Self-intersection local time of <span class="mathjax-formula">$(\alpha ,d,\beta )$</span>-superprocess

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Self-intersection local time of (α, d, β)-superprocess

L. Mytnik

, J. Villa

aTechnion-Israel Institute of Technology, Faculty of Industrial Engineering and Management, Haifa 32000, Israel bUniversidad Autónoma de Aguascalientes, Departamento de Matemáticas y Física, Av. Universidad 940,

C.P. 20100, Aguascalientes, Ags., Mexico Received 6 March 2006; accepted 25 July 2006

Available online 27 February 2007

Abstract

The existence of self-intersection local time (SILT), when the time diagonal is intersected, of the(α, d, β)-superprocess is proved ford/2< αand for a renormalized SILT whend/(2+(1+β)⁻¹) < αd/2. We also establish Tanaka-like formula for SILT.

©2007 Elsevier Masson SAS. All rights reserved.

Résumé

L’existence de temps locaux d’auto-intersection est établie pour certaines classes de superprocessus, ainsi qu’une formule ana- logue à celle de Tanaka.

©2007 Elsevier Masson SAS. All rights reserved.

MSC:primary 60G57; secondary 60H15

Keywords:Self-intersection local time; Infinite variance superprocess; Tanaka-like formula

1. Introduction and statement of results

This paper is devoted to the proof of existence of self-intersection local time (SILT) of(α, d, β)-superprocesses for 0< β <1. Let us introduce some notation. LetBb(R^d)(respectivelyCb(R^d)) be the family of all bounded (respec- tively, bounded continuous) Borel measurable functions onR^d, andMF(R^d)be the set of all finite Borel measures onR^d. The integral of a functionf with respect to a measureμis denoted byμ(f ). IfEis a metric space we denote by D([0,+∞), E)the space of all càdlàg E-valued paths with the Skorohod topology. We will use cto denote a positive and finite constant whose value may vary from place to place. A constant of the formc(a, b, . . .)means that this constant depends on parametersa, b, . . ..

Let(Ω,F,F_·, P)be a filtered probability space where the(α, d, β)-superprocessX= {Xt: t 0}is defined.

That is, byXwe mean aMF(R^d)-valued, time homogeneous, strong Markov process with càdlàg sample paths, such

✩ Research supported in part by the Israel Science Foundation.

* Corresponding author.

E-mail addresses:[email protected] (L. Mytnik), [email protected] (J. Villa).

1 Thanks go to UAA for allow this postdoctoral stay.

2 Research supported in part by Henri Gutwirth Fund for the Promotion of Research.

0246-0203/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpb.2006.07.005

that for any non-negative functionϕ∈Bb(R^d), E

exp

−Xt(ϕ)

|X0=μ

=exp

−μ Vt(ϕ)

,

whereμ∈MF(R^d)andVt(ϕ)denotes the unique non-negative solution of the following evolution equation v_t=S_tϕ−

t 0

S_t−s

(v_s)^1+β

ds, t0.

Here{St: t0}denotes the semigroup corresponding to the fractional Laplacian operatorα.

Another way to characterize the(α, d, β)-superprocessXis by means of the following martingale problem:

⎧⎨

⎩

For allϕ∈D(α)(domain ofα) andμ∈MF(R^d), X₀=μ, andM_t(ϕ)=X_t(ϕ)−X₀(ϕ)−_t

0X_s(_αϕ)ds, is aFt-martingale.

(1.1) Ifβ=1 thenM_·(ϕ)is a continuous martingale. In this paper we are interested in the case of 0< β <1,and here M_t(ϕ)is a purely discontinuous martingale. This martingale can be expressed as

Mt(ϕ)= t 0

R^d

ϕ(x)M(ds,dx), (1.2)

whereM(ds,dx)is a martingale measure, it and the stochastic integral with respect to such martingale measure is defined in [11] (or in Section II.3 of [8]).

The SILT is heuristically defined by γ_X(B)=

B

R^2d

δ(x−y)X_s(dx)Xt(dy)dsdt,

whereB⊂ [0,∞)× [0,∞)is a bounded Borel set andδ is the Dirac delta function. LetD= {(t, t ): t >0}be the time diagonal onR₊×R₊. Forβ=1,B∩D=∅andd7, Dynkin [6] proved the existence of SILT,γX, for a very general class of continuous superprocesses. Also, from the Dynkin’s works follows the existence of SILT when β=1,B∩D=∅andd3 (see [1]). Forβ=1,d=4,5 andB∩D=∅,Rosen [15] proved the existence of a renormalized SILT for the(α, d,1)-superprocess. A Tanaka-like formula for the local time of(α, d,2)-superprocess was established by Adler and Lewin in [3]. The same authors derived a Tanaka-like formula for self-intersection local time for(α, d,2)-superprocess (see [2]). In this paper we are going to extend the above results for the case of 0< β <1.

