Comparison with the related literature

IV.6. Comparison with the related literature

In this section we are going to compare our result with the existing literature. More precisely, the comparison will be made with Madan et al. [50] which is the most general result for discrete-time approximations of BSDEs. However, we need to underline that the work of Briand et al. [15] has greatly inspired the author. In the following we will translate the notation to the one we have used.

We start with the Condition(B1), i.e. the limit-filtration has to be quasi-left-continuous, which is common between the two works. Actually, we may recall that M´emin [53, Counter-example 3] which justifies the necessity of (B1). However, in our case we are not restricted in (square-integrable) L´evy martingale, but it can be an arbitrary (square-integrable) one.

We proceed to the conditions imposed on the sequence of driving martingales (ĎX^k,◦, X^k)_k∈N. In [50] the integrators can be multidimensional, while we have presented the real-valued case. However, in view of the results in Chapter III, we can readily adapt the dimension of the integrators to a higher one. The same holds true for the dimension of the solution of the BSDE, in view of the main theorem of Chapter II. In Chapter IV we preferred to present the result in an as simple as possible manner. In both cases, they are square-integrable martingales whose terminal values converge in mean to the respective terminal values of the natural pair of X^∞. Now we proceed to describe a few improvements that take place under our framework.

• The time horizon in our case can be infinite, while in [50] it is assumed finite.

• The terminal values in [50] need to converge in a stronger sense than inL²−mean, see [50, Conditions (2.2) and (2.5)], while in our case the convergence is in L²−mean. Actually, we have explained at the beginning of Subsection III.6.1 the role of Lemma III.39 in order to obtain this sharp convergence.

• In our framework we need M_μXk,

ΔX^k,◦ !P^Gk = 0 for every k ∈N, which is a weaker condition compared to independence assumption imposed in [50]. Let us argue about it. To this end, let us fix a finite time horizon T, a k ∈ N and a partition (t^m)m=0,...,M of [0, T]. We will assume that X^k,◦, X^k, have independent increments and are constant on the interval [t^m, t^m+1) form = 0, . . . , M−1. In this setting we have for every boundedG^k−predictable functionU

M_μXk,

U M_μXk,

ΔX^k,◦ !P^Gk =M_μXk,

UΔX^k,◦

=E^M−1

m=0

U(t^m,ΔX_t^k,m)ΔX_t^k,m^◦

= 0.

Finally, regarding the properties the generator should possess, we will restrict ourselves on mentioning that Madan et al. [50, Assumption 1] demands the generators to be uniformly Lipschitz. In Subsection IV.5.6 we have commented that the uniform Lipschitz condition translates the pointwise convergence into seem-ingly stronger kind of convergence. For this reason, we could say that we have used finally the usual convergence assumptions as they translate in a weaker framework.

APPENDIX A

Auxiliary results

A.1. Auxiliary results of Chapter II

Proof of Lemma II.13. Let (γ, δ)∈ Cβ.We shall begin by obtaining the critical points of the map Π^Φ.We have

∂

∂γΠ^Φ(γ, δ) = (2 + 9δ) e^(δ−γ)ΦΦγ²+ (2−δΦ)γ−δ γ²(δ−γ)² ,

∂

∂δΠ^Φ(γ, δ) =−9

δ² + e^(δ−γ)Φ

( 9 + (2 + 9δ)Φ (δ−γ)

γ(δ−γ)² − 2 + 9δ γ(δ−γ)²

) .

The only possible critical points for Π^Φare therefore such thatδ=−2/9 orγ= ^δΦ−2±

√4+δ²Φ²

2Φ . However, the values δ=−2/9 andγ= ^δΦ−2−

√4+δ²Φ²

2Φ are ruled out as negative. For 0< δ≤β we have δΦ−2 +√

4 +δ²Φ²

2Φ , δ

∈ Cβ. Let us define γ^Φ(δ) := ^δΦ−2+

√4+δ²Φ²

2Φ , for 0< δ≤β.It is easy to verify thatγ^Φ(δ)∈(0, δ).Then, some tedious calculations yield that

∂Π^Φ

∂δ γ¯^Φ(δ), δ

=−9

δ² −exp[ δ−¯γ^Φ(δ) Φ]

