Processus de Lévy en Finance - Laboratoire de Probabilités et ...

82 CHAPTER 2. THE CALIBRATION PROBLEM
and
E P [E(N ′′ ) T log E(N ′′ ) T ] = bT + e bT
∞ ∑
k=0
e
−λT (λT )k
k!
E[ ∏ s≤T
(1 + ∆N s ′′ ) ∑ log(1 + ∆N s ′′ )|k jumps]
s≤T
Since, under the condition that N ′′ jumps exactly k times in the interval [0, T ], the jump sizes
are independent and identically distributed, we find, denoting the generic jump size by ∆N ′′ :
E P [E(N ′′ ) T log E(N ′′ ) T ]
∑
∞
= bT + e bT e
k=0
−λT (λT )k
= bT + λT E[(1 + ∆N ′′ ) log(1 + ∆N ′′ )]
∫ ∞
= bT + T (1 + x) log(1 + x)ν ′′ (dx)
−∞
∫
= T (φ(x) log φ(x) + 1 − φ(x))ν P (dx).
R\I
k!
kE[1 + ∆N ′′ ] k−1 E[(1 + ∆N ′′ ) log(1 + ∆N ′′ )]
In particular, E P [E(N ′′ ) T log E(N ′′ ) T ] is finite if and only if the integral in the last line is finite.
Combining the above expression with (2.21) and (2.20) finishes the proof.
2.4.1 Properties of the relative entropy functional
Lemma 2.10. Let P, {P n } n≥1 ⊂ L + B for some B > 0, such that P n ⇒ P . Then for every r > 0,
the level set L r := {Q ∈ L : I(Q|P n ) ≤ r for some n} is tight.
Proof. For any Q ∈ L r , P Q denotes any element of {P n } n≥1 , for which I(Q|P Q ) ≤ r.
characteristic triplet of Q is denoted by (A Q , ν Q , γ Q ) and that of P Q by (A P Q, ν P Q, γ P Q). In
addition, we define φ Q := dνQ . From Theorem 2.9,
dν P Q
∫ ∞
−∞
(φ Q (x) log φ Q (x) + 1 − φ Q (x))ν P Q
(dx) ≤ r/T ∞ .
The
Therefore, for u sufficiently large,
∫
{φ Q >u}
∫
φ Q ν P Q
(dx) ≤
{φ Q >u}
2φ Q [φ Q log φ Q + 1 − φ Q ]ν P Q(dx)
φ Q log φ Q
≤
2r
T ∞ log u ,
which entails that for u sufficiently large,
∫
{φ Q >u}
ν Q (dx) ≤
2r
T ∞ log u