The usual way to give a rigorous definition of SILT is to take a sequence(ϕ_ε)_ε>0of smooth functions that converges in distribution toδ, define the approximating SILTs

γ_X,ε(B)=

B

R^2d

ϕ_ε(x−y)X_s(dx)Xt(dy)dsdt, B⊂R₊×R₊,

and prove that(γX,ε(B))ε>0converges, in some sense (it is usually takenL²(P), L^1+β(P),L¹(P), distribution or in probability), to a random variableγX(B). In what follows we chooseϕε=pε, wherepε is theα-stable density, given by

p_ε(x, y)= 1 (2π )^d

R^d

e^{−i(z·(x−y))−ε|z|α}dz, x, y∈R^d,

when 0< α <2 and pε(x, y)= 1

(2π ε)^d/2e^{−|x−y|2/2ε}, x, y∈R^d,

forα=2. In this paper we will consider the particular case whenB= {(t, s): 0stT}. Here we denoteγX,ε(B) byγX,ε(T ), that is

γX,ε(T )= T 0

t 0

R^2d

pε(x−y)Xs(dx)Xt(dy)dsdt, ∀T 0.

Moreover, we are going to consider the renormalized SILT

˜

γX,ε(T )=γX,ε(T )−e^λε T 0

R^d

Xs

G^λ,ε(x− ·)

Xs(dx)ds,

where

G^λ,ε(x)= ∞ ε

e^−λtp_t(x)dt, λ, ε0.

Notice thatG^λ,ε(x)↑G^λ,0(x)=G^λ(x)asε↓0 for allx∈R^d, and ifd > αthen

G(x)=G^0,0(x)=c(α, d)|x|^α−d, (1.3)

where

c(α, d)= ((d−α)/2) 2^α/2π^d/2(α/2),

andis the usual Gamma function.Gis called Green function of_α. Also notice that, forλ >0, we have

_αG^λ,ε(x)=λG^λ,ε(x)−e^−λεp_ε(x), x∈R^d. (1.4)

Now we are ready to present our main result.

Theorem 1.LetXbe the(α, d, β)-superprocess with initial measureX₀(dx)=μ(dx)=h(x)dx, wherehis bounded and integrable with respect to Lebesgue measure on R^d. Let M be the martingale measure which appears in the martingale problem(1.1)forX.

(a) Letd/2< α. Then there exists a processγ_X= {γ_X(T ): T 0}such that for everyT >0,δ >0 P

sup

tT

γ_X,ε(t )−γ_X(t )> δ

→0, asε↓0.

Moreover, for anyλ >0,

γ_X(T )=λ T 0

t 0

R^2d

G^λ(x−y)X_s(dx)Xt(dy)dtds− T 0

R^d

X_T

G^λ(x− ·)

X_s(dx)ds

+ T 0

R^d

X_s

G^λ(x− ·)

X_s(dx)ds+ T 0

R^d

t 0

R^d

G^λ(x−y)M(ds,dy)Xt(dx)dt, a.s. (1.5)

(b) Letd/(2+(1+β)⁻¹) < αd/2. Then there exists a processγ˜X= { ˜γX(T ): T 0}such that for everyT >0, δ >0

P

sup

tT

γ˜X,ε(t )− ˜γX(t )> δ

→0, asε↓0.

Moreover, for anyλ >0,

˜

γ_X(T )=λ T

0

t 0

R^2d

G^λ(x−y)X_s(dx)Xt(dy)dtds− T 0

R^d

X_T

G^λ(x− ·)

X_s(dx)ds

+ T 0

R^d

t 0

R^d

G^λ(x−y)M(ds,dy)Xt(dx)dt, a.s. (1.6)

The processesγ_Xandγ˜_Xare called SILT and renormalized SILT ofX, respectively, and(1.5)and(1.6)are called Tanaka-like formula for SILT.

