C([0, T ], R),  . 1 ) since (C([0, T ], R),  . 1 ) is everywhere  . 1 dense in L1T . Now, for everu nonempty open interval I the set {ξ ∈ C([0, T ], R), ξ(t) ∈ I} is clearly everywhere dense in (C([0, T ], R),  . 1 ) so that Ui (α) ∩ {ξ ∈ C([0, T ]), ξ(t) ∈ I} is a nonempty set. On the other hand, ξ →  ξ 1 and ξ → ξ(t) are both continuous as functionals on (C([0, T ], R),  . sup ) so that Ui (α) ∩ {ξ ∈ C([0, T ]), ξ(t) ∈ I} is a (nonempty) open set of (C([0, T ], R),  . sup ). Now, if Φi (a, t) = Φi (a , t) for some a < a , then Qi,r (Xt ∈ (a, a )) = 0. PX being continuous X − αi 1 > 0 P-a.s., hence Qi,r is equivalent to 1Ci (α) (X).P. Consequently

P({X ∈ Ui (α)} ∩ {Xt ∈ (a, a )}) = 0. This is impossible owing to the assumption on the support of PX . Consequently a → Φi (a, t) is strictly increasing and one concludes like in the case p > 1 to the existence of a continuous version of α on [0, T ] (or simply on (0, T ] if PX0 is not continuous). When X0 = x0 one must consider the behaviour at 0 more carefully. First note that for every a = x0 , limt→0 Φi (a, t) = Φi (a, 0) = sign(a − x0 )Qi,r (Ω) since X is pathwise continuous. Now let (tn ) be a sequence of positive real numbers going to 0 such that αi (tn ) > x0 + η for some η > 0. Then, Φi (x0 + η, tn ) ≤ Φi (αi (tn ), tn ) = 0 for every n ≥ 1 so that sign(η)Qi,r (Ω) = Φi (x0 + η, 0) ≤ 0 which is impossible. Hence lim supt→0 αi (t) ≤ x0 . One shows similarly that lim inf t→0 αi (t) ≥ x0 i.e. limt→0 αi (t) = x0 . (c) It is a consequence of Equation (2.13): |αi (t) − αi (s)| =

E((Xt − Xs )Li ) E(Li )

with

. Li = 1Ci (α) X − αi r−2 2

When r > 2, The Holder Inequality yields the announced result with ))(r−2)/(r−1) /(E(1Ci (α) X − αi r−2 )). CX,α := max (E(X − αi r−1 2 2 1≤i≤n

When r = 2, one sets accordingly CX,α := 1/ min1≤i≤n P(X ∈ Ci (α)).

♦

Examples: • The (Lr ,  . p ) stationary n-quantizers, 1 ≤ p ≤ r < +∞, of the standard Brownian motion and are made up with continuous functions which are null at 0, 1/2-Holder if p = 2. The same result holds for the Brownian bridge over [0, T ] and an obvious adaptation of the above proof shows that any of its stationary quantizers are 0 at T . • The (Lr ,  . p ) stationary n-quantizers, 1 ≤ p ≤ r < +∞, of the Brownian diffusion solution of the S.D.E. (1.1) are continuous provided that b and σ have linear growth. • The (Lr ,  . p ) stationary n-quantizers, 1 < p ≤ r < +∞, of the standard compound Poisson processes (with intensity λ > 0) are made up with continuous functions which are null at 0. When p = 2 they are 12 -Holder. When p = 1, the distributions PXt are purely discrete, so the theorem does not apply (and the regularity of (Lr ,  . 1 )-stationary quantizers is an open question).

15

A counter-example when p = +∞ (due to S. Graf [6]): We will exhibit a bounded pathwise continuous process X having a discontinuous (L1 ,  . ∞ )-optimal 1-quantizer. Set, for every n ≥ 1

ξn (t) :=

  0 if t ∈ [0, 1/2 − 2−n ]       2n (2t − 1) + 2 if t ∈ [1/2 − 2−n , 1/2 − 2−(n+1) ]   2n (1 − 2t)     

if t ∈ [1/2 − 2−(n+1) , 1/2]

−ξn (1 − t)

if t ∈ [1/2, 1]

Then, set

PX =



p n δ ξn

n≥1

where (pn )n≥1 is a probability distribution with 0 < pn < 1/2 for every n ≥ 1. Then set ξmin := 1[0,1/2) − 1(1/2,1] . One checks that, for every n ≥ 1, ξn − ξmin ∞ = 1/2 so that, for every ∀ r ∈ [1, +∞],

 X − ξmin ∞ Lr (P) = 1/2.

On the other hand, one shows like in the former counterexample that for every ξ ∈ L∞ ([0, T ]), EX − ξ∞ ≥ 1/2: one reproduces the string of inequalities starting at (2.5) once noticed that for every n = m, ξn − ξm ∞ = 1. Consequently, the  . Lr (P) -norm being nondecreasing as a function of r, ∀ r ∈ [1, +∞], e1,r (X, L∞ ([0, T ])) = e1,r ({ξmin }, X, L∞ ([0, T ])) = 1/2. The discontinuous function ξmin is an (Lr ,  . ∞ )-optimal 1-quantizer of the pathwise continuous process X, 1 ≤ r ≤ +∞. Note that t → Xt from [0, T ] into Lp (Ω, P) is continuous for any p ∈ [1, +∞) since X is pathwise continuous and uniformly bounded by 1. Consequently it follows from Theorem 3 that, as soon as 1 < p ≤ r < +∞, any (Lr ,  . p ) optimal n-quantizer of X consists of continuous functions. However, t → Xt from [0, T ] into L∞ (Ω, P) is not continuous (at t = 1/2), so the pathwise regularity of an optimal (Lr ,  . ∞ )-optimal n-quantizer of an L∞ (Ω, P)-contiguous process remains open. But the  . ∞ -norm being nowhere Gateaux-differentiable, the very notion of stationary quantizer no longer exists. So this would require to develop a new approach. 2.3.2

Convexity moment inequality (r = p ≥ 2)

When r = p = 2, Equation (2.12) simply reads: = E(X | X). X

(2.15)

This identity has many interesting consequences based on Jensen inequality (in particular some ≤ X for every ρ ≥ 1). moment inequalities like X ρ ρ The case p = r > 2: It is possible to derive from Equation (2.12) some similar – although less One defines for every t ∈ [0, T ] a probability Qp by straightforward – consequences for X. t

Qpt =

t |p−2 |Xt − X

E|Xt − X t |p−2 16

.P.

Then, for dt-almost every t ∈ [0, T ],




t = E p Xt | X . X Qt

(2.16)

This identity is the extension of (2.15). One still derives by the conditional Jensen inequality that, for every convex function Φ : R → R+ ,


t ) ≤ E p Φ(Xt ) | X Φ(X Qt



dt-a.e.

Finally, integrating with respect to Qpt ,









EP Φ(X t )|Xt − X t |p−2 ≤ EP Φ(Xt )|Xt − X t |p−2 ≤ +∞

3

dt-a.e.