¯

γ^Φ(δ) δ−¯γ^Φ(δ)2 ·2¯γ^Φ(δ)Φ + 9¯γ^Φ(δ) + 2 (¯γ^Φ(δ)Φ + 1)² <0

therefore Π^Φdoes not admit any critical point onCβ, for which 0< γ < δ < β. Hence, the infimum on this set is necessarily attained on its boundary. The cases where at least one amongδandγgoes to 0, or where their difference goes to 0, lead to the value∞. The only remaining case is therefore 0< γ < δ=β, where β is fixed. Then we get

d

dγΠ^Φ γ, β

= (2 + 9β) e^(β−γ)ΦΦγ²+ (2−βΦ)γ−β γ²(β−γ)² ,

and Π^Φ(γ, β) viewed as a function ofγ attains its minimum at its critical point given by γ^Φ(β), since

dΠ^Φ dγ γ, β

<0 on (0, γ^Φ(β)) and ^dΠ_dγ^Φ γ, β

>0 on (γ^Φ(β), β).

Now, we proceed to the second case, and start by determining the critical points of Π^Φ_?. It holds

∂

∂γΠ^Φ_?(γ, δ) =−8

γ² + 9δe^(δ−γ)ΦΦγ²−(δΦ−2)γ−δ γ²(δ−γ)² ,

∂

∂δΠ^Φ_?(γ, δ) =−9

δ²+ 9e^(δ−γ)Φ(1 +δΦ)(δ−γ)−δ γ(δ−γ)² .

Following analogous computations as above we can prove that, for (γ, δ)∈ Cβ, the equation

∂

∂γΠ^Φ_?(γ, δ) = 0⇔Pδ(γ) := 8(δ−γ)²−9δe^(δ−γ)Φ Φγ²−(δΦ−2)γ−δ

= 0

has a unique root, sayγ^Φ_?(δ),which moreover satisfiesγ^Φ_?(δ)∈(γ^Φ(δ), δ).This can be proved because the functionP_δ: (0, δ)→Ris decreasing, for each fixedδ∈(0, β), withP_δ γ^Φ(δ)

>0 andP_δ(δ)<0. Now observe that for γ > _1+δΦ^δ2Φ it holds _∂δ^∂ Π^Φ_?(γ, δ)<0 and that P_{δ 1+δΦ}^δ2Φ

>0. Using the monotonicity of P_δ we have thatγ^Φ_?(δ)> _1+δΦ^δ2Φ and therefore also _∂δ^∂ Π^Φ_?(γ^Φ_?(δ), δ)<0.Arguing as above we can conclude that the infimum is attained forδ=β at the point γ^Φ_?(β), β

.

Finally, the limiting statements follow by straightforward but tedious computations.

133

A.1.1. The non-exponential submultiplicative case. In this subsection we provide thea priori estimates of the semi–martingale decomposition (II.20) in the case hζ :R→[1,∞), where hζ(x) = 1 for x < 0 and h_ζ(x) := (1 +x)^ζ, for x >0 where ζ is a fixed number greater than 1, with the additional assumption that the process A defined in (F4) is P−a.s. bounded by Ψ. However, using completely analogous arguments thea priori estimates can be obtained for a submultiplicative function h. In [64, Proposition 25.4] (see the proof of (iii)) we can find an easy criterion for an increasing functionh:R→R⁺ to be submultiplicative. Namely, it suffices for the function h: R → (0,∞) to be flat on (−∞, b] and the function log(h(·)) to be concave on (b,∞), for some b ∈ R+. It is immediate that the functionhζ

as defined above is submultiplicative. Since ζ is fixed, we will simply denotehζ as hin the rest of the section.

Before we proceed to the proof we provide the properties of the submultiplicative function we will need, presented however for the function hζ, as well as some notation. For We start with properties.

(i) h:R+→[1,∞) andhis non-decreasing.

(ii) his submultiplicative,i.e. there existsch>0 such that

h(x+y)≤chh(x)h(y), for allx, y∈R⁺. (iii) It holdsx≤h(x) for everyx∈R+.

