Integration by parts setting

We consider a filtered probability space (Ω,G,(Gt)_t∈[0,1],P) endowed with a Poisson random measureµon [0,1]×E, where E is an open subset of R, with compensator υ on [0,1]×E and with compensated measure

˜

µ=µ−υ. We now consider the process (Y_t^β)_t∈[0,1], the solution of

Y_t^β=y₀+ Z t

0

a(Y_s^β, θ)ds+cσ Z t

0

Z

E

z˜µ(ds,dz), (A.1)

where the parameter β= (θ, σ)^T belongs toR×(0,∞),ais a real valued function andcis a constant.

This is the framework of Cl´ement and Gloter [5] and our aim is to give some explicit representation formulas for the density of Y₁^β and its derivative with respect toβ.

We assume that the following assumptions are fulfilled.

H: (a) The functionahas bounded derivatives up to order five with respect to both variables.

(b) The compensator of the Poisson random measureµis given byυ(dt,dz) = dt×g(z)dzwithg≥0 onE,C¹ onE and such that

∀p≥2, Z

E

|z|^pg(z)dz <∞.

We now recall the Malliavin operatorsLandΓ and their basic properties (see Bichteleret al.[2], Chapter IV, Section 8-9-10). For a test functionf : [0,1]×E7→R(f is measurable,C²with respect to the second variable, with bounded derivative, and f ∈ ∩_p≥1L^p(ν)) we set µ(f) =R1

0

R

Ef(t, z)µ(dt,dz). We introduce an auxiliary functionρ:E7→(0,∞) such thatρadmits a derivative andρ, ρ⁰andρ^g_g⁰ belong to∩p≥1L^p(g(z)dz). With these notations, we define the Malliavin operator L, on a simple functional µ(f), in the same way as in [5] by the following equations :

L(µ(f)) = 1 2µ

ρ⁰f⁰+ρg⁰

gf⁰+ρf⁰⁰

,

where f⁰ and f⁰⁰ are the derivatives with respect to the second variable. For Φ=F(µ(f1), .., µ(fk)),withF of classC², we set

LΦ=

k

X

i=1

∂F

∂xi

(µ(f1), ..., µ(fk))L(µ(fi)) +1 2

k

X

i,j=1

∂²F

∂xi∂xj

(µ(f1), ..., µ(fk))µ(ρf_i⁰f_j⁰).

These definitions permit to construct a linear operatorLon a spaceD⊂ ∩p≥1L^pwith the same basic properties as in equations. (i)–(iii), p.2322 from [5].

We associate toL, the symmetric bilinear operatorΓ:

Γ(Φ, Ψ) =L(ΦΨ)−ΦLΨ−Ψ LΦ.

Moreover, iff andhare two test functions, we have:

Γ(µ(f), µ(h)) =µ(ρf⁰h⁰).

These operators satisfy the following properties (see [2], Eq. (8-3)) LF(Φ) =F⁰(Φ)LΦ+1

2F⁰⁰(Φ)Γ(Φ, Φ), Γ(F(Φ), Ψ) =F⁰(Φ)Γ(Φ, Ψ),

Γ(F(Φ1, Φ2), Ψ) =∂Φ₁F(Φ1, Φ2)Γ(Φ1, Ψ) +∂Φ₂F(Φ1, Φ2)Γ(Φ2, Ψ). (A.2) The operator L and the operator Γ permit to establish the following integration by parts formula (see [2], Props. 8–10, p.103).

Proposition A.1. ForΦandΨ in D, andf bounded with bounded derivatives up to order two, we have Ef⁰(Φ)Ψ Γ(Φ, Φ) =Ef(Φ)(−2Ψ LΦ−Γ(Φ, Ψ)).

