Asymptotic Evaluation of The Poisson Measures For Tubes Around Jump Curves
Asymptotic Evaluation of The Poisson Measures For Tubes Around Jump Curves
Asymptotic Evaluation of The Poisson Measures For Tubes Around Jump Curves
ASYMPTOTIC EVALUATION OF
THE POISSON MEASURES
FOR TUBES AROUND JUMP CURVES
[145]
146 X. Bardina et al.
for a given function L(ẋ, x), then the above expression is called the Onsager–
Machlup functional , and can be interpreted as a likelihood ratio for the law of
the process X. Most of the studies on Onsager–Machlup’s functional concern
the case of a diffusion process X which is the solution of the stochastic
differential equation
dX(t) = b(X(t))dt + dW (t), X(0) = x0 , X(t) ∈ Rd ,
where x0 ∈ Rd and the coefficient b has some regularity. Then, for a large
class of norms on the Wiener space, and for functions in the Cameron–
Martin space, it can be shown that the Onsager–Machlup function exists
and is given by
d d
1X 1 X ∂bi
L(φ̇, φ) = − |φ̇i − bi (φ)|2 − (φ).
2 i=1 2 i=1 ∂xi
We refer to Ikeda and Watanabe (1981) and Shepp and Zeitouni (1992) for
some basic results in that direction, and to Lyons and Zeitouni (1999) and
Capitaine (2000) for theorems concerning the consistency of the Onsager–
Machlup functional with respect to the norm considered on the Wiener
space. Note also that the case of SDEs in infinite dimensions driven by a
Gaussian noise have been considered in Mayer–Wolf and Zeitouni (1993),
and Bardina, Rovira and Tindel (2000, 2001).
A natural (but to our knowledge unadressed in the literature) problem
is to find if this kind of result still holds on the Poisson space. That is,
if N is a standard Poisson process on [0, 1], X the solution to a stochastic
differential equation driven by N , and φ : [0, 1] → R a deterministic function,
we would like to evaluate P {kX − φk ≤ ε} for various norms. For the sake
of computations, we deal with two simple cases: the case X = N , and the
case when X is the solution to a linear equation of the form
t
Xt = Xs ds + Nt , t ∈ [0, 1].
0
In those two examples, some fundamental differences with respect to the
Gaussian case can already be observed:
1. The Girsanov transform, which is an essential tool in the computation
of the Onsager–Machlup functional on the Wiener space, is of little help in
our case, since it transforms the Poisson process to a general semi-martingale
that cannot be handled easily.
Asymptotic evaluation of Poisson measures 147
2. It seems natural to deal with jump functions, which are the closest to
the a.s. paths of the Poisson process. For this reason, we will evaluate the
probability of some tubes around functions of the type
k
X
ht = I[Si ,1) (t),
i=1
where k ≥ 0 and 0 < S1 < . . . < Sk < 1 are the jump points of h in the case
X = N , and of the type
t
φht = φhs ds + ht
0
where 0 < S1 < . . . < Sk < 1 are the jump points. Then, for ε > 0 small
enough,
X εj
k−1
k −1 εk
P {kN − hk1 ≤ ε} = 2 e ε
e − = 2k e−1 + O(εk+1 ),
j=0
j! k!
ε2k
P {kN − hk2 ≤ ε} = 2k e−1 + O(ε2k+2 ).
k!
Remark 3.2. Note that the probability depends only on the number of
jumps of the function h.
Proof. We will only develop the L2 case. The proof for the L1 norm can
be done using the same arguments.
Since 0 < S1 < . . . < Sk < 1, there exists ε0 such that Si+1 − Si > ε20
for all i ∈ {1, . . . , k − 1}, S1 > ε20 and 1 − Sk > ε20 . Along the proof we will
consider ε < ε0 .
Asymptotic evaluation of Poisson measures 149
We have
∞
X e−1
P {kN − hk2 ≤ ε} = P {kN − hk2 ≤ ε | N1 = j} .
j=0
j!