(iv) The indefinite integral R

h(s)^−γdscan be explicitly calculated. We define forγ > ¹_ζ H_γ(x) :=

Z

(0,x]

h^−γ(s) ds= 1 1−γζ

(1 +x)^1−γζ−1

= 1

−γζ+ 1

(1 +x)^−γζ+1−1

= 1

−γζ+ 1

h(x)^−γζ+1_ζ

−1

= 1

−γζ+ 1

h(x)−γ+¹_ζ

−1

= 1

−γζ+ 1

h^ϑ(γ)(x)−1

, (A.1)

where ϑ(γ) :=−γ+¹_ζ <0.Moreover, Z

(t,u]

h^−γ(s)ds=Hγ(u)−Hγ(t) =|Hγ(t)| − |Hγ(u)|

= 1

| −γζ+ 1|[h^ϑ(γ)(t)−h^ϑ(γ)(u)] = 1

γζ−1[h^ϑ(γ)(t)−h^ϑ(γ)(u)]. (A.2) (v) The set ΓH :=

γ∈R+, Hγ :R+→(−∞,0] is increasing is non–empty. In view of the restriction in (iv), the set ΓH is indeed non-empty,i.e.

ΓH = (1

ζ,∞). (A.3)

(vi) There existsβ ∈R+such that E

Z

(0,T]

h^β(A_s)kf_sk²₂ α²_s dC_s

<∞.

By [64, Proposition 25.4] we can calculate the submultiplicative constant c_h= exp

ln h(2·0)

−ln h(0)

= 1.

Lemma A.1. Assume the framework of Lemma II.16 except for the definition of the β−norms, where we substitute the exponential function with the functionh, forζ≥1;e.g see(vi). Moreover, assume that A is bounded by the positive constant Ψ. Then, it holds for ¹_ζ < γ < δ <βˆ

kαyk²_H2 δ

+kηk²_H2 δ ≤

"

2h(Ψ)h^δ(Φ) + 9 +9[h(Ψ)]^1−1ζ[h(Φ)]^δ−1ζ δ−¹_ζ + 1

# kξk²_L2

δ

+

"

2[h(Ψ)]^1+ζ1[h(Φ)]^δ−γ+1ζ

δ−γ+_ζ¹+ 1 +9h(Ψ)[h(Φ)]^δ−γ δ−γ+ 1 + 9

δζ−1

# f α H²_δ

, and

kyk²_S2 δ +kηk²

H²δ ≤

8 + 2h(Ψ)h^δ(Φ) kξk²

H²δ

A.1. AUXILIARY RESULTS OF CHAPTER II 135

+

"

[h(Ψ)]^1ζ

γζ−1 +2[h(Ψ)]^1+1ζ[h(Φ)]^δ−γ+1ζ δ−γ+¹_ζ + 1

# f α

2

H²_δ

. Proof. Recall the identity

yt=ξ+ Z

(t,T]

fsdCs− Z

(t,T]

dηs=E

"

ξ+ Z

(t,T]

fsdCs

Gt

#

, (A.4)

and introduce the anticipating function

F_t= Z

(t,T]

f_sdC_s. Forγ∈ΓH, we have by the Cauchy-Schwarz inequality,

kFtk²₂ ≤ Z

(t,T]

h^−γ(A_s) dA_s Z

(t,T]

h^γ(A_s)kf_sk²₂ α²_s dC_s

= Z

(At,AT]

h^−γ(AL_s) ds Z

(t,T]

h^γ(As)kfsk²₂ α²_s dCs Lemma I.34

≤

(i)

Z

(A_t,A_T]

h^−γ(s) ds Z

(t,T]

h^γ(A_s)kf_sk²₂ α²_s dC_s

(A.2)

= 1

γζ−1[h^ϑ(γ)(At)−h^ϑ(γ)(AT)]

Z

(t,T]

h^γ(As)kfsk²₂ α²_s dCs

≤ 1

γζ−1h^ϑ(γ)(At) Z

(t,T]

h^γ(As)kfsk²₂

α²_s dCs. (A.5)

Forδ∈R+ and by integrating (A.5) w.r.t. h^δ(A_t) dA_tit follows Z

(0,T]

h^δ(At)kFtk²₂dAt (A.5)

