A NOTE ON THE FUNCTIONAL LAW OF THE

M. N’ Z I (Abidjan)

M. E D D A H B I (Marrakech)

A NOTE ON THE FUNCTIONAL LAW OF THE

ITERATED LOGARITHM FOR L´ EVY’S AREA PROCESS

Abstract. By using large deviation techniques, we prove a Strassen type law of the iterated logarithm, in H¨older norm, for L´evy’s area process.

1. Introduction. In the last years there have been several attempts to study the Brownian motion and diffusion processes by endowing the path- space with stronger topologies than the uniform one. For example, Baldi, Ben Arous and Kerkyacharian [2] showed that the large deviations principle for the Brownian motion still holds under the topology induced by any H¨older norm with exponent α < 1/2. As a consequence of this result they deduced Strassen’s law, in H¨older norm, for the Brownian motion.

The aim of this short note is to prove an analogue of this law for L´evy’s area process which is the stochastic analogue of the area contained in a lens-shaped domain. More precisely, let ξ = {(ξ 1 (t), ξ 2 (t)) : t ≥ 0} be a 2-dimensional Gaussian process with independent components. L´evy’s stochastic area process L = {L(t) : t ≥ 0} associated with ξ is defined by

L(t) = 1 2

 ^t

0

ξ 1 (u) ξ 2 (du) −

t

\

0

ξ 2 (u) ξ 1 (du) 

, t ≥ 0.

This process has been thoroughly studied in recent years (see e.g. Ikeda, Kusuoka and Manabe [7] and Chan et al. [4]) and plays an important role in the study of various problems in analysis, geometry, mathematical physics and statistics. For example, let B = {(B 1 (t), B 2 (t)) : t ≥ 0} be a 2- dimensional Brownian motion and let ξ be the stationary Gaussian process

1991 Mathematics Subject Classification: 60F05, 60H05.

Key words and phrases: law of the iterated logarithm, Brownian motion, L´evy’s area process, maximum likelihood.

[223]

defined by the stochastic differential equation ξ(dt) = Aξ(t)dt + B(dt) and such that ξ(0) is independent of B, where

A =

 −θ 1 −θ 2

θ 2 −θ 1

 ,

θ 1 > 0 and −∞ < θ 2 < ∞ being some unknown parameters to be estimated from the observations {ξ(t) : 0 ≤ t ≤ T } of ξ until time T ≥ 0.

Liptser and Shiryaev [9, p. 212] have proved that ξ has independent com- ponents and that the maximum likelihood estimates b θ 1 (T, ξ) and b θ 2 (ξ, T ) of θ ₁ and θ ₂ respectively are given by the equations

1

θ b 1 (T, ξ) − b θ 1 (T, ξ)

T

\

0

(ξ ₁ ² (t) + ξ ₂ ² (t)) dt =

T

\

0

ξ 1 (t) ξ 1 (dt) +

T

\

0

ξ 2 (t) ξ 2 (dt),

b θ ₂ (T, ξ) =

T

T

0 ξ 1 (t) ξ 2 (dt) −

T

T

0 ξ 2 (t) ξ 1 (dt)

T

T

0 (ξ ² ₁ (t) + ξ ² ₂ (t))dt .

Here, we study the asymptotic behaviour of L´evy’s area associated with a Brownian motion B. We shall use a recent result of Ben Arous and Ledoux [3] on large deviations, in H¨older norm, for diffusion processes.

The paper is organized as follows: in Section 2 we state the results and in Section 3 we give the proofs. Before closing this section, let us note that the asymptotic behaviour of L´evy’s area process via the law of the iterated logarithm can be found in Helmes, R´emillard and Theodorescu [5], N’zi, R´emillard and Theodorescu [10] and R´emillard [11].

2. Strassen’s law in H¨ older norm for the area process. Let us denote by C (resp. C ^α ) the space of all continuous functions f : [0, 1] → R ² (resp. f : [0, 1] → R) such that f (0) = 0 endowed with the uniform (resp.