Morover, if Γ(Φ, Φ)is invertible andΓ⁻¹(Φ, Φ)∈ ∩_p≥1L^p, we have

Ef⁰(Φ)Ψ =Ef(Φ)HΦ(Ψ), (A.3)

with

HΦ(Ψ) =−2Ψ Γ⁻¹(Φ, Φ)LΦ−Γ(Φ, Ψ Γ⁻¹(Φ, Φ)) (A.4)

=−2Ψ Γ⁻¹(Φ, Φ)LΦ− 1

Γ(Φ, Φ)Γ(Φ, Ψ) + Ψ

Γ(Φ, Φ)²Γ(Φ, Γ(Φ, Φ)). (A.5)

A.2 Representation of the density of Y

₁^β

and its derivative

The integration by parts setting of the preceding section permits to derive the existence of the density ofY₁^β given by (A.1), and gives a representation of this density as an expectation. From Bichteleret al.([2], Sect. 10, p.130) we know that ∀t >0, the variable Y_t^β, the solution of (A.1), belongs to the domain of the operator L, and we can computeLY_t^β and Γ(Y_t^β, Y_t^β) as in [5]. We recall the representation formula for the density ofY₁^β (see [5]).

Theorem A.2. [Cl´ement and Gloter [5]]: Let us denote byq^β the density ofY₁^β. We assume thatH holds and that the auxiliary function ρsatisfies:

lim inf

u→∞

1 lnu

Z

E

1_{{ρ(z)≥1/u}}g(z)dz= +∞. (A.6)

Then,

q^β(u) =E(1_{Yβ

1≥u}H_Yβ 1

(1)), with,

H_Yβ 1

(1) = Γ(Y₁^β, Γ(Y₁^β, Y₁^β))

Γ(Y₁^β, Y₁^β)² −2 LY₁^β

Γ(Y₁^β, Y₁^β) = W₁^β

(U₁^β)² −2LY₁^β

U₁^β , (A.7)

where the processes (LY_t^β)and(U_t^β) =Γ(Y_t^β, Y_t^β)are solutions of the linear equations:

LY_t^β = Z t

0

a⁰(Y_s^β, θ)LY_s^βds+1 2

Z t 0

a⁰⁰(Y_s^β, θ)U_s^βds+cσ 2

Z t 0

Z

E

ρ⁰(z) +ρ(z)g⁰(z) g(z)

µ(ds,dz), (A.8)

U_t^β = 2 Z t

0

a⁰(Y_s^β, θ)U_s^βds+c²σ² Z 1

0

Z

E

ρ(z)µ(ds,dz). (A.9)

The process (W_t^β) =Γ(Y_t^β, U_t^β)is the solution of the linear equation:

W_t^β= 3 Z t

0

a⁰(Y_s^β, θ)W_s^βds+ 2 Z t

0

a⁰⁰(Y_s^β, θ)(U_s^β)²ds+c³σ³ Z t

0

Z

E

ρ(z)ρ⁰(z)µ(ds,dz). (A.10) In [5], the authors studied the derivative of q^β with respect to the drift parameterθ only. Here, we intend to study the derivative of q^β with respect to both parameters θ and σ. We first remark that (Y_t^β)_t admits derivatives with respect to θ and σ (see [2], Thm. 5.24 p.51), denoted by (∂θY_t^β)t and (∂σY_t^β)t respectively.

Moreover, (∂θY_t^β)t, (∂σY_t^β)tare respectively the unique solutions of

∂_θY_t^β= Z t

0

a⁰(Y_s^β, θ)∂_θY_s^βds+ Z t

0

∂_θa(Y_s^β, θ)ds, (A.11)

∂σY_t^β = Z t

0

a⁰(Y_s^β, θ)∂σY_s^βds+c Z t

0

Z

E

zµ(ds,˜ dz). (A.12)

By iterating the integration by parts formula, since Y₁^β admits derivatives with respect to θ and σ, one can prove, under the assumptionH, the existence and the continuity inβof∇βq^β(see Thm. 4-21 in [2]), moreover, we will represent it as an expectation in Theorem A.4. The next result extends the result of Theorem 5 in [5], by giving an expression for the logarithm derivatives of the density w.r.t. (θ, σ) in terms of a conditional expectation.