≤ 1

γζ−1 Z

(0,T]

h^δ+ϑ(γ)(At) Z

(t,T]

h^γ(As)kfsk²₂

α²_s dCsdAt

= 1

γζ−1 Z

(0,T]

h^γ(As)kfsk²₂ α²_s

Z

(0,s)

h^δ+ϑ(γ)(At) dAtdCs

≤ 1

γζ−1 Z

(0,T]

h^γ(As)kfsk²₂ α²_s

Z

(0,s]

h^δ+ϑ(γ)(At) dAtdCs

≤ 1

γζ−1 Z

(0,T]

h^γ(As)kfsk²₂ α²_s

Z

(A0,As]

h^δ+ϑ(γ)(AL_t) dtdCs, (A.6) where in the equality we used Tonelli’s Theorem. By [64, Proposition 25.4] we have that h^δ+ϑ(γ) re-mains submultiplicative if δ+ϑ(γ) > 0, which is equivalent to δ > |ϑ(γ)|. Therefore, we can apply Lemma I.34.(vii) for the functiong(x) =h^δ+ϑ(γ)(x). Then, Inequality (A.6) becomes

Z

(0,T]

h^δ(A_t)kF_tk²₂dA_t^{Lemma I.34}≤ c^δ+ϑ(γ)_h h^δ+ϑ(γ)(Φ) Z

(0,T]

h^γ(A_s)kfsk²₂ α²_s

Z

(A₀,A_s]

h^δ+ϑ(γ)(t) dtdC_s

c_h=1

= h^δ+ϑ(γ)(Φ) δ+ϑ(γ) + 1

Z

(0,T]

h^γ(As)kfsk²₂

α²_s h^δ+ϑ(γ)+1(As)−h^δ+ϑ(γ)+1(A0) dCs A0=0

≤

h(A0)=1

h^δ+ϑ(γ)(Φ) δ+ϑ(γ) + 1

Z

(0,T]

h(A_s)δ+¹_ζ+1kf_sk²₂

α²_s dC_s, (A.7)

which is integrable ifδ+¹_ζ + 1≤β; see (vi). Define

%:=δ+1

ζ + 1 and Λ^γ,δ,Φ:= h^δ+ϑ(γ)(Φ) δ+ϑ(γ) + 1. To sum up, for

γ∈ΓH, δ > γ−1

ζ and 0< %≤β, (A.8)

we have

E hZ

(0,T]

h^δ(A_t)kF_tk²₂dA_ti

≤Λ^γ,δ,Φ f α

2 H²%

. (A.9)

For the estimate of kαyk_H2

δ we first use the fact that kαyk²

H²_δ =E hZ

(0,T]

h^δ(A_t)ky_tk²₂dA_ti

≤2E hZ

(0,T]

E

h^δ(At)kξk²₂+h^δ(At)kFtk²₂ Gt

dAt

i

= 2E hZ

(0,∞)

E

h^δ(A_t)kξk²₂+h^δ(A_t)kFtk²₂ Gt

dA^T_ti

=2E hZ

(0,∞)

h^δ(At)kξk²₂+h^δ(At)kFtk²₂dA^T_ti

= 2E hZ

(0,T]

h^δ(At)kξk²₂dAt

i + 2E

hZ

(0,T]

h^δ(At)kFtk²₂dAt

i

= 2E hZ

(A0,AT]

h^δ(AL_s)kξk²₂dsi + 2E

hZ

(0,T]

h^δ(At)kFtk²₂dAt

i

= 2c^δ_hh^δ(Φ)E hkξk²₂

Z

(A₀,A_T]

h^δ(s) dsi + 2E

hZ

(0,T]

h^δ(At)kFtk²₂dAt

i

=

ch=1







2h^δ(Φ)E

hkξk²₂h^δ(λ)(AT −A0)i + 2E

hR

(0,T]h^δ(At)kFtk²₂dAt

i

, forλ∈[A0, AT] or

2

δζ+1h^δ(Φ)E

hkξk²₂ h^δ+1ζ(AT)−h^δ+1ζ(A0)i + 2E

hR

(0,T]h^δ(At)kFtk²₂dAt

i

(A.10)

(i),(iii)

≤

A0=0,(A.9)











2h^δ(Φ)kξk²

L²_δ+1+ 2Λ^γ,δ,Φ,ch

f α

2 H²%

.

2

δζ+1h^δ(Φ)kξk²

L²_{δ+ 1}

ζ

+ 2Λ^γ,δ,Φ,ch

f α

2 H²%

.