(2.17)

Functional Quantization of Brownian diffusions

We come now to the Brownian diffusion model (1.1) described in the introduction, assuming that it admits at least one weak solution(5 ). In the main two theorems of this section, we will rely on the following assumption on the drift b and the diffusion coefficient σ 

(i)

σ ∈ C 1 ([0, T ] × R, R),

(3.2)

(ii) ∀ (t, x) ∈ [0, T ]× R, |b(t, x)| ≤ C(1+|x|) and 0 < σ(t, x) ≤ C(1+|x|),

This assumption will be the starting point to “remove” the diffusion coefficient of X: we will introduce a new diffusion Yt := S(t, Xt ) which will satisfy a new S.D.E. whose diffusion coefficient will be constant equal to 1. This function S is often called in the literature the Lamperti transform. As emphasized by the computations below this is a mainly 1-dimensional procedure. The Lamperti transform is defined for every (t, x) ∈ [0, T ] × R by  x

S(t, x) := 0

dξ . σ(t, ξ)

Under this assumption, the function is defined on [0, T ] × R and is C 1,2 ([0, T ] × R) with ∂S (t, x) = − ∂t

  x 1 ∂σ 0

σ 2 ∂t



(t, ξ)dξ,



1 ∂σ ∂S 1 ∂2S (t, x) = − (t, x). (t, x) = and 2 ∂x σ(t, x) ∂x σ 2 ∂x

5

Many criterions for the existence of weak solutions can be found in the literature (see e.g. Theorem 4.22, p. 323, in [11]). Let us mention simply among others the Skorokhod-Stroock-Varadhan Theorem (see e.g. [11]): a weak solution exists as soon as b and σ are bounded and continuous on Rd . On the other hand a unique strong solution (hence weak) solution exists as soon as b and σ are Lipschitz continuous in x uniformly in t ∈ [0, T ] and |b(., 0)| + |σ(., 0)| is bounded (see, e.g., Theorem A.3.3 in [2]). In 1-dimension other criterions are available like the Engelbert-Schmidt criterion (see [11], Theorem 5.15 p.341): If b(t, x) = b(x) and σ(t, x) = σ(x) are Borel functions with at most linear growth such that ∀ x ∈ R, σ 2 (x) > 0



x+ε

and x−ε

1 + |b(y)| dy < +∞ for some ε > 0 σ 2 (y)

then SDE (1.1) has a unique weak solution starting from a given distribution µ.

17

(3.1)

Furthermore, for every t ∈ [0, T ], x → S(t, x) is continuous and strictly increasing. It follows from (3.2)(ii) that |St (x)| ≥ C1 log(1 + |x|). Hence, for every t ∈ [0, T ], S(t, . ) is one-to-one from R onto R and admits a (continuous) inverse function denoted St−1 satisfying St−1 (0) = 0. Furthermore, St−1 is differentiable with a derivative (St−1 ) (y) satisfying 0 < (St−1 ) (y) = σ(t, St−1 (y)) ≤ C(1 + |St−1 (y)|) so that, by the Gronwall Inequality, for every y ∈ R, |St−1 (y)| ≤ eC|y| − 1. Consequently 

|St−1 (y  ) − St−1 (y)| ≤ CeC(|y|∨|y |) |y − y  |.

∀ t ∈ [0, T ], ∀ y, y  ∈ R,

(3.3)

Furthermore, when σ is bounded over [0, T ] × R by σsup , one easily checks that St−1 is σsup Lipschitz continuous, namely |St−1 (y  ) − St−1 (y)| ≤ σsup |y − y  |

∀ t ∈ [0, T ], ∀ y, y  ∈ R,

(3.4)

Consequently, if one sets Yt := S(t, Xt ), Itˆ o’s formula yields Yt

=

β(t, Yt ) dt + Wt 

with

β(t, y) :=

b − σ

Y0 = S(0, x0 ) =: y0

  x 1 ∂σ 0

σ 2 ∂t

(3.5)



(t, ξ)dξ −

1 ∂σ (t, St−1 (y)). 2 ∂x

(3.6)

Note that, (t, y) → St−1 (y) is continuous on [0, T ] × R since both sets ≥

≥

{(t, y) : St−1 (y) ≤ c} = {(t, y) : y ≤ S(t, c)} are closed for every c ∈ R. Therefore, if Assumption (3.2) holds, β : [0, T ] × R → R is a Borel function, continuous as soon as b is.

3.1

Brownian diffusions: the Lipschitz setting

Theorem 4 Assume that the coefficients b and σ of the diffusion X satisfy Assumption (3.2). Assume furthermore that the function β defined by (3.6) satisfies β is Lipschitz continuous in y uniformly in t ∈ [0, T ]

(3.7)

(with a Lipschitz coefficient denoted [β]Lip ). (a) Then ∀ p ∈ [1, +∞),



en (X, Lp ) ≤ Cb,σ,T,p en (W, Lp ),

(3.8)

with p = 2 if 1 ≤ p < 2 and any p > p + 2 if p ≥ 2, so that




en (X, Lp ) = O (log n)− 2 . 1

When σ is bounded by σsup , one may set p = p and Cb,σ,T,p = σsup e2 still for bounded σ, Assumption (3.7) is satisfied as soon as (t, x) → 1 ∂σ 2 ∂x (t, x)

is Lipschitz continuous in x uniformly in t ∈ [0, T ]. 18

p−1 [β]p T p−1 /p Lip

b σ (t, x)

. Moreover,

 
− 0x σ12 ∂σ ∂t (t, ξ)dξ −

(b) If, furthermore, σ > ε0 > 0, en (X, Lp ) ≥ so that

21−1/p (1

ε0 en (W, Lp ) + [β]pLip T p )1/p

en (X, Lp ) ≈ (log n)−1/2

(3.9)

n → ∞.

as

Some extensions and examples: • Homogeneous case (σ(t, x) = σ(x) and b(t, x) = b(x)). If b is differentiable and σ is twice differentiable, then β is differentiable. Then Assumption (3.7) is equivalent to β  bounded that is b − b

σ 1 − σσ” σ 2

is bounded.

(3.10)

Note that, under this assumption and those of Theorem 4, the diffusion Equation (1.1) admits a unique strong solution (see Exercise 2.20 in [11]). • Still in the homogeneous case, one may relax the regularity assumption on σ into σ(y) − σ(x) = Then, S  (y) − S  (x) =

 y  σ (u)

σ 2 (u)

x

 y

σ  (u)du

for

x

σ  ∈ L1loc (dx)

du so that the extended Itˆ o’s formula applies to get (3.5) (see [18],

IV-45). • Theorem 4 admits a natural extension to the case where the diffusion Xt a.s. lives in an open interval I of the real line on which σ(t, . ) > 0 for every t ∈ [0, T ]. The function S needs to be appropriately modified by setting  x

∀ t ∈ [0, T ], ∀ x ∈ I,

S(t, x) := x1

dξ σ(t, ξ)

where x1 is an arbitrary fixed value lying inside I. One modifies Assumption (3.2) into 

(i)

σ ∈ C 1 ([0, T ] × I, R),

(ii) ∀ (t, x) ∈ [0, T ]×I, |b(t, x)| ≤ C(1+|x|) and 0 < σ(t, x) ≤ C(1+|x|). Furthermore, assume that ∀ t ∈ [0, T ],

 [x1 ,+∞)∩I

dξ = +∞ σ(t, ξ)



and (−∞,x1 ]∩I

dξ = +∞. σ(t, ξ)

Then, the conclusion of Theorem 4 still holds if Assumption (3.7) is satisfied. The adaptation of the proof is straightforward. • The Black & Scholes model: The above extension shows that the rate obtained in Theorem 4 applies to the classical Black & Scholes Equation (geometric Brownian motion). dXt = rXt dt + ϑ Xt dWt ,

19

X0 = x0 > 0

where r ∈ R and ϑ > 0 are real numbers. It suffices to consider the open interval I = (0, +∞) (then β(t, y) := r/ϑ − ϑ/2). • The local volatility model: More generally, it applies – still with I = (0, ∞) – to usual some usual extensions like the models with local volatility dXt = rXt dt + ϑ(Xt ) Xt dWt ,

