Exponential integrability of Itô's processes

J. Math. Anal. Appl. 358 (2009) 427–433

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Exponential integrability of Itô’s processes Wenliang Huang a,b,∗ a b

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China School of Management, University of Shanghai for Science and Technology, Shanghai 200093, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 6 November 2008 Available online 8 May 2009 Submitted by M. Ledoux

In this note we prove the exponential integrability of super-norms of general Itô’s processes under certain assumptions, and then apply it to the diffusion processes determined by stochastic differential equations. In particular, a conjecture in [Y. Hu, Exponential integrability of diffusion processes, in: Contemp. Math., vol. 234, 1999, pp. 75–84] is solved. © 2009 Elsevier Inc. All rights reserved.

Keywords: Exponential integrability Itô’s process Stochastic differential equations

1. Introduction and main result Let { w t , t ∈ [0, T ]} be an m-dimensional standard Brownian motion on a filtered probability space (Ω, F , P; (Ft )t ∈[0, T ] ). The classical Fernique theorem says that for some α > 0

Ee α ·supt ∈[0,T ] | w t | < +∞. 2

This type of exponential integrability has numerous applications in probability theory. In [2], Y. Hu studied the exponential integrability for a large class of diffusion processes determined by stochastic differential equations. In this note we extend Hu’s result to the general Itô’s process. In particular, we solve a conjecture given in [2, p. 83]. More materials about the exponential functionals of Brownian motion is referred to the monograph of M. Yor [4]. Let { M t , t ∈ [0, T ]} be a positive Itô process with the following form:

t Mt = M 0 +

t ξs dw s +

0

ηs ds, 0

where M 0 is an F0 -measurable random variable, and ξt , Our main result is stated as follows:

ηt are (Ft )-adapted Rm and R-valued stochastic processes.

Theorem 1.1. Suppose that there exist γ ∈ [1, 2] and λ1 , λ2 > 0 such that for all s ∈ [0, T ] 2 −γ

ξs 2  λ1 M s

*

ηs  λ2 M s1−γ ,

,

P-a.s.

Address for correspondence: Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China. E-mail address: [email protected].

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2009.05.006

©

2009 Elsevier Inc. All rights reserved.

(1)

428

W. Huang / J. Math. Anal. Appl. 358 (2009) 427–433 1+γ 2

(I) Case 1 < γ  2: If there is a 1 < q <

Ee

α M 0q

such that for all α > 0

< +∞,

then for all α > 0 q

Ee α ·supt ∈[0,T ] Mt < +∞. (II) Case γ = 1: If for some β > 0

Ee β M 0 < +∞, then for some α > 0

Ee α ·supt ∈[0,T ] Mt < +∞. This theorem will be proved in Section 2. In Section 3, we shall apply it to the diffusion process defined by stochastic differential equations. 2. Proof of Theorem 1.1 Let p  γ ∧ 2. By Itô’s formula we have p

t

p

Mt = M 0 + p

p −1

Ms

ξs dw s + N t ( p ),

(2)

0

where

t

p −1 Ms s ds

N t ( p ) := p

η

p ( p − 1)

+

t

0

p −2

Ms

2

ξs 2 ds.

0

Noting that by (1) p −γ

dE N t ( p )/dt  λ2 p E M t

+

λ1 p ( p − 1 ) 2

p −γ

EMt

p −γ

 (λ2 + λ1 ) · p 2 E M t

,

(3)

and taking expectations for both sides of (2), we get for any p  2 p −γ

p

dE M t /dt = (λ2 + λ1 ) p 2 E M t

(4)

.

(Case 1 < γ  2): In what follows, for the simplicity of notations, we write

M t∗ := sup M s . 0st

Letting p = n in (2) and by (1), we also have

t M tn  M n0 + n

M ns −1 ξs dw s + (λ2 + λ1 ) · n2

0

t

n −γ

Ms

ds.

0

By Doob’s maximal inequality (cf. [3]) we have

 ∗ nq

EMt

nq EM0

C

q q    t s        n −γ n −1 2q  + E sup n M u ξu dw u  + n E M s ds   s∈[0,t ]  0

0

q  t  t     q(n−γ ) nq n −1 2q EM s ds  C E M 0 + En M u ξu dw u  + n   

0

 (2 )

=C

nq EM0

0

 q + E M n − M n − N t (n) + n2q t

t

0

0

 q(n−γ ) EM s ds

W. Huang / J. Math. Anal. Appl. 358 (2009) 427–433

 nq EM0

C

nq + EMt

nq EM0

C

+n

 q(n−γ ) EM s ds

0



(4 )

t 2q

t

nq−γ EM s

2

+ (nq)

nq

 C 0 EM0 + C 1



t ds + n

q(n−γ ) EM s ds

2q

0

t

429

0

 2

nq 

n EM s

nq−γ nq

n −γ  nq  n + n2q E M s ds.