(A.11)

In the second equality we have used that the processes kξk²₂1Ω×[0,∞](·) and |F(·)|² are uniformly in-tegrable, hence their optional projections are well defined. Indeed, using (A.5) and recalling that E[kF0k²₂]<∞, we can conclude the uniform integrability of |F(·)|². Then, it holds that

Π^G_o h^δ(A_·)kξk²₂+h^δ(A_·)|F(·)|²

t=h^δ(A_t)Π^G_o kξk²₂1Ω×[0,∞](·)

t+h^δ(A_t)Π^G_o |F(·)|²

t

=h^δ(At)E kξk²₂

Gt

+h^δ(At)E kFtk²₂

Gt

=E

h^δ(At)kξk²₂+h^δ(At)kFtk²₂ Gt

, which justifies the use of Corollary I.23.

For the estimate of kyk_S2

δ we have kyk_S2

δ =E h

sup

0≤t≤T

h^δ2(At)kytk2²i

≤E h

sup

0≤t≤T

h^δ2(At)E

kξk2+kFtk2

Gt²i

=E h

sup

0≤t≤TE

h^δ2(At)kξk2+h^δ2(At)kFtk2

Gt²i

≤2E

"

sup

0≤t≤TE q

h^δ(At)kξk²₂+h^δ(At)kFtk²₂

Gt

²#

(A.5)

≤ 2E



 sup

0≤t≤TE

" s

h^δ(A_T)kξk²₂+ 1

γζ−1h^δ+ϑ(γ)(A_t) Z

(t,T]

h^γ(A_s)kf_sk²₂ α²_s dC_s

Gt

#²



≤ 2E



 sup

0≤t≤TE

" s

h^δ(A_T)kξk²₂+ 1 γζ−1

Z

(t,T]

[h(A_s)]^γ+δ+ϑ(γ)kfsk²₂ α²_s dC_s

G_t

#²



≤ 2E



 sup

0≤t≤TE

" s

h^δ(AT)kξk²₂+ 1 γζ−1

Z

(0,T]

[h(As)]^δ+1ζkfsk²₂ α²_s dCs

Gt

#²



A.1. AUXILIARY RESULTS OF CHAPTER II 137

≤ 8E

"

h^δ(AT)kξk²₂+ 1 γζ−1

Z

(0,T]

[h(As)]^δ+1ζkfsk²₂ α²_s dCs

#

= 8kξk²_H2 δ

+ 1

γζ−1 f α

2

H²_{δ+ 1}

ζ

. (A.12)

where in the second and third inequalities we used the inequalitya+b≤p

2(a²+b²) and (A.5) respec-tively.

What remains is to controlkηk_H2

δ. We remind once more the reader that Z T

t

dηs=ξ−yt+Ft, hence

E

kξ−yt−Ftk²₂ Gt

=E

"

Z

(t,T]

dTr[hηis]

Gt

#

. (A.13)

In addition, we have

h^δ(As) = (1 +As)^δζ =δζ Z

(A0,As]

(1 +x)^δζ−1dx+ 1 =δζ Z

(A0,As]

h^δ−1ζ(x)dx+ 1 (A.14)

Z

(0,T]

h^δ(As) dTr[hηis]

(A.14)

≤ δζ Z

(0,T]

Z

(A₀,A_s]

h^δ−1ζ(t) dtdTr[hηis] + Tr[hηiT]

Lemma I.34

≤

(i)

δζ Z

(0,T]

Z

(A0,As]

h^δ−1ζ(AL_t) dt dTr[hηis] + Tr[hηiT]

= δζ

Z

(0,T]

Z

(0,s]

h^δ−1ζ(At) dAtdTr[hηis] + Tr[hηiT]

≤ δζ Z

(0,T]

h^δ−1ζ(A_t) Z

(t,T]

dTr[hηi_s]dA_t+ Tr[hηi_T], so that

kηk_H2 δ ≤δζE

"

Z

(0,T]

h^δ−1ζ(At) Z

(t,T]

dTr[hηis] dAt

#

+E[Tr[hηiT]]. (A.15) We now need an estimate for E

hR

(0,T]dTr[hηis]i

, which is given by E[Tr[hηiT]] = E

|ξ−y0+F(0)|²

≤3E

kξk²₂+ky0k²₂+kF0k²₂

(A.4)

≤ 9E kξk²₂

+ 9E

kF0k²₂^(A.5)

≤

(i)

9kξk_L2

δ+ 9 1 δζ−1

f α

H²_δ

forδ∈ΓH (A.5)

≤ 9kξk_L2

δ+ 9

δζ−1 f α

H²_δ

, (A.16)

where we used the fact thatE ky0k²₂

≤2E

kξk²₂+kF0k²₂ .