X0 = x0 > 0

where ϑ : R → (ε0 , +∞) is a bounded (and bounded away from 0) twice differentiable function C C satisfying |ϑ (x)| ≤ 1+|x| and |ϑ (x)| ≤ 1+|x| 2 . This follows from the above Criterion (3.10). Remarks: • A careful reading of the proof below shows that the inequalities established in Theot rem 4 still hold true if the Brownian motion W is formally replaced by a Wiener integral 0 c(s)dWs , c ∈ L2T . • Basically, the proof consists in two steps: the first is to pass from X to Y using some estimates for the Lamperti transform S, the second is to pass from Y to the Brownian motion W . In both cases we will adopt a constructive approach, exhibiting some sequences of n-quantizer whose Lp quantization errors do achieve the announced rate. In particular, as far as en (Y, Lp ) is concerned, the sequence which provides the upper-bound (3.8) is obtained from a sequence of optimal Lp¯ nquantizer of the Brownian motion (with p¯ > p). The essential ingredient will be that an Lp¯ optimal n-quantizer is always stationary in the sense of (2.12) so that it satisfies Inequality (2.17) (when p¯ > 2) or (2.15) when p¯ = 2. Proof: Preliminary step One constructs an elementary quantizer y ∈ LpT as the solution (in LpT ) of the integral equation  t β(s, y(s))dt + h(t) (3.11) y(t) = y0 + 0

where h ∈ T is an elementary quantizer for the Brownian motion (by “elementary quantizer” we mean simply an element of a quantizer as a set). The existence and uniqueness in Lp ([0, T ]), dt) of a solution for (3.11) under a Lipschitz assumption for the function β classically follows from the approach used for Ordinary Differential Equations. The mapping Φp : LpT → LpT finite by Lp

Φp (y) ≡ t →

 t

β(s, y(s))ds + h(t) 0

is contracting as soon as T [β]Lip < 1. Hence Φp has a unique fixed point since LpT is a Banach space. A global solution of (3.11) in LpT can be constructed by induction for any T > 0 by simply sticking pieces of solution starting at the appropriate values. Note that when h is continuous the solution y is continuous too. Using the standard Gronwall techniques and (u + v)p ≤ 2p−1 (up + v p ), one classically shows the lemma below whose proof is left to the reader. Lemma 1 Let Y be defined by (3.5), let h ∈ Lp ([0, T ], R) and let y ∈ LpT be the solution of the above ODE (3.11). Then, for every p ∈ [1, +∞),  T

c([β]Lip , T ) with c([β]Lip , T ) =

0

|Ws − h(s)| ds ≤ p

1 2p−1 (1+[β]pLip T p )

 T 0

 T

|Ys − y(s)| ds ≤ C([β]Lip , T ) p

and C([β]Lip , T ) := e2 20

p−1 [β]p T p−1 Lip

.

0

|Ws − h(s)|p ds

(3.12)

(a) Case 1 ≤ p < 2: Let αW = (α1W , . . . , αnW ) be an L2 -optimal n-quantizer for the Brownian motion (in L2T ) and let (Ci (αW ))1≤i≤n be a Voronoi partition induced by αW . Then set n 

 := W

αiW 1Ci (αW ) (W )

i=1



e2n (W, L2 )

so that

E

=



 T

|Wt −

min

1≤i≤n 0

 T

=E

αiW (t)|2 dt

t |2 dt. |Wt − W

0

It is known (see [14]) that any optimal quadratic quantizer set of a Gaussian process lie in its self-reproducing space, so αW is a subset of the Cameron-Martin space. In particular, it is made up with continuous functions over [0, T ] (this also follows from Theorem 3). Consequently, the functions yi solutions of (3.11) with h := αiW , 1 = 1, . . . , n are continuous too and so are the functions (S. )−1 (yi ). In particular, all these functions lie in Lp ([0, T ], dt). epn (X, Lp ) ≤ epn ({(S. )−1◦ yi ), i = 1, . . . , n}, X, Lp ) 

= E

 T

min

1≤i≤n 0



≤ C pE

−1

|(St )

 T

min

1≤i≤n 0

−1

(Yt ) − (St )

 p

(yi (t))| dt 

epC(|yi (t)|∨|Yt |) |Yt − yi (t)|p dt

(3.13)

using Inequality (3.3). Now 

E



 T

epC(|yi (t)|∨|Yt |) |Yt − yi (t)|p dt

min

≤E

1≤i≤n 0

 n  i=1



 T

pC(|yi (t)|∨|Yt |) 1{W |Yt − yi (t)|p dt . =αW } e i

(3.14)

0

Since β is Lipschitz continuous, it has linear growth so that, for every t ∈ [0, T ], 

|Yt | ∨ |yi (t)| ≤ e

Kβ T



|αiW (s)|)

|y0 | + Kβ T + sup (|Ws | ∨ s∈[0,T ]

,

i = 1, . . . , n

for some positive real constant Kβ . Following Lemma 1, one then gets 

E min



 T

pC(|yi (t)|∨|Yt |)

e

1≤i≤n 0

|Yt − yi (t)| dt p

p−1





 T

t |) Kp,β,T supt∈[0,T ] (|Wt |∨|W

≤ Cβ,T,p E e

0

t |p dt |Wt − W

(3.15)

Kβ T

(|y0 |+Kβ T ) and K Kβ T . with Cβ,T,p := e2 [β]Lip /p epCe β,T,p = pCe Now, αW is stationary as an optimal quadratic quantizer i.e.




∀ t ∈ [0, T ],



t = E(Wt | W ) W

P-p.s.,

consequently, elementary conditional Jensen inequality yields 

     ≤ exp Kβ,T,p sup |Wt | + Kβ,T,p E( sup |Ws | | W ) exp Kβ,T,p sup (|Ws | ∨ |Ws |) s∈[0,T ]

s∈[0,T ]

) ≤ Z E(Z | W 

where

s∈[0,T ]



Z = exp Kβ,T,p sup |Ws | . s∈[0,T ]

21

Let q¯ := 

2 2−p

be the conjugate exponent of 2/p. H¨ older Inequality implies that 

 s |) T Kβ,T,p sups∈[0,T ] (|Ws |∨|W

E e

0

t |p dt |Wt − W

 ) q¯ T ≤ Z E(Z | W L (P ) p Z2L2¯q (P) T 1− 2

≤

1− p2

  T

p

t |2 dt E |Wt − W

2

0

epn (W, L2 ).

Combining this with Inequalities (3.24) and (3.15) yields p

epn (X, Lp ) ≤ Cβ,T,p Z2L2¯q (P) T 1− 2 epn (W, L2 ). The conclusion follows from the classical fact that the Laplace transform of sup |Ws | is finite on s∈[0,T ]

the whole real line so that ZL2¯q (P) is finite.



Case p ≥ 2: Let p ∈ (p, +∞) be a real number to specified further on. One considers an Lp  optimal n-quantizer in LTp for the Brownian motion, still denoted αW . By Theorem (3), the αiW lie in C([0, T ], R). Let y := {y1 , . . . , yn } denote the (continuous) quantizer obtained by solving (3.11)  (with p and h = αiW ). The yi ’s clearly lie in LTp and Lp (0, T ], dt). The same formal computations as those carried out in the former case yield 

epn (X, Lp )

≤ C E p

≤ Cp E where Y :=

n  i=1

 T

min

1≤i≤n 0

 T



pC(|yi (t)|∨|Yt |)

e

t |∨|Yt |) pC(|Y

e 0

|Yt − yi (t)| dt p



|Yt − Yt |p dt

(3.16)

1{W =αW } yi is a (non Voronoi) quantization process. i

Now, setting Y t := sup |Ys | and Φr (u) := (ueu )r for every r > 0 yields

E

 T

0≤s≤t

t |∨|Yt |) pC(|Y

e 0



|Yt − Yt | dt p



≤C

−p

E e

2pCY T



 T 0

Φp (C|Yt − Yt |)dt .