Theorem A.3. Under the assumptions of TheoremA.2,

∇βq^β q^β (u) =





∂_θq^β q^β (u)

∂_σq^β q^β (u)



=E H_Yβ

1

∇βY₁^β

|Y₁^β=u

, (A.13)

where

H_Yβ 1

∇_βY₁^β :=



 H_Yβ

1

∂θY₁^β H_Yβ

1

∂σY₁^β



=−2 ∂θY₁^β

∂σY₁^β

!LY₁^β

U₁^β + ∂θY₁^β

∂σY₁^β

! W₁^β

U₁^β2 − 1 U₁^β



 Γ

Y₁^β, ∂θY₁^β Γ

Y₁^β, ∂σY₁^β



, (A.14)

LY₁^β,U₁^β andW₁^β are given in Theorem A.2, the process(V_t^θ) =Γ

Y_t^β, ∂θY_t^β

is the solution of

V_t^θ= 2 Z t

0

a⁰ Y_s^β, θ

V_s^θds+ Z t

0

U_s^β

(∂_θa)⁰ Y_s^β, θ

+a⁰⁰ Y_s^β, θ

∂_θY_s^β

ds, (A.15)

and the process (V_t^σ) =Γ

Y_t^β, ∂σY_t^β

is the solution of

V_t^σ= 2 Z t

0

a⁰ Y_s^β, θ

V_s^σds+ Z t

0

a⁰⁰ Y_s^β, θ

∂_σY_s^βU_s^βds+c²σ Z t

0

Z

E

ρ(z)µ(ds,dz). (A.16)

Proof. Theorem A.3 is an extension of Theorem 5 in [5] where the main novelty is the expression for ^∂σ_q^qβ^β. For the computation of the new termH_Yβ

1

(∂σY₁^β), we apply Theorem 10-3 in [2] to the stochastic differential equation satisfied by the vector (Y_t^β, U_t^β, ∂_σY_t^β)^T, this gives the above expression for (V_t^σ).

We end this subsection with an explicit representation of∇βq^β(u) which gives a computation of the iterated Malliavin weight H_Yβ

1 (H_Yβ

1 (∇βY₁^β)).

Theorem A.4. Under the assumptions of TheoremA.2,

∇βq^β(u) = ∂_θq^β(u)

∂σq^β(u)

!

=Eh

1{^Y1^β≥u}H^Y1^β

H_Yβ 1

∇βY₁^βi

, (A.17)

where

H_Yβ 1

H_Yβ 1

∇βY₁^β

=−2H_Yβ 1

∇βY₁^βLY₁^β U₁^β +H_Yβ

1

∇βY₁^β W₁^β

U₁^β2 −



 Γ

Y₁^β,H_Yβ 1

∂θY₁^β Γ

Y₁^β,H_Yβ 1

∂_σY₁^β



 1

U₁^β, (A.18)

where∂θY₁^β, ∂σY₁^βare respectively given by equations (A.11),(A.12)andU₁^β, W₁^βare computed in TheoremA.2, H_Yβ

1(∇βY₁^β)is given in TheoremA.3.

Proof. Letf be a smooth functions with compact support. Then,

∇βEh f

Y₁^βi

= Z

R

du∇βq^β(u)f(u).

On the other hand, using the integration by parts formula of the Malliavin calculus, we have

∇_βEh f

Y₁^βi

=Eh f⁰

Y₁^β

∇_βY₁^βi

=Eh f

Y₁^β H_Yβ

1

∇βY₁^βi

=Eh F

Y₁^β H_Yβ

1

H_Yβ 1

∇βY₁^βi ,

where F denotes a primitive function off. Iff converges to Dirac mass at some point u, from the estimates above, we can deduce (A.17). Moreover, from (A.5) we also get (A.18).

To complete the result of Theorem A.4, we give the expressions for Γ

Y₁^β,H_Yβ 1

∂_θY₁^β and Γ

Y₁^β,H_Yβ 1

∂σY₁^β .

Lemma A.5. Under the assumptions of TheoremA.2, are computed in Theorem A.3, H_Yβ

1

Proof. From the basic properties of the operators L and Γ (linearity and the chain rule property) stated in Section A.1, we get that

Γ

Similarly, we have

Then, from (A.7) and the above estimates, we get the formula (A.19), after some calculus and the proof is complete.

Lemma A.6. Under the assumptions of Theorem A.2, there are versions of the processes (D_t^β) = solutions of the linear equations:

D^β_t = 2

Proof. The proof of LemmaA.6is a direct consequence of Theorem 10-3 in [2]. Indeed, considering the stochastic differential equation satisfied by the vector

Y_t^β, LY_t^β, U_t^β, W_t^β, V_t^θ, V_t^σ, ∂θY_t^β, ∂σY_t^βT

and using Theorem 10-3 in [2], we prove that the processes

D^β_t

t are solutions of linear equations, respectively, given by (A.20)–(A.23).

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