Remark 1.It is interesting to note that our bound on dimensions d <

2+(1+β)⁻¹ α,

for renormalized SILT does not converge, asβ↑1, to the boundd <3αestablished by Rosen [15] for finite variance superprocess (β=1). For example, simple algebra shows that for 5/3< α <2, there is a SILT for finite variance superprocess in dimensionsd5. However for anyβ∈((3α−5)/(5−2α),1)we get the existence of SILT only in dimensionsd4 and this bound not improve tod5 ifβ↑1. So, our bound is more restrictive, and we believe that it is related to the fact that(α, d, β)-superprocess (withβ <1) has jumps. Our conjecture is that forβ <1, the renormalized SILT defined by (1.6) does not exist in dimensions greater than(2+(1+β)⁻¹)α.

The common ways to prove the existence of SILT for the finite variance superprocesses (see e.g. [2,15]) do not work here. The reason for this is that such proofs strongly rely on the existence of high moments ofX (at least of order four), and(α, d, β)-superprocessX has moments of order less than 1+β. To overcome this difficulty let us consider the path properties ofXmore carefully. It is well known (see Theorem 6.1.3 of [4]) that, for 0< β <1,the (α, d, β)-superprocessX is a.s. discontinuous and has jumps of the formX_t=rδ_x, for somer >0, x∈R^d. Here δxdenotes the Dirac measure concentrated atx. Let

N_X(dx,dr,ds)=

{(x,r,s):X_s=rδ_x}

δ_(x,r,s), (1.7)

be a random point measure onR^d×R₊×R₊with compensator measureN_Xgiven by

N_X(dx,dr,ds)=ηr^−2−βdrXs(dx)ds, (1.8)

where

η=β(β+1) (1−β).

LetK >0 fix. From [7] and [4] we have that the(α, d, β)-superprocess X has the following decomposition: Let ϕ∈D(_α),t0,

Xt(ϕ)=μ(ϕ)+ t 0

Xs(αϕ)ds−Cβ(K) t 0

Xs(ϕ)ds+ t 0

K 0

R^d

rϕ(x)NX(dx,dr,ds)

+ t 0

∞ K

R^d

rϕ(x)N_X(dx,dr,ds), (1.9)

whereN_X=N_X−N_Xis a martingale measure and C_β(K)= η

βK^β.

As we have mentioned already, one of the problems of working with the(α, d, β)-superprocessXis dealing with “big”

jumps. In fact, the “big” jumps produce the infinite variance of the process and they appear in the term corresponding

to the integral with respect toNX on (1.9). So, the first step in the establishing the existence of SILT for(α, d, β)- superprocessXis to “eliminate” those jumps. This is achieved via introducing the following auxiliary process.

Let us considerer the canonical space, Ω^◦=D([0,∞), M_F(R^d)), F^◦=B(Ω^◦)andFt^◦=σ{Y_r^K: 0rt}, whereY_r^K(ω^◦)=ω^◦(r). For any μ∈M_F(R^d)there exists (see [4]) a measureQ_μon(Ω^◦,F^◦), such that for any non-negative functionϕ∈Bb(R^d)

Eμ

exp

−Y_t^K(ϕ)Fs^◦

=exp

−Y_s^K

V_t^K_−s(ϕ)

, ∀0< st, (1.10)

and E_μ

exp

−Y_t^K(ϕ)

=exp

−μ

V_t^K(ϕ)

, ∀t >0 (1.11)

(notice that the expectation is taken here with respect to the measureQ_μ).V_t^Kis the unique non-negative solution for the non-linear equation

∂v^K_t /∂t=(α−Cβ(K))v_t^K−Φ^K(v_t^K),

v^K₀ =ϕ, (1.12)

where

Φ^K(x)=η K 0

e^−ux−1+ux

u^−β−2du. (1.13)

Note that whenK= ∞the resulting processY^∞and the regular(α, d, β)-superprocessXhave the same distribution.

Now, for anyK >0, define the stopping time τ_K=inf

t >0: |X_t|> K

. (1.14)

In Section 2 we will show that if we define the process which evolves as X up to timeτ_K and then continues to evolve asY^K starting atXτ_K−, then this process has the same law asY^K. This together with the fact thatτK↑ ∞as K→ ∞(see Lemma 2) implies that it is enough to show existence of the SILT for the processY^K. This task will be accomplished in Section 3, modulo some technical moment estimates that will be derived in Section 4. The main steps leading to the proof of Theorem 1 will be described in the next section.