(3.17)

 that It follows from the very definition of Y and W

|Yt − Yt | ≤ [β]Lip

 t 0

t | |Ys − Ys |ds + |Wt − W

so that by the Gronwall Lemma |Yt − Yt | ≤

 t 0

s |µt (ds) |Ws − W

µt (ds) = [β]Lip e[β]Lip (t−s) ds + δt (ds).

where

The function Φp (C, . ) being (nondecreasing and) convex since p > 1, Jensen Inequality implies that  t
  s | µt (ds) Φp µt ([0, t])C|Ws − W Φp (C|Yt − Yt |) ≤ µt ([0, t]) 0  t

= [β]Lip

0 −t[β]Lip

+e

≤ [β]Lip

 T 0






s | ds e−s[β]Lip Φp Cβ,T |Ws − W









t | Φp Cβ,T |Wt − W






s | ds + Φp Cβ,T |Wt − W t | Φp Cβ,T |Ws − W

22

with Cβ,T := C eT [β]Lip . Integrating w.r.t. the above inequality over [0, T ] and using that Φp is non decreasing, one gets  T 0

Φp (C|Yt − Yt |) dt

≤ (T [β]Lip + 1)

 T






t | dt Φp Cβ,T |Wt − W

0

p pT [β]Lip

 T

= (T [β]Lip + 1)C e ≤ Cp,β,T

 T 0




0






t | |Wt − W t |p dt exp pCeT [β]Lip |Wt − W 

t | |Wt − W t |p dt exp Cp,β,T |Wt − W

for some appropriate real constant Cp,β,T . Consequently  T

Φp (C|Yt − Yt |) dt ≤ Cp,β,T exp (Cp,β,T W T )

0

 T 0






t | |Wt − W t |p dt. exp Cp,β,T |W

(3.18)

On the other hand, one derives classically from equation (3.11) and the Gronwall Lemma that




Y T ≤ eKβ T |y0 | + Kβ T + W T . Let ρ, θ > 1, 1/ρ + 1/θ = 1 be two conjugate real numbers. Then, 

E e

2pCY T



 T

Φp (|Yt − Yt |)dt

0



≤ Cp,β,T,y0 E e

Cp,β,T W T

 T

t| Cp,β,T |W

e 0

≤ Cp,β,T,y0 eCp,β,T |W |T Lρ (P) ×T

1− θ1

 T 0



t| θCβ,T,p |W

E e

 t | dt |Wt − W p

  θ1 pθ  |Wt − Wt | dt

applying twice H¨ older Inequality and once Fubini’s Theorem. Now, set p = pθ + 2 so that p > 2. Consequently, Inequality (2.17) applied to the convex function u → eθCβ,T,p u implies that 

t| θCβ,T,p |W

E e

t |pθ |Wt − W






t |pθ ≤ E eθCβ,T,p |Wt | |Wt − W



dt-a.s.

On the other hand, eCp,β,T W T Lρ (P) < +∞ since the Laplace transform of W T is finite on the whole real line. Hence, 

E e

2pCY T



 T 0

Φp (|Yt − Yt |)dt

  T

≤ Cp,θ,β,T,y0 E

1 θCβ,T,p |Wt |

e 0

t | dt |Wt − W pθ

One applies once again H¨ older Inequality, now with the exponents µ := 1 + This yields 

E e

2pCY T

 T 0

Φp (|Yt − Yt |)dt

 θCβ,p,T |Wt |

≤ Cp,θ,β,T,y0 e

− p2

≤ Cp,θ,β,T,y0 (log n)

23

.

  T

Lλ (P⊗dt) E

0

2 pθ

=

p pθ

θ

. and λ :=

p

t | dt |Wt − W

 p p

pθ 2 .

since one knows from [9] that








en (W, Lp ) = O (log n)− 2 . 1

Plugging this estimate in (3.17) and then in (3.16) yields the expected inequality. The σ bounded case: The diffusion coefficient σ being bounded, (St )−1 is σsup -Lipschitz continuous by (3.4). Hence, for every set α := {α1 , . . . , αn } of Borel functions defined in LpT , every (S. )−1 (αi ) ∈ LpT and en ({(S. )−1 ◦ αi , i = 1, . . . , n}, X, Lp ) ≤ σsup en (α, Y, Lp ) ≤ +∞ (3.19) Consequently, if α is an Lp -optimal n-quantizer (in LpT ) for the process Y en (X, Lp ) ≤ en ({(S. )−1◦ αi ), i = 1, . . . , n}, X, Lp )  

E

=

 T

min

1≤i≤n 0

 1

|(Ss )−1 (Ys ) − (Ss )−1 (αi (s))|p ds

p

≤ σsup en (α, Y, Lp ) = σsup en (Y, Lp ). Now, let αW := {α1W , . . . , αnW } be an Lp -optimal n-quantizer for the standard Brownian motion. Now, for every i ∈ {1, . . . , n}, set yi as the solution of ODE (3.11) with right hand side term αiW . Then it follows from the right inequality in Lemma 1 that y := {y1 , . . . , yn } is a subset of LpT such that en (Y, Lp ) ≤ en (y, Y, Lp ) ≤ e2 = e2

p−1 [β]p T p−1 /p Lip

p−1 [β]p T p−1 /p Lip

en (αW , W, Lp )

en (W, Lp ).

The combination of the above two inequalities yields the announced inequality. (b) If one assumes that ∀ (t, x) ∈ [0, T ] × R, σ(t, x) ≥ ε0 > 0

(uniform ellipticity)

then, one derives as a straightforward consequence that S is (1/ε0 )-Lipschitz continuous in x uniformly in t ∈ [0, T ]. Consequently if h ∈ LpT , t → S(t, h(t)) ∈ LpT . As a consequence, it follows that, for every subset α := {α1 , . . . , αn } of LpT , |α| ≤ n, en (α, X, Lp ) ≥ εp0 en ({S(., αi ), i = 1, . . . , n}, Y, Lp ) so that en (X, Lp ) ≥ εp0 en (Y, Lp ).

(3.20)

Then, let y := {y1 , . . . , yn } be an Lp -optimal n-quantizer for the process Y . Set αi := yi −

 .

β(s, yi (s)) ds,

i = 1, . . . , n.

0

The αi ’s lie in LpT hence the left inequality in Lemma 1 yields en (Y, Lp ) = en (y, Y, Lp ) ≥ Cβ,T en (α, W, Lp ) ≥ Cβ,T en (W, Lp ) with Cβ,T =

1 . 21−1/p (1+[β]pLip T p )1/p

Combined with (3.20) this yields the announced result. 24

♦

Construction of quantizers with the optimal rate: When 1 ≤ p < 2, a careful reading of the proof shows that one may replace the sequence of optimal quadratic quantizers of the Brownian motion used to get the upper-bound by any sequence of stationary quadratic quantizers of the 1 Brownian motion whose quadratic quantization error goes to 0 at O(log− 2 n)-rate. The main interest of such a remark is that it is possible to exhibit and compute numerically such a sequence of quadratic quantizers of the Brownian motion as it is emphasized in the introduction: it simply needs to optimize in an appropriate way (see [14] for details) the size of the marginal quantizers of the “product n-quantizers” described in (1.9). Then a time discretization of the transform “αW → αX ” makes feasible a numerical procedure providing (good approximations) of Lp n1 quantizers αX,n , n ≥ 1, for X whose Lp -quantization error goes to zero at a O((log n)− 2 )-rate.