2
Sn+1 +ε
= (n + 1) n! dt1 . . . dtn dtn+1 ,
Sn+1 −ε2 Aε
150 X. Bardina et al.
where
n n
X o
Aε = t1 < . . . < tn ; |ti − Si | ≤ ε2 − |tn+1 − Sn+1 | .
i=1
ε2
= (n + 1)2 2 (ε2 − u)n du = 2n+1 ε2(n+1) ,
n
nX
k j
X o
≤P |Ti − Si | + (1 − Ti ) ≤ ε2 N1 = j .
i=1 i=k+1
Thus,
∞
X e−1
P {kN − hk2 ≤ ε} = P {kN − hk2 ≤ ε | N1 = j}
j=0
j!
ε2k
+ O(ε2k+2 ),
= 2k e−1
k!
which completes the proof of the theorem.
As in the case of Poisson process, we can assume that there exists ε0 > 0
such that Si+1 − Si > ε20 for all i ∈ {1, . . . , k − 1}, S1 > ε20 and 1 − Sk > ε20 .
From now on we will assume ε ≤ ε0 .
Before the proof of the theorem we will show a preliminary lemma.
152 X. Bardina et al.
Lemma 4.2. (a) For fixed α < ε0 , if there exists i ∈ {1, . . . , k} such that
|Si − Ti | > α2 , then
kX − φh k22 > C1 α2
where C1 denotes a universal constant.
(b) There exists ε1 > 0, depending only on k, such that for fixed α < ε1 ,
if for some i > k, Ti < 1 and 1 − Ti > α2 , then
kX − φh k22 > C2 α2
where C2 is another universal constant.
Proof. (a) Set n := inf{i ∈ {1, . . . , k} : |Si − Ti | > α2 }. Then
Sn +α2
kX − φh k22 ≥ |X(t) − φh (t)|2 dt.
Sn −α2
If we define
n−1
X
dn = (e−Ti − e−Si ) + I{Tn <Sn } e−Tn ,
i=1
Then
1 1
kX − φh k22 > |X(t) − φ(t)|2 dt = e2t |d∞ (t)|2 dt
1−α2 1−α2
− kα2 )2 2 (e
−1 2
≥(e − e2(1−α ) ) ≥ C2 α2 ,
2
by similar arguments to the proof of (a) and for α small enough.
Asymptotic evaluation of Poisson measures 153
Remark 4.3. Note that from Lemma 4.2 we can assume, for ε small
enough, that for all i, l ∈ {1, . . . , k} such that i < l, Si < Tl and Ti < Sl .
Proof of Theorem 4.1. As in the case of a Poisson process we only develop
the L2 case. We have
X∞
h e−1
P {kX − φ k2 ≤ ε} = P {kX − φh k2 ≤ ε | N1 = j} .
j=0
j!
1 h X −Ti −Tl 2
k
= {e (e − e2(Ti ∨Tl ) ) − e−Ti −Sl (e2 − e2(Ti ∨Sl ) )
2
i,l=1
i
− e−Si −Tl (e2 − e2(Si ∨Tl ) ) + e−Si −Sl (e2 − e2(Si ∨Sl ) )}
1 h X 2−Ti −Tl
k
(2) = {e − e|Ti −Tl | − 2e2−Ti −Sl + 2e|Ti −Sl |
2
i,l=1
i
+ e2−Si −Sl − e|Si −Sl | } .
Notice that if we put δi := Ti − Si , then
e2−Si −Sl + e2−Ti −Tl − 2e2−Ti −Sl = e2−Si −Sl (1 + e−δi −δl − 2e−δi )
2−(Si +Sl ) (δi + δl )2 ηi,l
1
2 ηi1
=e δi − δ l + e − δi e ,
2
by Taylor’s decomposition, where ηi1 , ηi,l
1
∈ [−2ε2 /C1 , 2ε2 /C1 ].