2. Proof of Theorem 1

The processY^K whose Laplace transform is given by (1.10), (1.11) has the following decomposition:

Y_t^K(ϕ)=μ(ϕ)+ t

0

Y_s^K(_αϕ)ds−C_β(K) t

0

Y_s^K(ϕ)ds+ t 0

K 0

R^d

rϕ(x)N_YK(dx,dr,ds), ∀t0, (2.1)

whereN_YK=N_YK−N_YK, and N_YK(dx,dr,ds)=

{(x,r,s):Y_s^K=rδx}

δ_(x,r,s),

N_YK(dx,dr,ds)=η1(0,K](r)r^−2−βdrY_s^K(dx)ds.

Note thatN_YK is defined in a way analogous to (1.7), however it does not have jumps “greater” thanK.

In the following lemma we are going to construct the probability space where Y^K coincides with X up to the stopping timeτ_K.

Lemma 1.There exists a probability space on which a pair of processes(Y¨^K, X)is defined and possesses the following properties:

(a) Y¨^K coincides in law withY^K, (b) Y¨_t^K=X_t, ∀t < τ_K.

Proof. Define

Ω≡Ω×Ω^◦, F≡F×F^◦, Ft≡Ft×Ft^◦, and let

Y¨_t^K(w, w^◦)≡

Xt(w), t < τK(w), w^◦(t−τK(w)), tτK(w).

Define the measureP on(Ω,F):

P (B×C)=

Ω

1B(w)P^τK(w)(C)P(dw), where

P^τK(w)(C)=QX_{τK (ω)−}

w^◦∈Ω^◦: Y¨^K(w, w^◦)∈C

. (2.2)

Letϕ∈Dom(α)andt >0. From the definition ofY¨^Kwe have Y¨_t^K(ϕ)=μ(ϕ)+

t 0

Y¨_s^K(_αϕ)ds−C_β(K) t 0

Y¨_s^K(ϕ)ds+ t 0

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds), (2.3)

whereN_Y¨K is defined by (1.7) fort < τK and N_Y¨K(dx,dr,ds)=

{(x,r,s):Y¨_s^K=rδx}

δ_(x,r,s),

fortτ_K. Let us check that_t

0

_K

0

R^drϕ(x)N_Y¨K(dx,dr,ds)is anFt-martingale: For anyt > u,B∈Fu,C∈Fu^◦, we obtain by using the definition (2.2) ofP^τK(w)

P

1B×C

t 0

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds)− u 0

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds)

=

B

P^τK(w)

1C

t 0

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds)− u 0

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds)

P(dw)

=

B

P^τK(w)

1C

t 0

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds)−

(t∧τK(w))∨u 0

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds)

P(dw)

+

B

P^τK(w)

1C

(t∧τ_K(w))∨u 0

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds)− u 0

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds)

P(dw)

=

B

P^τK(w)

Q_X

τK (w)−

1C

t (t∧τK(w))∨u

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds)

F_t^◦_∧τK(w))∨u

P(dw)

+

B

P^τK(w)

1C

(t∧τ_K(w))∨u u

K−

0

R^d

rϕ(x)NX(dx,dr,ds)

P(dw)

=

B

P^τK(w)

1CQ_X

τK (w)−

t (t∧τK(w))∨u

K 0

R^d

rϕ(x)N_Y¨K(dx,dr,ds)

F_(t^◦_∧τK(w))∨u

P(dw)

+

B

P^τK(w)(C)

(t∧τK(w))∨u u

K−

0

R^d

rϕ(x)N_X(dx,dr,ds)P(dw)

=

B

1C

X_·∧u(ω)

1_{u<τ_K}P

(t∧τ_K(w))∨u u

K−

0

R^d

rϕ(x)NX(dx,dr,ds) Fu

P(dw),

where in the last equality for the first term we have used the fact that for P-a.s. ω, N_Y¨K is a (Q_X

τK (w)−,Ft)- martingale measure onR^d×R₊× [τ_K∧t, t]. As for the second term we have used the simple identity

P^τK(w)(C)1_{u<τ_K}=1C

X_·∧u(ω) 1_{u<τ_K}

for anyC∈F_u^◦. Now use the fact thatN_Xis a(P,F_t)-martingale measure to get that

P

(t∧τK(w))∨u u

K−

0

R^d

rϕ(x)N_X(dx,dr,ds) Fu^◦

=0,

and the proof thatt 0

K 0

R^drϕ(x)N_Y¨K(dx,dr,ds)is a martingale is complete.