For the first summand on the right-hand-side of (A.15), we have E

"

Z

(0,T]

h^δ−1ζ(At) Z

(t,T]

dTr[hηis] dAt

#

Lemma I.23

= E

"

Z

(0,T]

h^δ−ζ1(At)E

"

Z

(t,T]

dTr[hηis]

Gt

# dAt

#

(A.13)

= E

"

Z

(0,T]

h^δ−1ζ(At)E

|ξ−yt+Ft|² Gt

dAt

#

≤ 3E

"

Z

(0,T]

h^δ−1ζ(At)E

kξk²₂+kytk²₂+kFtk²₂ Gt

dAt

#

(A.4)

≤ 3E

"

Z

(0,T]

h^δ−1ζ(A_t)kξk²₂dA_t

# + 3E

"

Z

(0,T]

h^δ−1ζ(A_t)kFtk²₂dA_t

#

+ 6E

"

Z

(0,T]

h^δ−ζ1(At)E

kξk²₂+kFtk²₂ Gt

dAt

#

Lem. I.23

= 9E

"

Z

(0,T]

h^δ−1ζ(At)kξk²₂dAt

# + 9E

"

Z

(0,T]

h^δ−1ζ(At)kFtk²₂dAt

#

(γ∈Γ_H)

= 9E

"

Z

(A₀,A_T]

h^δ−1ζ(AL_t)kξk²₂dt

# + 9E

"

Z

(0,T]

h^δ−1ζ(At)kFtk²₂dAt

#

Cor. I.36

≤

(A.9)

9h^δ−1ζ(Φ)E

"

kξk²₂ Z

(A₀,A_T]

h^δ−1ζ(t) dt

#

+ 9Λ^{γ,δ−1ζ,Φ} f α H²δ+1

forδ+ 1≤β

≤ 9h^δ−1ζ(Φ) δ−¹_ζ + 1E

hkξk²₂

h^δ−1ζ+1(AT)−1 dti

+ 9Λ^{γ,δ−1ζ,Φ} f α

H²_δ+1

≤ 9h^δ−1ζ(Φ) δ−¹_ζ + 1 kξk_δ−1

ζ+1+ 9Λ^{γ,δ−1ζ,Φ} f α

H²_δ+1

. (A.17)

In total, Inequality (A.15) can be written kηk_H²

δ ≤ δζE

"

Z T 0

h^δ−1ζ(A_t) Z

(t,T]

dTr[hηis] dA_t

#

+E[Tr[hηiT]]

(A.16)

≤

(A.17)

9h^δ−1ζ(Φ) δ−¹_ζ + 1 kξk_δ−1

ζ+1+ 9Λ^{γ,δ−ζ1,Φ} f α

H²δ+1

+ 9kξk_L²

δ+ 9

δζ−1 f α

H²δ

(A.18) In order to be able to construct a contraction, we need to have on the right hand side of (A.11), resp.

(A.12), and (A.18) only k^f_αk_H2

δ. One way to overcome this problem is to assume that A is P−a.s.

bounded, say by Ψ.Then, the Inequality (A.11) can be written as

kαyk²

H²_δ≤











2h^δ(Φ)kξk²

L²_δ+1+ 2Λ^γ,δ,Φ,ch

f α

2 H²%

2

δζ+1h^δ(Φ)kξk²

L²_{δ+ 1}

ζ

+ 2Λ^γ,δ,Φ,ch

f α

2 H²%

≤











2h(Ψ)h^δ(Φ)kξk²

L²_δ+^2[h(Ψ)]

1+ 1ζ[h(Φ)]^δ+ϑ(γ) δ+ϑ(γ)+1

f α

H²_δ

,

2

δζ+1h^ζ1(Ψ)h^δ(Φ)kξk²

L²δ

+^2[h(Ψ)]

1+ 1ζ[h(Φ)]^δ+ϑ(γ) δ+ϑ(γ)+1

f α

H²δ

.