(5)

0

Here and below, the constants C and C 0 , C 1 > 1 are independent of n, and may be different in different occasions. Set ∗ nq

g (t ) := E M t



, γ

γ



ρ (x) := C 1 n2 x1− nq + n2q x1− n . By (5) we have

t

nq



g (t )  C 0 · E M 0 +



ρ g (s) ds. 0

By Bihari’s inequality (cf. [1] or [5]), we have

 

nq 

g (t )  G −1 G C 0 · E M 0

 +t ,

(6)

where

x G (x) =

1

ρ ( y)

dy =

0

1 C1

x

1 γ 1− n2 y nq

0

γ

+ n2q y 1− n

dy .

Note that for x > 1 γ

ρ (x)  2C 1n2q x1− nq =: ρ˜ (x). Hence nq C 0 · EM0

g (t ) 

t + Cn

2q





ρ˜ g (s) ds.

+ 0

Observing that

G˜ (x) :=

x

γ

1

ρ˜ ( y )

0

dy =

nq · x nq 2γ C 1n2q

,

by Bihari’s inequality again, we find that

g (t ) 



nq C 0 · EM0

1+γ 2

In view of q <

q(2q − 1)

γ

<

+ Cn

2q

γ  nq

+

2T γ C 1n2q−1 q

 nq γ

nq

2γ

−1 

nq

nq

C 0 E M 0 + Cn2q + C 2 n γ

nq(2q−1)

γ



.

 32 , we have 2q

γ

.

Thus, there exists N ∈ N large enough such that for all n  N

g (t )  n

2nq

γ

(7)

.

Now we proceed to estimate g (t ) by starting from (6). It is easy to see that

x G (x) = 0

x

1 C 1 (n2 y

γ 1− nq

γ

+ n2q y 1− n )

dy 

γ

1 γ

0

C 1 n2q y 1− n

dy =

xn C 1 γ n2q−1

,

430

W. Huang / J. Math. Anal. Appl. 358 (2009) 427–433

and for x  n

2nq

γ

x

γ

1

G (x) 

dy =

γ

2C 1n2q y 1− n

0

xn 2C 1 γ n2q−1

.

Therefore, by (7) and (6) we have for n  N γ

γ

n  (C 0 E M nq 0 )  G g (t )  + T, 2q−1



g (t ) n 2C 1 γ n2q−1

C1γ n

which then produces that ∗ nq

EM T

n γ n(2q−1)  n n   −1  γn nq  nq γ .  2 C 0 E M 0 n + 2C 1 γ T n2q−1 γ  2 γ 2 C 0EM0 + C 3 n γ

By Stirling’s formula n! ∼ nn e −n ∗q

Ee α M T =

√

2π n, we have

∗ nq

α EM t n ! n n

n

n

n(2q−1)

αn 4 γ C 0 E M nq αn C γ n 3 0 +  n! n! n n n

 γ1 q  ∼ C 0 E eα4 M0 + Since

2q−1

γ

< 1 and E(e α C

αn C 3γ en n (1 −

n

n

γ

(2q−1)

γ

)√

. 2π n

1

γ Mq 0

) < +∞ for all α > 0, we have for all α > 0

∗q

Ee α M T < +∞. (Case γ = 1): In this case, we have

t

t

Mt = M 0 +

t

ξs dw s + 0

ηs ds  M 0 + 0

˜t ξs dw s + λ2 T =: M

0

and

˜ s. ξs2  λ1 M s  λ1 M Hence, without any loss of generality, we may assume that

t Mt = M 0 +

ξs dw s . 0

Letting p = n in (4) we get



dE M tn /dt  λ1 · n2 E M tn−1  λ1 · n2 E M tn

n−n 1

.

Solving this differential inequality, we find

E M tn 



E M n0

 n1

+ λ1nT

As above, there exists 0 < α0 <

Ee α0 Mt < +∞,

n

   2n−1 E M n0 + λn1nn T n .

β

such that

2

∀t ∈ [0, T ].

(8)

Define

t Z tα := exp

ξs dw s −

α 0

α2



t ξs2 ds

2

.

0

α By Kazamaki’s theorem, Z tα is an exponential martingale for 0 < α < 20 .

W. Huang / J. Math. Anal. Appl. 358 (2009) 427–433

Now we take 0 < α <

α0

431

sufficiently small such that

2

2α T < 1. Then by Doob’s inequality and Hölder’s inequality, we have

  ∗ E eα(M T −M0 )  E



 sup

t ∈[0, T ]

Z tα · exp

α2



t 2

ξs  ds

2 0

  T 1/2    α 2 1/2 2    E sup Z t M s ds E exp α t ∈[0, T ]

0

  2 1/2    2  1/2 E exp α T M ∗T  C E Z Tα   1/2   α M ∗  α T E e2 T  C Ee 2α M T , where the last step is due to T  α 2  Z  = e 2α ( M T − M 0 )−α 2 0 ξs 2 ds  e 2α M T . T

Set ∗

f (α ) := Ee α M T . Then



∗

f (α /2)  Ee α ( M T − M 0 )

12  α M 12  1/4  α T /2  α M 12 · Ee 0  C Ee 2α M T f (α /2) · Ee 0 .