α-H¨older) norm kf k = sup

t∈[0,1]

|f (t)|,



resp. kf k α = sup

0≤s,t≤1

|f (t) − f (s)|

|t − s| ^α

 .

For every h in the Cameron–Martin space H, i.e. the space of all absolutely continuous functions null at the origin with square integrable derivatives, we put

|h| ² H =

1

\

0

| ˙h| ² ds,

where ˙h denotes the derivative of h.

For every A ∈ C ^α , we put Λ(A) =

 inf  ₁

2 |h| ² _H : h ∈ H, F (h) ∈ A

if F ⁻¹ (A) 6= ∅,

∞ otherwise,

where

F (h)(t) = 1 2

 ^t

0

h 1 (u) ˙h 2 (u) du −

t

\

0

h 2 (u) ˙h 1 (u) du 

, h = (h 1 , h 2 ).

In particular, for every g ∈ C ^α , we denote Λ({g}) by λ(g). We also set K = {g ∈ C ^α : λ(g) ≤ 1}.

For every u ≥ 0, let us put φ(u) =

 log log u if u ≥ 3, 1 if 0 < u < 3, and

Z u = L(u ·) uφ(u) .

Now, we state the main result of this paper. From now on, we assume that 0 < α < 1/2.

Theorem 2.1. The process {Z _u : u > 0} is P -a.s. relatively compact and has K as set of limit points in the H¨ older topology.

The proof of Theorem 2.1 follows the classical lines in Baldi [1], which consists in proving the two propositions below:

Proposition 2.2. For every ε > 0, there exists u ⁰ > 0 P -a.s. such that if u > u ⁰ then d(Z u , K) < ε, where

d(g, K) = inf

h∈K kg − hk α .

Proposition 2.3. Let g ∈ K. Then, for every ε > 0, there exists c = c ε ∈ (1, ∞) such that

P (kZ _c ^j − gk α ≤ ε i.o.) = 1.

3. Proofs of Propositions 2.2 and 2.3. Let us first state a large

deviations principle, in H¨older norm, for L, which is an immediate conse-

quence of the main theorem in Ben Arous and Ledoux [3] and the scaling

property of L in Helmes and Schwane [6].

Theorem 3.1. For every A ∈ C ^α , we have

−Λ(A ^◦ ) ≤ lim inf

ε→0 ε ² log P (ε ² L ∈ A) ≤ lim sup

ε→0

ε ² log P (ε ² L ∈ A) ≤ −Λ(A) where A ^◦ and A are respectively the interior and the closure of A in the H¨ older topology.

P r o o f o f P r o p o s i t i o n 2.2. We divide the proof in three steps.

S t e p 1. We first prove that for every c ∈ (1, ∞) and every ε > 0, there exists j ⁰ ∈ N such that if j ≥ j ⁰ then d(Z c ^j , K) < ε.

Let K ε = {g ∈ C ^α : d(g, K) ≥ ε}. In view of the Borel–Cantelli Lemma, we only have to check

X

j

P (Z _c ^j ∈ K ε ) < ∞.

By virtue of the scaling property of L, we have (3.1) P (Z c ^j ∈ K ε ) = P

 L

φ(c ^j ) ∈ K ε

 .

Now, let us prove that Λ(K ε ) > 1. Since K is compact and λ is lower semicontinuous, there exists g 0 ∈ K ε such that λ(g 0 ) = inf g∈K ε λ(g). If Λ(K ε ) ≤ 1 then λ(g 0 ) ≤ 1. Therefore, g 0 ∈ K, which contradicts g 0 ∈ K ε .

Let δ > 0 be such that Λ(K ε ) > 1 + 2δ. In view of Theorem 3.1, we have for j large

P

 L

φ(c ^j ) ∈ K ε



≤ exp(−(1 + δ)φ(c ^j )) = cte j ^1+δ , which leads to the conclusion by virtue of (3.1).