3.2

The Girsanov approach

Let us come back to the SDE satisfied by Y = S(t, Xt ) with drift β i.e. dYt = β(t, Yt ) dt + dWt ,

Y0 = y0 := S(0, x0 ).

Note that, in view of Skorokhod-Stroock-Varadhan criterion, Equation (3.5) always has a weak solution as soon as β is bounded has at most linear growth (and so has the original Equation (1.1)). In the homogeneous case (β(t, x) = β(x)), linear growth would be sufficient in view of the EngelbertSchmidt criterion (see [11], Theorem 5.15 p.341 or the footnote in Section 3). As a preliminary step, one notes that, for every p ≥ 1, every integer n ≥ 1, every x0 , y0 ∈ R, en (X, Lp ) = en (X − x0 , Lp )

and

en (Y, Lp ) = en (Y − y0 , Lp )

so that one may assume without loss of generality throughout this section that x0 = 0 and y0 = S(0, 0) = 0. One introduces for every λ ∈ R, the Dol´eans exponentials  (λ) Lt

:= exp −λ

 t 0

λ2 β(s, Ys )dWs − 2

 t

 2

β (s, Ys )ds 0

(λ)

The process (Lt )t∈[0,T ] is a positive local martingale. Classical Novikov criterion implies that (λ)

(Lt )t∈[0,T ] is a true martingale as soon as 

λ2 E exp 2

 T

 2

β (t, Yt )dt

< +∞.

0

In particular, it is classically fulfilled when the drift function β is bounded which will be our assumption throughout this section. Remark. It is possible to relax partially the assumption of the Novikov criterion (see Benes’s Theorem in [11] p.199-200 or [17] p.332), but we did not succeeded so far in taking advantage of this refinement. Then, if (Lt )t∈[0,T ] is a true martingale, Girsanov theorem says that, if one sets P∗ := LT .P, then P∗ is a probability measure and Y is a P∗ -Brownian motion (on [0, T ]). (1)

(1)

25

Lemma 2 Let p ≥ 1 and r > 1. Assume that the drift function β is bounded. Let Ψ : C([0, T ], R) × [0, T ] → R+ be a bi-measurable nonnegative functional. Then    1/r T β2 T  sup  1−1/r pr    T e 2(r−1) E Ψ(W, t) dt ,  ≤ T 0 p   E Ψ(Y, t) dt r  T T β2  sup 0  −(r−1) − p/r  2r  e E Ψ(W, t) dt .  ≥ T

(3.21)

0

The proof is standard and relies on Girsanov Theorem (namely Y is a P∗ -Brownian motion) and H¨ older Inequality. It is deferred to Annex 2 at the end of the paper. Now we are in position to derive the second functional quantization rate result for the original diffusion, essentially based now on the boundedness of the function β involved in the Lamperti transform. Theorem 5 Assume that b and σ satisfy Assumption (3.2) and that furthermore b (t, x) −→ (t, x) − σ

  x 1 ∂σ

σ2

0

∂t

(t, ξ)dξ −

1 ∂σ (t, x) 2 ∂x

is bounded over [0, T ] × R.

(a) Then ∀ p ∈ [1, +∞),






en (X, Lp ) ≤ Cb,σ,T,p en (W, Lp ) = O (log n)− 2 1

(3.22)



where p ≥ 2 is like in Theorem 4(a). When σ is bounded by σsup , then one may set p = p and Cb,σ,T,p = σsup . (b) If, furthermore, σ > ε0 > 0, then, for every p ∈ (1, +∞) and every r ∈ (1, p), en (X, L ) ≥ ε0 T p

so that

− r−1 − p

T β2 sup

e

2rp

en (W, Lp/r )

(3.23)

en (X, Lp ) ≈ (log n)−1/2 .

Remark. This approach does not provide a lower bound when p = 1. Proof: We still use the notations used in the proof of Theorem 4. The above Assumption (3.22) implies that the drift function β of Y is bounded. (a) Case 1 ≤ p < 2: let αW = {α1W , . . . , αnW } be an L2 -optimal n-quantizer (in L2T ) for the Brownian motion. epn (X, Lp ) ≤ epn ({(S. )−1◦ αiW ), i = 1, . . . , n}, X, Lp ) 

= E

 T

min

1≤i≤n 0



≤ C E p

≤ C E p

−1

|(St )

 T

min

1≤i≤n 0

 T

−1

(Yt ) − (St )



(αiW (t))|p dt 

pC(|αW i (t)|∨|Yt |)

e

t |∨|Yt |) pC(|Y

e 0

26

|Yt −

αiW (t)|p dt

(3.24)



|Yt − Y t |p dt

(3.25)

where Y :=

n 

αiW 1Ci (αW ) (Y ) is the (Voronoi) αW -quantization process of Y . Applying Lemma 2

i=1

with r ∈ (1, 2/p) to the functional Ψ(ω, t) :=

n 

W (t)|∨|ω(t)|)

eC(|αi

|αiW (t) − ω(t)|1Ci (αW ) (ω)

i=1

leads to

E

 T

t |∨|Yt |) pC(|Y

e 0



|Yt − Y t |p dt

≤ T

1− r1

≤ T

1− r1

T β2 sup

e

2(r−1)

T β2 sup

e

2(r−1)

E

 T

t |) rpC(|Wt ∨|W

e 0



1 t |pr dt |Wt − W

E e

0

Now one proceeds like in Theorem 4: H¨older inequality applied with p¯ =

E

 T

t |∨|Yt |) pC(|Y

e 0



|Yt − Y t |p dt

 T

t |)) rpC(supt∈[0,T ] (|Wt ∨|W

≤ Cβ,p,r,T

  T

E

2 pr

r

. 1 r

t |pr dt |Wt − W

and q¯ =

2 2−pr

.

yields

p t |2 dt |Wt − W

0

2

.



Case p ≥ 2: Let αW denote an Lp -optimal n-quantizer for some p > p. The beginning of the proof is similar to that in Theorem 4 and one easily gets that 

en (X, L ) ≤ E p

 T

min

1≤i≤n 0

 pC(|αW i (t)|∨|Yt |)

e

|Yt −

αiW (t)|p dt

.

Then, set Y as a Voronoi quantization process of αW . One shows like in the case 1 ≤ p < 2, one derives successively that 

E

 T

min

1≤i≤n 0

 pC(|αW i (t)|∨|Yt |)

e

|Yt −

αiW (t)|p dt

≤E ≤T ≤T

 T

t |∨|Yt |) pC(|Y

e 0

1− r1

1− r1

e

T β2 sup 2(r−1)

T β2 sup

e

2(r−1)

E



|Yt − Y t | dt p

 T 0

1



t |pr dt erpC(|Wt ∨|Wt |) |Wt − W



 T

rpC|W |T

E e

t| rpC|W

e 0

r

1 t | dt |Wt − W

r

pr

for any r > 1. The rest of the proof is identical to that of the Lipschitz setting. The σ bounded case works the same way round. (b) We simply need to lower bound en (Y, Lp ) with some appropriate minimal quantization error of the Brownian motion. Let r > 1 such that p/r > 1 and let αY be an Lp -optimal n-quantizer for Y . Set for every (ω, t) ∈ C([0, T ], R) × [0, T ], Ψ(ω, t) :=

n 

|ω(t) − αiY (t)|1Ci (αY ) (ω)

i=1

27

where (Ci (αY ))1≤i≤n denotes a Voronoi partition of αY in LpT . Then it follows from the lower bound in (3.21) that epn (Y, Lp )