This gives that

 

f (α /2)  C Ee 2α M T

1/4  α M 12  2−2α T (8) · Ee 0 < +∞,

and the proof is complete. 3. Exponential integrability of diffusion process Consider the following SDE



d X t = σ (t , X t ) dw t + b(t , X t ) dt , X 0 = x ∈ Rd ,

(9)

where W t is an m-dimensional standard Brownian motion, and and Rd respectively. Here dw t denotes the Itô differential. The generator of this equation is

L t f (x) =

d 1

2

σ , b are continuous functions from R+ × Rd to Rd × Rm

∂2 f ∂f (x) + b i (t , x) (x), ∂ xi ∂ x j ∂ xi d

ai j (t , x)

i , j =1

i =1

where

ai j (t , x) =

m

σki (t , x)σkj (t , x).

k =1

Let φ be a positive C 2 -function on Rd satisfying

φ(x)  c |x|β

(as x → ∞).

Suppose that for some

γ ∈ [1, 2]

(C1) Γt (φ)(x)  λ1 φ 2−γ (x) for some λ1 > 0 and all t > 0, x ∈ Rd , where Γ is the “carré du champ” operator defined by

Γt (φ)(x) =

d i , j =1

(C2) L t φ(x)  λ2 φ

1−γ

ai j (t , x)

∂φ ∂φ (x) (x). ∂ xi ∂xj

(x) for some λ2 > 0 and all t > 0, x ∈ Rd .

432

W. Huang / J. Math. Anal. Appl. 358 (2009) 427–433

Then we have: Theorem 3.1. Let { X t , t ∈ [0, T ]} be the solution to SDE (9). Under (C1) and (C2), we have (a) if γ > 1, then for 1 < q <

1+γ 2

and all α > 0

 q E e α ·supt ∈[0,T ] φ( Xt ) < +∞; (b) if γ = 1, then for some α > 0

  E e α ·supt ∈[0,T ] φ( Xt ) < +∞. Proof. Applying Itô’s formula to φ( X t ), we have

t φ( X t ) = φ( X 0 ) +

  ∇φ( X s ), σ (s, X s ) dw s Rd +

0

t L s φ( X s ) ds. 0

Note that

 ∗  σ (s, x)∇φ(x)2 = Γs (φ)(x)  λ1 φ 2−γ (x). The result follows by Theorem 1.1.

2

Remark 3.2. In [2, Theorem 1.1], Hu proved that the above (a) holds for γ > 3/2 and (b) holds for γ = 3/2. He also conjectured in [2, p. 83] that the critical exponent γ should be 1. In particular, Theorem 3.1 solves this conjecture. Let us consider the following one-dimensional SDE

d X t = X tδ dw t ;

X 0 = x  0,

where δ ∈ [0, 1]. The corresponding formal generator is

L f (x) =

1 2

x2δ f (x),

 2 Γ ( f )(x) = x2δ f (x) . Let φ : R → R+ be a C 3 -function and when |x| > 1, φ(x) = |x|2−2δ , where β > 0. It is easy to see that when |x| > 1

L φ(x) =

β(β − 1) 2

,

Γ (φ)(x) = β 2 |x|2−2δ = β 2 φ(x). Thus we get by Theorem 3.1: Corollary 3.3. When 0  δ < 1, we have for some α > 0 2−2δ

Ee α supt ∈[0,T ] | Xt |

< +∞.

In particular, when δ = 0, this is the classical Fernique theorem. Remark 3.4. If δ = 1, the solution to SDE (10) is the geometric Brownian motion and is explicitly given by t

Xt = x · e w t − 2 . It is easy to see that for any positive β > 0

Ee | Xt | = +∞. β

(10)

W. Huang / J. Math. Anal. Appl. 358 (2009) 427–433

433

References [1] I. Bihari, A generalization of a lemma of Belmman and its application to uniqueness problem of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1956) 71–94. [2] Y. Hu, Exponential integrability of diffusion processes, in: Contemp. Math., vol. 234, 1999, pp. 75–84. [3] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Grundlehren Math. Wiss., vol. 293, Springer-Verlag, 1998. [4] M. Yor, Exponential Functionals of Brownian Motion and Related Processes, Springer-Finance, Berlin, 2001. [5] X. Zhang, J. Zhu, Non-Lipschitz stochastic differential equations driven by multi-parameter Brownian motions, Stoch. Dyn. 6 (3) (2006) 329–340.