S t e p 2. Now, we want to prove that for every ε > 0, there exists c ε > 1 such that for every 1 < c < c ε there exists j 0 = j 0 (ω) such that Y j (ω) ≤ ε for every j ≥ j 0 , where

Y j = sup

c ^j ≤u≤c ^j+1

1

c ^j φ(c ^j ) kL(u·) − L(c ^j ·)k α .

By virtue of the Borel–Cantelli Lemma, we only have to show that X

j

P (Y _j > ε) < ∞.

By using the scaling property of L we obtain P (Y _j ≥ ε) = P

 sup

c ^j ≤u≤c ^j+1

u

c ^j φ(c ^j ) kL − L(c ^j /u ·)k _α ≥ ε



≤ P

 sup

c ^j ≤u≤c ^j+1

1

φ(c ^j ) kL − L(c ^j /u ·)k α ≥ ε/c



= P

 sup

1≤v≤c

1

φ(c ^j ) kL − L(·/v)k α ≥ ε/c



= P

 1

φ(c ^j ) L ∈ A



where

A = {g ∈ C ^α : sup

1≤v≤c

kg − g(·/v)k α ≥ ε/c}.

By virtue of Theorem 3.1 and since A is closed, for every δ > 0 and j sufficiently large, we have

P

 1

φ(c ^j ) L ∈ A



≤ exp(−(Λ(A) − δ)φ(c ^j )).

It remains to show that we can choose δ such that for c small, Λ(A) >

1 + δ. Let g ∈ A be such that λ(g) < ∞. There exist 1 ≤ v ≤ c and (s, t) ∈ [0, 1] ² such that

ε

c |t − s| ^α ≤ |(g(t) − g(t/v)) − (g(s) − g(s/v))|

(3.2)

=   

t

\

s∨(t/v)

˙g(u) du −

s∧(t/v)

\

s/v

˙g(u) du   

≤ (|t − s ∨ (t/v)| ^1/2 + |s ∧ (t/v) − s/v| ^1/2 )k ˙gk _L ² where ˙g is the derivative of g.

Now, let f ∈ H be such that λ(g) = ¹ ₂ |f | ² _H and F (f ) = g. Since it is easy to prove that k ˙gk L ² ≤ |f | ² _H , we deduce from (3.2) that

λ(g) ≥ ε 2c

 |t − s| ^α

(|t − s ∨ (t/v)| ^1/2 + |s ∧ (t/v) − s/v| ^1/2

 . By virtue of Lemma 3.4 in Baldi [1], we obtain

λ(g) ≥ ε

4c (c − 1) ^α−1/2 . Therefore, we have Λ(A) ≥ _4c ^ε (c − 1) ^α−1/2 .

Since α < 1/2, it follows that for c small we have Λ(A) > 1, which ends Step 2.

S t e p 3. For every c ^j ≤ u ≤ c ^j+1 we have d(Z _u , K) ≤

Z ^u − c ^j φ(c ^j ) uφ(u) Z _c ^j

_α (3.3)

+    1 −

c ^j φ(c ^j ) uφ(u)

 kZ ^{c j} k α + d(Z _c ^j , K).

Let us deal with the right member of (3.3). In view of Step 2, the first term

is ≤ ¹ ₃ ε. Now, Step 1 implies that kZ _c ^j k α is bounded for j large. Since

j→∞ lim    1 −

c ^j φ(c ^j ) uφ(u)

    =

c − 1 c ,

for c close to 1 and j large we see that the second term is ≤ ¹ ₃ ε. Step 1 also implies that the third term is ≤ ¹ ₃ ε. The assertion of Proposition 2.2 follows immediately.

P r o o f o f P r o p o s i t i o n 2.3. Let g ∈ K and f ∈ H be such that

1

2 |f | ² _H = λ(g) and F (f ) = g. By virtue of the Proposition in Ben Arous and Ledoux [3] and the scaling property of L, for δ small and j large, we have

P



kZ _c ^j − gk α > ε,

B(c ^j ·) p c ^j φ(c ^j ) − f

≤ δ



≤ exp(−2φ(c ^j )).

It follows that X

j

P



kZ _c ^j − gk α > ε,

B(c ^j ·) p c ^j φ(c ^j ) − f

≤ δ



< ∞.