= E

 T

Ψ(Y, t)p dt

0

≥ T

−(r−1) −

= T

−(r−1) −

e e

  T

T β2 sup

E

2r



T β2 sup

r

Ψ(W, t)  T

1≤i≤n 0

≥ T −(r−1)e−

T β2 sup 2r

Examples: • Let

dt

0

E min

2r

p/r

epn (W, Lp/r ).

r

|Wt −

αiY (t)|p/r dt

♦

σ(t, x) = 2 + sin(x2 )/x.

b(t, x) = 0,

Since b and σ are differentiable enough Assumption (3.7) reduces to the boundedness condition  (3.10). Now one checks that b , b σσ are bounded whereas σσ  is unbounded so that the condition is not fulfilled. • Theorem 5 includes cases where the drift b(t, x) = b(x) in (1.1) is discontinuous since Assumption (3.22) only requires boundedness of β (the existence and uniqueness in distribution of a weak solution to equations (1.1) and (3.5) follows from Engelbert-Schmidt criterion (3.1)).

4

Diffusions with a Gaussian martingale term

In this section we investigate the functional quantization of diffusion whose diffusion coefficient σ(t, x) = σ(t) is only a function of time. So, as soon as σ ∈ L2T , the martingale term of the diffusion is a Gaussian martingale  t

Mt =

σ(s)dWs . 0

If one looks carefully at the proofs of the former sections, one gets easily convinced that all  the various inequalities established between en (X, Lp ) and en (W, Lp ) still hold if one replaces the Brownian motion W by any continuous time martingale (Mt )t∈[0,T ] such that sup[0,T ] |Mt | has a finite Laplace transform over the whole real line. This is clearly the case of the above Gaussian martingales since, by the Dambins-Dubins-Schwarz Theorem, M = B . σ2 (s)ds . 0

where B is a standard Brownian motion built on the same probability space. So, we will assume throughout this section that the drift of the diffusion is zero i.e. we will only consider the case of Gaussian martingales: the extension to the more general case dXt = b(t, Xt )dt + σ(t)dWt is obvious in view of the former theorems. Another question concerning these Gaussian martingale diffusions is to exhibit some quantizers that achieve this rate. We will try to answer both questions. 28

4.1

The main result for the rates

Set for every t ∈ [0, T ],

 t

ρ(t) :=

σ 2 (s)ds.

0

Here is the main result of this section. Proposition 4 Let σ ∈ L2 ([0, T ]), dt) and Mt :=  T

the Wiener integral. Assume that ρ(T ) =

 t

σ(s)dWs the Gaussian martingale given by 0

σ 2 (t)dt > 0.

0

(a) For every p ∈ [1, +∞), 1

en (M, Lp ) ≤ T p and

− 12

ρ(T )1/2 en,p (W, L∞ ) = O((log n)− 2 ) 1



en,1 (M, L∞ ) ≤

ρ(T )en,1 (W, L∞ ).

2p

(b) Let p ∈ [1, +∞). If σ ∈ LTp−1 , then en (M, Lp ) ≈ (log n)− 2 . 1

If, furthermore, σ ∈ L2+η for some η > 0, then T en,1 (M, L∞ ) ≈ (log n)− 2 . 1

Proof: (a) The key identity here is still the Dambins-Dubins-Schwarz Theorem: let α be n. Set quantizer lying in L∞ T αi



:=

The αi ’s obviously lie in LpT . Then 

E

 T

min

1≤i≤n 0



|Mt − αi (t)|p dt

ρ(T ) T

1



2

αi



T ρ( . ) , i = 1, . . . , n. ρ(T )

    1   p  T 2   T ρ(t)   Wρ(t) − ρ(T ) = E  min αi dt 1≤i≤n 0  T ρ(T )        1   p 2  ρ(T ) uT   ≤ T E  min ess supu∈[0,ρ(T )] Wu − αi 1≤i≤n T ρ(T )    p     p     )  Ws ρ(T   ρ(T ) 2  T     = T E  min ess sups∈[0,T ] 
1 − αi (s)   1≤i≤n T  ρ(T ) 2   T    

= T 1−p/2 ρ(T )p/2 E 

since

Wsρ(T )

√

ρ(T )

 s∈[0,T ]

(4.1)

min ess sups∈[0,T ] |Ws − αi (s)|p .

1≤i≤n

(4.2)

is still a standard Brownian motion. Then, it follows from L∞ ⊂ LpT that T 1

en (M, Lp ) ≤ T p

− 12

ρ(T )1/2 en,p (W, L∞ ).

29

Then, one derives from [4] (see also Section 3.1 in [9]) that




en (M, Lp ) = O (log n)− 2 . 1

This bound for en,p (W, L∞ ) is derived from a small  . sup -ball probability estimate for W . (b) Assume first that p ∈ (1, +∞). One applies the reverse H¨older inequality with s = − η2 < 0 and η r = 2+η ∈ (0, 1), one gets 

E min



 T

1≤i≤n 0

|Mt −

αi (t)|pdt



= E

 T

min

1≤i≤n 0



≥ E min ≥

% % 0

%

& 1%

T

0

≥ 

since r ∈ (0, 1). Now

Wuρ(T )

√

ρ(T )

0

T

dρ(t) σ 2s (t)

&1  s



0

&

dt

1 s

σ 2(s−1) (t)



E  min

&− 2 η

σ 2+η (t)dt

σ

0

&1   T pr r 1 E  min Wuρ(T ) − ρ(T ) 2 αi (u) ρ(T )du  1≤i≤n 0

(t)dt

pr & 1  T  r  Wuρ(T )  min − αi (u) du '  1≤i≤n 0  ρ(T )



η

2+η

   pr & r1  1 s Ws − ρ(T ) 2 αi  ds   ρ(T ) 

ρ(T )

%

&− 2 %

T

%

1≤i≤n

0

%

r

|Mt − αi (t)|pr dρ(t)

T

=

σ (t)

dt

T

=



&1  %  &1   pr s r T  1 ρ(t)  dρ(t)  Wρ(t) − ρ(T ) 2 αi  min E  1≤i≤n ρ(T )  σ 2(s−1) (t)

T

0

|Mt −

%

1≤i≤n

dρ(t) αi (t)|p 2

1+ pr 2

ρ(T )

0

E



is still a standard Brownian motion by the classical scaling u∈[0,T ]

property so that 

E i.e.

 T

min

1≤i≤n 0



|Mt −

αi (t)|p dt

≥ ρ(T )

1 + p2 r

%

T

σ

&− 2 %  η

2+η

(t)dt

0 1

en (M, α , Lp ) ≥ ρ(T ) p

2 + pη + 12

%

E

&−

T

σ 2+η (t)dt

2 pη

& 1

 T

min

1≤i≤n 0

pr

|Wu − αi (u)| du

r

pη

en (W, α, L 2+η ).

0

pη 2 > 0 so that 2+η = 1. Then, one considers an Lp -optimal sequence of nNow, set η = p−1 quantizers α for M and the sequence of n-quantizers α ⊂ L1T for the Brownian motion associated by (4.1). This provides the expected lower bound since on the one hand en (W, α, L1 ) ≥ en (W, L1 ) and on the other hand it is established in [9] that

en (W, L1 ) ≈ (log n)− 2 . 1

Finally this shows that en (M, Lp ) = Ω((log n)− 2 ). 1

30

If p = 1, the same arguments show in a simpler way (namely without the Schwarz Inequality), en (M, L1 ) ≥

3 1 ρ(T ) 2 en (W, L1 ). 2 σsup

If p = +∞, then, consider a large enough real number q such that σ ∈ L2+η , with η := T qη η 2+η = 1). Then, set once again r = 2+η (so that qr = 1). First, note that

E



min M − αi sup



1≤i≤n

 1 q

≥ T E min

%

1≤i≤n

2 q−1

(so that

&1  q |Mt − α (t)|q dt  .