(A.19)

the Inequality (A.12) can be written as kyk_S2

δ ≤8kξk²

H²δ+ 1 γζ−1

f α

2

H²_{δ+ 1}

ζ

≤8kξk²

H²δ+[h(Ψ)]^1ζ γζ−1

f α

2

H²_δ

(A.20) and the Inequality (A.18) can be written as

kηk_H2

δ ≤ 9 + 9[h(Ψ)]^1−ζ1[h(Φ)]^δ−1ζ δ−¹_ζ + 1

! kξk_δ+

9h(Ψ)Λ^{γ,δ−1ζ,Φ}+ 9 δζ−1

f α

2 H²_δ

= 9 + 9[h(Ψ)]^1−ζ1[h(Φ)]^δ−1ζ δ−¹_ζ + 1

! kξk_δ+

9h(Ψ)[h(Φ)]^δ−γ δ−γ+ 1 + 9

δζ−1

f α

2 H²_δ

. (A.21)

A.2. Auxiliary results of Chapter III

Lemma A.2. Let(L^k)k∈Nbe a sequence ofR^p−valued processes and(N^k)k∈Nbe a sequence ofR^q−valued processes such that

(i) Tr

[L^k]_∞

k∈Nis uniformly integrable and (ii) Tr

[N^k]_∞

k∈N is bounded inL¹(G;R),i.e. sup

k∈N

E Tr

[N^k]_∞

<∞.

A.2. AUXILIARY RESULTS OF CHAPTER III 139

Then

Var [L^k, N^k]

∞

₁

k∈N is uniformly integrable.

Proof. By Corollary I.29.(iv), it is sufficient to prove that the sequence Var [L^k,i, N^k,j]

∞

k∈N

is uniformly integrable, for every i = 1, . . . , p and j = 1, . . . , q. We will use Theorem I.25 in order to prove the required property. Let i, j be arbitrary but fixed. The L¹–boundedness of the sequence (Var([L^k,i, N^k,j])_∞)_k∈N is obtained by means of the Kunita–Watanabe inequality in the form I.26 and the L¹−boundedness of the sequences [L^k,i]_∞

k∈N and [N^k,j]_∞

k∈N; the former is L¹−bounded as uniformly integrable. By the Kunita–Watanabe inequality again, but now in the form (I.24), and the Cauchy–Schwarz inequality, we obtain for anyA∈ G

Z

A

Var [L^k,i, N^k,j]

∞dP

K-W ineq.

≤ Z

A

[L^k,i]

1

∞2[N^k,j]

1

∞2 dP

C-S ineq.

≤ Z

A

[L^k,i]∞ dP ¹₂Z

A

[N^k,j]∞dP ¹₂

≤ Z

A

[L^k,i]∞ dP ¹₂

sup

k∈N

Z

Ω

[N^k,j]∞ dP ¹₂

≤ CZ

A

[L^k,i]_∞ dP ¹₂

, (A.22)

where C² := sup

k∈N E hTr

[N^k]_∞i

. Note that for the third inequality we used that [N^k,j]_∞ ≥ 0 for all k∈N.

Now we use [35, Theorem 1.9] for the uniformly integrable sequence [L^k,i]∞

k∈N. For everyε >0, there existsδ >0 such that, wheneverP(A)< δ, it holds

sup

k∈N

Z

A

[L^k,i]_∞ dP< ε²

C² ⇐⇒sup

k∈N

Z

A

[L^k,i]_∞ dP 1/2

< ε C. This further implies sup

k∈N

E

1AVar [L^k,i, N^k,j]_∞^(A.22)

< ε, which is the required condition.

The proof for the predictable quadratic variation is completely analogous. We state the result, however, for our convenience.

Lemma A.3. Let(L^k)_k∈Nbe a sequence ofR^p−valued processes and(N^k)_k∈Nbe a sequence ofR^q−valued processes such that

(i) hL^kiandhN^kiare well-defined for every k∈N, (ii) Tr

hL^ki_∞

k∈Nis uniformly integrable and (iii) Tr

hN^ki_∞

k∈N is bounded inL¹(G;R),i.e. sup

k∈N

E Tr

[N^k]_∞

<∞.

Then

Var hL^k, N^ki

∞

₁

k∈N is uniformly integrable.

In the following we provide some results which can be seen as simple exercises in measure theory.

Lemma A.4. LetD be a countable and dense subset ofR.Then every openU ⊂Ris the countable union of intervals with endpoints in D, i.e. there exists a sequence of intervals (ak, bk)

k∈N with ak, bk ∈D, for every k∈N,such that U =∪_k∈N(ak, bk).