Then, due to the uniqueness of the decomposition ([7], Theorem 7) we conclude thatY¨^Khas the same distribution asY^K. 2

Convention.Based on the above lemma, from now on we will assume thatY^K, Xare defined on the same probability space andY_t^K=X_t,∀t < τ_K.

Now we are going to show that timeτ_K can be made greater than any constantT with probability arbitrary close to 1 by takingKsufficiently large.

Lemma 2.For everyT >0andε >0, there existsK >0such thatP (τ_KT )ε.

Proof. LetZ_T^Kthe number of jumps of height greater thanKin[0, T]×R^d, that isZ_T^K=N ([K,+∞)×[0, T]×R^d).

Then there exists (see [13], page 1430) a standard Poisson processA^K_t such that Z^K_T =A^K

cβK^−1−βT

0 Xs(R^d)ds,

for some positive constantcβ. Then from the Markov inequality we have, P (τ_KT )=P

Z^K_T 1

=P

A^K

cβK^−1−βT

0 Xs(R^d)ds1

=P

A^K

c_βK−1−βT

0 X_s(R^d)ds1, c_β K^1+β

T 0

Xs

R^d

dsK⁻¹

+P

A^K

cβK^−1−βT

0 Xs(R^d)ds1, c_β K^1+β

T 0

X_s R^d

ds < K⁻¹

P

T 0

X_s R^d

c_β⁻¹K^β

+P

A^K_K−11

c_βK^−βE_μ T

0

X_s R^d

+ 1−P

A^K

K⁻¹=0

=c_βT μ R^d

K^−β+

1−exp

−K⁻¹ .

The result follows, since the right-hand side goes to 0 asK→ ∞. 2 Now the proof of Theorem 1 relies on the following proposition.

Proposition 1.LetK >0andY^K be the truncated(α, d, β)-superprocess with initial measureY₀^K(dx)=μ(dx)= h(x)dx, wherehis bounded and integrable with respect to Lebesgue measure dxonR^d.

(a) Ford/2< αthere exists a processγ^K

Y^K= {γ^K

Y^K(T ): T 0}such that limε↓0E

sup

t <T

γ_Y^KK,ε(t )−γ_Y^KK(t )=0, ∀T >0, and for anyλ >0,

γ_Y^K_K(T )=

λ−C_β(K)^T

0

t 0

R^2d

G^λ(x−y)Y_s^K(dx)Y_t^K(dy)dtds− T 0

R^d

Y_T^K

G^λ(x− ·) Y_s^K(dx)

+ T 0

R^d

Y_s^K

G^λ(x− ·)

Y_s^K(dx)ds+ T 0

R^d

t 0

R^d

G^λ(x−y)M^K(ds,dy)Y_t^K(dx)dt, a.s.

(b) Ford/(2+(1+β)⁻¹) < αd/2there exists a processγ˜^K

Y^K = { ˜γ^K

Y^K(T ): T 0}such that limε↓0E

sup

t <T

γ˜^K

Y^K,ε(t )− ˜γ^K

Y^K(t )=0, ∀T >0, and for anyλ >0,

˜

γ_Y^KK(T )=

λ−Cβ(K)^T

0

t 0

R^2d

G^λ(x−y)Y_s^K(dx)Y_t^K(dy)dtds− T 0

R^d

Y_T^K

G^λ(x− ·)

Y_s^K(dx)ds

+ T 0

R^d

t 0

R^d

G^λ(x−y)M^K(ds,dy)Y_t^K(dx)dt, a.s.

Proof is postponed.

The above proposition immediately yields:

Proof of Theorem 1. Fix arbitraryε, δ >0 and letd/2< α. SinceX_t=Y_t^Kfor anyt < τ_K, we immediately get that γX(t )=γ_YK(t ), ∀t < τK,

andγX(t )satisfies Tanaka formula (1.5) fort < τK. Moreover, since by Lemma 2,τK↑ ∞, asK→ ∞, there is no problem to defineγ_X(t )satisfying (1.5) for anyt >0.

Now let us check the convergence part of the theorem. For anyT >0, by Lemma 2, we can fixK >0 such that P (τ_KT )δ. Then

εlim₁↓0P

sup

tT

γX,ε₁(t )−γX(t )> ε lim

ε₁↓0P

sup

tT

γX,ε₁(t )−γX(t )> ε, τK> T

+P (τKT )

lim

ε₁↓0P

sup

tT

γ_YK,ε1(t )−γ_YK(t )> ε, τ_K> T +δ

=δ,

and sinceδ, ε >0 were arbitrary the proof of convergence is complete.