Now, since there exists c = c ε such that P



B(c ^j ·) p c ^j φ(c ^j ) − f

≤ δ i.o.



= 1, we deduce that P (kZ _c ^j − gk α ≤ ε i.o.) = 1 for c = c ε .

R e m a r k s. (i) Theorem 3.1 gives a stronger result than the law of the iterated logarithm obtained by Helmes, R´emillard and Theodorescu [5].

(ii) Theorem 3.1 can be easily generalized to Brownian functionals F (B) satisfying the following conditions:

(H1) For every a ≥ 0 the restriction of F to K a = 

f ∈ H : ¹ ₂ |f | ² _H ≤ a is continuous;

(H2) For every R > 0, ̺ > 0, a > 0 there exist ε 0 > 0, β > 0 such that for every f ∈ K _a ,

P (kF (εB) − F (f )k α > ̺, kεB − f k ≤ β) ≤ exp(−R/ε ² );

(H3) There exists δ > 0 such that for every ε > 0 and every (u, t) ∈ [0, ∞) ² ,

F (εB(u·))(t) = ε ^δ F (B)(ut);

(H4) For every ε > 0, there exists c ε > 1 such that for every 1 < c ≤ c ε , we have Λ(A) > 1, where

A = {g ∈ C ^α : sup

1≤v≤c kg − g(·/v)k α ≥ ε/c}.

Let us note that the class of Brownian functionals satisfying (H1)–(H4) contains the iterated stochastic integrals considered by Baldi [1] and the stochastic integrals in R´emillard [11].

References

[1] P. B a l d i, Large deviations and functional iterated logarithm law for diffusion pro- cesses, Probab. Theory Related Fields 71 (1986), 435–453.

[2] P. B a l d i, G. B e n A r o u s and G. K e r k y a c h a r i a n, Large deviations and Strassen theorem in H¨ older norm, Stochastic Process. Appl. 42 (1992), 171–180.

[3] G. B e n A r o u s and M. L e d o u x, Grandes d´eviations de Freidlin–Wentzell en norme h¨ olderienne, in: Lecture Notes in Math. 1583, Springer, 1994, 293–299.

[4] T. C h a n, D. D e a n, K. J a n s o n s and L. R o g e r s, Polymer conformations in elon- gational flows, Comm. Math. Phys. 160 (1994), 239–257.

[5] K. H e l m e s, B. R ´em i l l a r d and R. T h e o d o r e s c u, The functional law of the it- erated logarithm for L´ evy’s area process, in: Lecture Notes in Control and Inform.

Sci. 96, Springer, 1986, 338–345.

[6] K. H e l m e s and A. S c h w a n e, L´evy’s stochastic area formula in higher dimensions, J. Funct. Anal. 54 (1983), 117–192.

[7] N. I k e d a, S. K u s u o k a and S. M a n a b e, L´evy’s stochastic area formula for Gaus- sian processes, Comm. Pure Appl. Math. 4 (1994), 329–360.

[8] P. L´ev y, Le mouvement brownien plan, Amer. J. Math. 62 (1940), 487–550.

[9] R. S. L i p t s e r and A. N. S h i r y a e v, Statistics of Random Processes, Vol. 2, Sprin- ger, New York, 1977.

[10] M. N’ z i, B. R ´em i l l a r d and R. T h e o d o r e s c u, Between Strassen and Chung nor- malizations for L´ evy’s area process, Statist. Probab. Letters, to appear.

[11] B. R ´em i l l a r d, On Chung’s law of the iterated logarithm for some stochastic inte- grals, Ann. Probab. 22 (1994), 1794–1802.

M’hamed Eddahbi Modeste N’zi

Department of Mathematics Department of Mathematics

Cadi Ayyad University University of Abidjan

BP S15, Marrakech, Morocco 22 BP 582 Abidjan 22, Ivory Coast E-mail: nziy@ci.refer.org

Received on 4.1.1996;

revised version on 15.6.1996