T

0

i

then, following the approach for the case p < +∞ leads to

E



min M − αi sup

1≤i≤n



≥ T

= T

1 q

%

&−

T

σ 2+η (t)dt

ρ(T )

1 1 + qr 2

0 1 q

%

T

σ

&− 2+η

1 q

%

T

σ

3 2

& 1 −1 q

2+η

(t)dt

qr & 1  %   qr T W  uρ(T )  E  min − αi (u) du  '  1≤i≤n 0  ρ(T )



2 qη

ρ(T ) E

(t)dt

0

≥ T



2 qη



 T

min

1≤i≤n 0

|Wu − αi (u)| du

3

ρ(T ) 2 en,1 (W, L1 ).

♦

0

Remarks: • Note that no strong ellipticity assumption is required on σ. • In former works about quadratic functional quantization of Gaussian processes (see [14], [15], [9]), most results concerning both upper and lower bounds were based on the rate of convergence to 0 of the eigenvalues of the covariance operator of the process, combined when necessary with some comparison theorem for positive trace operators. For Gaussian martingales, this covariance operators CM reads on functions f ∈ L2T CM (f ) ≡ t →

 T 0

 s∧t

f (s)ΓM (s, t)ds

with

ΓM (s, t) :=

σ 2 (u)du.

0

Set ΓW (s, t) := s ∧ t. If σ is bounded by a real constant σsup then, σ2sup ΓW − ΓM is positive definite so that the eigenvalues of CM and Cε0 W satisfy λM,n ≤ λσsup W,n = σ2sup λW,n . Consequently, (see Lemma 4.11 in [14]), en (M, L2 ) ≤ σsup en (W, L2 ). One shows by the same arguments that if |σ| ≥ ε0 > 0, then en (M, L2 ) ≥ ε0 en (W, L2 ).

4.2

Some further results about quasi-optimal quantizers

• When p ∈ [1, 2), one may construct some sequences of quasi-optimal quantizers in the (Lp extended) sense of (1.7) for M from a sequence of quasi-optimal quadratic quantizers for the Brownian motion, following the approach developed in the proof of (b). In fact, one may use a

31

similar approach to exhibit some quantizers that do achieve this rate, provided that we have a bit of integrability around the singularities of 1/σ, namely that  T

dt

0

σ η (t)

< +∞

for some

η > 0.

Set s = 1 + η2 and r = 1 + η2 its H¨older conjugate exponent. It follows from H¨ older inequality applied to the positive finite measure ρ(dt) = σ 2 (t)dt that 

E

 T

min

1≤i≤n 0



|Mt −

αi (t)|p dt



= E

 T

min

1≤i≤n 0



%

≤ E  min

1≤i≤n

≤

% 0

T

s

T

0

Consequently

en (α , M, L ) ≤ ρ(T ) p

& 1 % &1  r T ρ(dt) s  |Mt − α (t)|rp ρ(dt)

E

σ 2s (t) 1 + p2 r

σ (t)

&1 % 

%

η + 12 p(2+η)



i

0

T ρ(dt)

= ρ(T ) 

|Mt −

ρ(dt) αi (t)|p 2

min

1≤i≤n 0

0

T

|Wu − ρ(T )

1/2

&1 % s

σ 2(s−1) (t) %



 ρ(T )

dt

dt η σ (t)

σ 2s (t)

0

E min 2 p(η+2)

& 1

&1

 T

1≤i≤n 0

&

αi



u |rp du ρ(T ) pr

|Wu − αi (u)| du p(1+ η2 )

en (α, W, L

r

.

).

2η It follows from the above inequality that, as soon as p ≤ 2+η , this approach produces a sequence of Lp -quantizers for M from a sequence of quasi-optimal quadratic quantizers for W in the sense of (1.7): one can always make η smaller so that p(1 + η2 ) = 2 and σ1 ∈ LηT .

• When σ is Lipschitz continuous, there is another method to produce some quasi-optimal Lp quantizers for M , given some quasi-optimal quadratic quantizers for W , which works for p ∈ [1, 2], without any integrability assumption on σ1 . It does not rely on the Dambins-Dubins-Schwarz Theorem. One considers a n-quantizer α of the Brownian motion W whose components lie in the Cameron-Martin space H 1 (this is the case e.g. of any L2 -optimal n-quantizer in L2T for the  Brownian motion, see [14]). So αi (t) = 0t α˙ i (s)ds, αi ∈ L2 , i = 1, . . . , n. Then, one sets  t

xi (t) =

σ(s)α˙ i (s)ds. 0

An integration by parts shows that Mt − xi (t) =

 t 0

σ(s)(dWs − α˙ i (s)ds) = (Wt − αi (t))σ(t) −

 t 0

(Ws − αi (s))dσ(s)

where dσ(t) denotes the Stieltjes measure of σ as a function with finite variations. Let p ∈ [1, 2]. The easy inequality ∀ a, b > 0, ∀ ρ > 0,

p

p

p

(a + b)p = [(a + b)2 ] 2 ≤ (1 + ρ) 2 ap + (1 + 1/ρ) 2 bp

32

r

and |dσ(t)| ≤ [σ]Lip dt successively yield  T 0

 

 T



p t 1 p T |Wt − αi (t)|p |σ(t)|p dt + (1 + ) 2 |Ws − αi (s)dσ(s)| dt ρ 0 0 0 p  T  T t p p 1 p p p |Ws − αi (s)|ds dt ≤ (1 + ρ) 2 σsup |Wt − αi (t)| dt + (1 + ) 2 [σ]Lip ρ 0 0 0  T  T  t p 1 p ≤ (1 + ρ) 2 σpsup |Wt − αi (t)|p dt + (1 + ) 2 [σ]pLip tp−1 |Ws − αi (s)|p dsdt ρ 0 0 0   T p p T 1 p |Wt − αi (t)|p dt. ≤ (1 + ρ) 2 σpsup + (1 + ) 2 [σ]pLip ρ p 0 p

|Mt − xi (t)|p dt ≤ (1 + ρ) 2

An optimization in ρ yields 

 T 0







[σ]Lip T

2p

2+p +  |Mt − xi (t)|p dt ≤ σsup

(p + 1)

2p 2+p

 1

p

1+ p2  

 T 0

|Wt − αi (t)|p dt

and in turn, setting x := {x1 , . . . , xn },  

 2p

2+p +  en (x, M, Lp ) ≤ σsup

 



[σ]Lip T (p + 1)

1 p

 2p

2+p +  ≤ σsup

 

[σ]Lip T (p + 1)

1 p

2p 2+p



2p 2+p

 12 + p1  

en (α, W, Lp )

 12 + p1  

en (α, W, L2 )

since p ≤ 2.