The proof of part (b) of the theorem goes along the same lines. 2 3. Existence of SILT forY^K — proof of Proposition 1

Fix arbitraryK >0. First, we derive very useful moment estimates forY^K. Let{S_t^K: t0}denote the solution of the partial differential equation

∂v_t^K

∂t =

_α−C_β(K) v_t^K.

That is,{S^K_t : t0}is the semigroup defined as

S_t^K=e^−Cβ(K)tS_t. (3.1)

Notice thatS_t^KϕStϕ, for all non-negative bounded measurable functionsϕ.

Following Theorem 3.1 of [9] we have that forϕ, ψ∈Bb(R^d), E

exp

−Y_t^K(ϕ)− t 0

Y_s^K(ψ )ds

Y₀^K=μ

=exp

−μ

V_t^K(ϕ, ψ )

, (3.2)

whereμ∈MF(R^d)andV_t^K(ϕ, ψ )denotes the unique non-negative solution to the following evolution equation v_t^K=S^K_t ϕ+

t 0

S_s^K(ψ )ds− t 0

S_t^K_−s Φ^K

v^K_s

ds, t0, (3.3)

where

Φ^K(x)=^∞

m=2

(−1)^m

m! χ (m)x^m (3.4)

and

χ (m)= ηK^m−1−β

m−1−β. (3.5)

Now we are going to calculate the first two moments ofY^K.

Lemma 3.Letϕbe a non-negative function onBb(R^d)andt >0. Then E_μ

Y_t^K(ϕ)

=μ S_t^Kϕ

, (3.6)

E_μ

Y_t^K(ϕ)2

= μ

S_t^Kϕ2

+χ (2)μ t

0

S_t^K_−s S_s^Kϕ2

ds

.

Proof. From (3.2) we have E_μ

e^−λYtK(ϕ)

=e^{−μ(vKt (λ))} (3.7)

where

v_t^K(λ)=λS_t^Kϕ− t 0

S_t^K_−s Φ^K

v^K_s (λ)

ds. (3.8)

Using the elementary inequality e^−x−1+xx²/2,x0, and (1.13) we have Φ^K(x)χ (2)

2 x², x0.

Let · _∞be the supremum norm, then 0v^K_t (λ)λS_t^Kϕλϕ_∞and the previous inequality implies t

0

S_t^K_−s Φ^K

v^K_s (λ)

dsχ (2)ϕ²_∞t 2 ×λ². Further from (3.8) we get

limλ↓0

λS_t^Kϕ−v_t^K(λ)

λ =lim

λ↓0

1 λ

t 0

S_t^K_−s Φ^K

v_s^K(λ) ds

lim

λ↓0

χ (2)ϕ²_∞t

2 ×λ=0, and we write this like

v_t^K(λ)=λS_t^Kϕ−o(λ). (3.9)

This implies Eμ

Y_t^K(ϕ)

=lim

λ↓0

1−E_μ[e^−λYtK(ϕ)] λ

=lim

λ↓0

1−e^{−λμ(SKt ϕ)+o(λ)} λ

=lim

λ↓0

1−e^{−λμ(SKt ϕ)+o(λ)} λμ(S_t^Kϕ)−o(λ) lim

λ↓0

λμ(S_t^Kϕ)−o(λ)

λ =μ

S_t^Kϕ .

Now, to calculate the second moment we follow the ideas used in the proof of Proposition 11 of Chapter II from [10]:

E_μ

Y_t^K(ϕ)2

=lim

λ↓0

2 λ²E_μ

e^−λYtK(ϕ)−1+λY_t^K(ϕ)

=lim

λ↓0

2 λ²

e^{−μ(vKt (λ))}−1+λμ S_t^Kϕ

=lim

λ↓0

2 λ²

e^{−λμ(StKϕ)+μ(0tSt−sK (ΦK(vKs (λ)))ds)}−1+λμ S_t^Kϕ

=lim

λ↓0

2 λ²

_∞

n=0

1 n!

μ

t 0

S_t^K_−s Φ^K

v_s^K(λ) ds

−λμ

S_t^Kϕn

−1+λμ S_t^Kϕ

.

Using the series expansion (3.4) forΦ^K and (3.9) we obtain t

0

S_t^K_−s Φ^K

v^K_s (λ) ds=

t 0

S_t^K_−s Φ^K

λS_s^Kϕ−o(λ) ds