On the other hand, by Remark 4.3, for the other term involved in the
sum (2) we have three situations:
154 X. Bardina et al.
If i < l, then
2e|Ti −Sl | − e|Ti −Tl | − e|Si −Sl | = eSl −Si (−eδl −δi + 2e−δi − 1)
Sl −Si (δl − δi )2 ηi,l
2
2 ηi2
=e −δl − δi − e + δi e ,
2
where ηi2 , ηi,l
2
∈ [−2ε2 /C1 , 2ε2 /C1 ].
If i = l, then
3
2e|Ti −Sl | − e|Ti −Tl | − e|Si −Sl | = 2e|Ti −Si | − 2 = 2|δi | + δi2 eηi ,
where ηi3 ∈ [−2ε2 /C1 , 2ε2 /C1 ].
Finally, if i > l, then
2e|Ti −Sl | − e|Ti −Tl | − e|Si −Sl | = eSi −Sl (−eδi −δl + 2eδi − 1)
(δi − δl )2 ηi,l
4 4
= eSi −Sl δi + δl − e + δi2 eηi .
2
4
where ηi4 , ηi,l ∈ [−2ε2 /C1 , 2ε2 /C1 ].
So, when N1 = k,
k
X
(3) kX − φh k22 = |δi | + Z,
i=1
where
k
X
−2(Si +Sl ) (δi + δl )2 ηi,l
1 1
Z= e e − δi2 eηi
2
i,l=1
k
X X
3 (δl − δi )2 ηi,l
2
2 ηi2
+ δi2 eηi
+ e − Sl −Si
e + δi e
i=1
2
i<l
X
Si −Sl (δi − δl )2 ηi,l
4
2 ηi4
+ e − e + δi e .
2
i>l
and using the results proved in Theorem 3.1 we get from (3)
X k
4
h 2 2 ε
(4) P {kX − φ k2 ≤ ε | N1 = k} ≤ P |δi | ≤ ε + C3 k 2 N1 = k
i=1
C1
k
ε2
= 2ε2 1 + C3 k 2 2 ≤ 2k ε2k + ck,1 ε2k+2
C1
where ck,1 depends only on k and C1 . On the other hand, for fixed ω such
Pk
that i=1 |δi | ≤ ε2 we clearly have |Z(ω)| < C3 k 2 ε4 . So, again from (3) we
Asymptotic evaluation of Poisson measures 155
get
nX
k o
(5) P {kX − φh k2 ≤ ε | N1 = k} ≥ P |δi | ≤ ε2 − C3 k 2 ε4 N1 = k
i=1
= (2ε (1 − C3 k 2 ε2 ))k ≥ 2k ε2k + ck,2 ε2k+2
2
where ck,2 depends only on k and C1 . Putting (4) and (5) together we obtain
(6) P {kX − φh k2 ≤ ε | N1 = k} = (2ε2 )k + O(ε2k+2 ).
• To deal with the case j > k notice that, again by Lemma 4.2, if
kX(ω) − φh k2 ≤ ε then |Ti (ω) − Si | ≤ ε2 /C1 for all i ∈ {1, . . . , k} and
|1 − Ti (ω)| ≤ ε2 /C2 for all i ∈ {k + 1, . . . , j}. So, using Proposition 2.2 we
have
P {kX − φh k2 ≤ ε | N1 = j}
≤ P {{ max |Ti − Si | ≤ ε2 /C1 } ∩ { max |1 − Ti | ≤ ε2 /C2 } | N1 = j}
1≤i≤k k+1≤i≤j
j!2k ε2j
≤ ,
C1k C2j−k
and then
∞
X e−1
(7) P {kX − φh k2 ≤ ε | N1 = j}
j!
j=k+1
∞
X e−1 2k ε2j 1 ε2k+2
≤ = e−1 2k · .
j=k+1 C1k C2j−k C2 C1k 1 − ε2 C2
Putting (1), (6) and (7) together we finish the proof of the theorem.
References
L. Shepp and O. Zeitouni (1992), A note on conditional exponential moments and Onsager–
Machlup functionals, Ann. Probab. 20, 652–654.