References [1] Beauzamy B., Introduction to Banach spaces and their geometry, 2nd edition, North-Holland Mathematics Studies, 68, North-Holland Publishing Co., Amsterdam, 1985, 338 p. ´pingle D., Numerical Methods for stochastic processes, Wiley-Interscience, New [2] Bouleau N., Le York, 1994, 359p. [3] Cohort P., Sur quelques probl`emes de quantification, th`ese de l’Universit´e Paris 6 (France), 2000. [4] Dereich S., Fehringer F., Matoussi A., Scheutzow M., On the link between small ball probabilities and the quantization problem for Gaussian measures on Banach spaces, J. Theoretical Probab., 16, 2003, pp.249-265. [5] Fehringer F., Kodierung von Gaussmassen, PhD thesis, TU Berlin, 2001. [6] Graf S., personal communication, 2003. [7] Graf S., Luschgy H., Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics n0 1730, Springer, 2000, 230 p. [8] Graf S., Luschgy H., Asymptotics of the quantization errors for self-similar probabilities, Real Analysis Exchange,26, 2001, pp.795-810. `s G., Functional Quantization and small ball probabilities for Gaussian [9] Graf S., Luschgy H., Page processes, to appear in J. Theoretical Probab..

33

[10] Herrndorf N., Approximation of vector-valued random variables by constants, J. Approx. Theory, 37, 1983, pp.175-181. [11] Karatzas I., Shreve S., Brownian Motion and Stochastic calculus, Springer, New York, 1988 (2nd edition 1991), 470p. [12] Ledoux M., Talagrand M.,Probability on Banach spaces: Isoperimetry and Processes, Springer, Berlin, 1991 (2nd printing, 2002), 480p. [13] Lindenstrauss J., Tzafriri L., Classical Banach Spaces I, Springer, Berlin, 1977. `s G., Functional quantization of Gaussian processes, J. Funct. Anal., 196, [14] Luschgy H., Page pp.486-531, 2002. `s G., Sharp asymptotics of the functional quantization problem for Gaussian pro[15] Luschgy H., Page cesses, pre-print LPMA-752 (Univ. Paris 6, France), 2002, To appear in Annals of Probability. `s G., Printems J., Optimal quadratic quantization for numerics: the Gaussian case, Monte [16] Page Carlo Methods and Applications, 9, n0 2, 2003, pp.135-166. [17] Revuz D., Yor M., Continuous martingales and Brownian Motion, Springer, Berlin, 3rd edition 1999, 599p. [18] Rogers L., Williams D., Diffusions, Markov processes, and martingales: Vol. 2 Itˆ o Calculus, Wiley, 1987, 475p. [19] Schaefer H., Banach Lattices and Positive Operators, Springer, Berlin, 1974.

Annex 1: proof of Theorem 2 We first need the following equivariance properties contained in the lemma below. Lemma 3 Let E1 and E2 be Banach spaces and let X be a Radon E1 -valued random vector satisfying

EXr < ∞. If T : E1 → E2 is a bounded linear operator, then

en,r (T (X), E2 ) ≤ T en,r (X, E1 ). If T : E1 → E2 is a bijective linear isometry, c > 0 and u2 ∈ E2 , then en,r (c T (X) + u2 , E2 ) = c en,r (X, E1 ) and Cn,r (c T (X) + u2 , E2 ) = c T (Cn,r (X, E1 )) + u2 . Proof of Lemma 3. Let us prove e.g. the first assertion. For any α ⊂ E1 with 1 ≤ |α| ≤ n, en,r (T (X), E2 ) ≤ (E min T (X)) − T (a)r )1/r a∈α

≤ and thus the assertion.

T (E min X − ar )1/r a∈α

♦

Proof of Theorem 2. (a) The inequality en,r (X, E) ≥ en,r (X, E ∗∗ ) is obvious. To prove the converse inequality assume first that supp(PX ) is finite. Let α ∈ Cn,r (X, E ∗∗ ) and let U denote the linear subspace of E ∗∗ spanned by supp(PX ) ∪ α. Since U is finite-dimensional, there exists

34

by local reflexivity of E, for every ε > 0, a bounded linear operator S : U → E satisfying S ≤ 1 + ε and S(x) = x for every x ∈ U ∩ E. (cf. [13] Lemma 1.e.6). Using Lemma 3, one derives en,r (X, E)r

≤

E min X − br = E min S(X) − S(a)r a∈α b∈S(α) r

≤ (1 + ε) en,r (X, E ∗∗ )r en,r (X, E) ≤ en,r (X, E ∗∗ ).

Hence

m : Ω → E of X, |X m (Ω)| ≤ m, for sufficiently large m For general X and ε > 0, choose a quantization X such that m 1∧r X − X Lr (P) ≤ ε. Then,

m , E))r∧1 | ≤ ε |(en,r (X, E))r∧1 − (en,r (X

and

m , E ∗∗ ))r∧1 | ≤ X − X m 1∧r |(en,r (X, E ∗∗ ))r∧1 − (en,r (X Lr (P) ≤ ε.

Since |supp(P



Xm

m , E) = en,r (X m , E ∗∗ ). This yields )| ≤ m < ∞, we have en,r (X |(en,r (X, E))r∧1 − (en,r (X, E ∗∗ ))r∧1 | ≤ 2ε.

Hence en,r (X, E) = en,r (X, E ∗∗ ). Furthermore, since Cn,r (X, E ∗∗ ) = ∅ by Corollary 1, it follows from Proposition 1(a) that (ej,r (X, E ∗∗ ))1≤j≤n is strictly decreasing provided |supp(PX )| ≥ n. (b) The inclusion Cn,r (X, E) ⊂ Cn,r (X, E ∗∗ ) follows from (a). To prove the converse inclusion, we may assume dim E ≥ 1. The assumption implies that there exists a linear projection Π from E ∗∗ onto E with Π ≤ 1. Let α ∈ Cn,r (X, E ∗∗ ) and set β = Π(α). Then β ∈ Cn,r (X, E) and for every x ∈ E, min x − b ≤ min x − a. a∈α

b∈β

This implies that the closed set A := {x ∈ E : min x − b = min x − a} a∈α

b∈β

satisfies PX (A) = 1. Therefore, A = E and in particular, β ⊂ A. One obtains mina∈α b − a = 0 for every b ∈ β and hence, β ⊂ α. By Proposition 1(a), we have |α| = |β| = n which yields β = α. Hence α ∈ Cn,r (X, E). ♦

Annex 2: Proof of Lemma 2 E





T p

Ψ(Y, t) dt

=

0

E

∗

(1) −1

(LT ) 

=

=

=

≤







T p

Ψ(Y, t) dt 0

   T T 1 E∗ exp β(s, Ys )dWs + β 2 (s, Ys )ds Ψ(Y, t)p dt 2 0 0 0      T T 1 T 2 ∗ p E exp β(s, Ys )dYs − β (s, Ys )ds Ψ(Y, t) dt 2 0 0 0      T T 1 T 2 E exp β(s, Ws )dWs − β (s, Ws )ds Ψ(W, t)p dt 2 0 0 0     q1   r  r1  T T q T 2 E exp q β(s, Ws )dWs − β (s, Ws )ds E Ψ(W, t)p dt 2 0 0 0 T

35

using H¨ older Inequality, where

+

1 r


 T = exp q 0 β(s, Ws )dWs −

(q)

= 1. Now Λt

P-martingale, it follows that

[0, T ], being a

E

1 q





T p

Ψ(Y, t) dt

≤

T

0 1 q

E ΛT

≤ T exp

(q)





q(q − 1) exp 2

q−1 T β2sup 2

 q1  

T 2

β (s, Ws )ds 0

 

E

T

Ψ(W, t)pr dt

E

q2 2

T 0

 β 2 (s, Ws )ds , t ∈  r1

T pr

Ψ(W, t) dt 0

 r1 .

0

The converse inequality follows the same way round by inverting the rˆ ole played by Y and W .

36

♦