Proof. LetUbe an open subset ofR.For everyx∈Uthere exists anrx>0 such thatB(x, rx)⊂U, whereB(y, r) ={z∈R,|y−z|< r}.Due to the density ofDwe can chooseax, bx∈D such thatax< bx

and (ax, bx) ⊂ B(x, rx) ⊂U. Then clearly ∪_x∈U(ax, bx) ⊂ U. On the other hand, D is countable and thereforeD×Dis also countable. Moreover, we can naturally associate to each ordered pair{{a},{a, b}}

the interval (a, b) ifa < bor the empty set ifa≥b, which proves that there are at most countably many

intervals with endpoints inD.

Corollary A.5. Let U ⊂R\ {0} be open. Then for everyi= 1, . . . , `there exists a countable subfamily of I(X^∞,i), which has been introduced in Section III.4, name(U^i,k)k∈N,such that U =∪k∈NU^i,k.

Proof. LetU be an open subset ofR\ {0} andi∈ {1, . . . , `}. Then, there exist open setsU⁺, U⁻ such thatU⁺⊂(0,∞),U⁻⊂(−∞,0) andU =U⁺∪U⁻.On the other hand, by [41, Lemma VI.3.12] we can conclude that the setI(X^∞,i) is at most countable. Therefore, the complement ofI(X^∞,i), denote it byI(X^∞,i)^c, is dense inR.Indeed, if I(X^∞,i)^c was not dense, there would exist an open subsetV of Rsuch thatI(X^∞,i)^c∩V =∅ or equivalentlyV ⊂ I(X^∞,i). However this is a contradiction, sinceV is uncountable and I(X^∞,i) is countable.

The above allow us to apply Lemma A.4 for D =I(X^∞,i) in order to find two countable families (U^+,i,k)_k∈N and (U^−,i,k)_k∈Nsuch that

U⁺= [

k∈N

U^+,i,kandU⁻= [

k∈N

U^−,i,k.

The required countable family is (U^+,i,k)k∈N∪(U^−,i,k)k∈N. Lemma A.6. It holdsσ J(X^∞)

=B(R^`), whereJ(X^∞)has been introduced in Subsection III.4 and σ J(X^∞)

denotes the σ−algebra inR^` generated by the familyJ(X^∞).

Proof. The spaceR^{`</sup>is a finite product of the second countable metric spaceR. Therefore, it holds B(R<sup>`}) =B(R)⊗ · · · ⊗ B(R)

| {z }

`−times

and to this end, it is sufficient to prove thatσ(I(X^∞,i)) =B(R) for everyi= 1, . . . , `.

By Corollary A.5 we have

σ(I(X^∞,i)) =σ({U ⊂R\ {0}, U is open}) =σ({U ⊂R\ {0}, U is open} ∪ {R} ∪ {0})

=σ({U ⊂R, U is open}) =B(R),

which allows us to conclude.

Lemma A.7. Let (Σ,S)be a measurable space and %1, %2 be two finite signed measures on this space.

Let Abe a family of sets with the following properties:

(i) Ais aπ−system,i.e. if A, B∈ A, thenA∩B∈ A.

(ii) It holdsσ(A) =S

(iii) %1(A) =%2(A) for everyA∈ A.

Then it holds%₁=%₂.

Proof. Let %⁺_i , %⁻_i be the (positive) measures obtained by the Jordan decomposition of %i, for i= 1,2.By (iii) we obtain

%⁺₁(A)−%⁻₁(A) =%⁺₂(A)−%⁻₂(A), for everyA∈ A.

However, the above equality can be translated into equality of measures, i.e.

%⁺₁(A) +%⁻₂(A) =%⁺₂(A) +%⁻₁(A), for everyA∈ A.

The class

C={A∈ S, %⁺₁(A) +%⁻₂(A) =%⁺₂(A) +%⁻₁(A)}

can be proven to be aλ−system.¹ Now we can conclude that the (positive) measures%⁺₁+%⁻₂ and%⁺₂+%⁻₁ coincide on D(A)⊂ C, where we have denoted by D(A) the λ−system generated by A.Using now the property (i) and theπ−λLemma,²see [4, Lemma 4.11], we can conclude that the (positive) measures

%⁺₁ +%⁻₂ and%⁺₂ +%⁻₁ coincide onσ(A) =S.

Let, now, P1, N1, resp. P2, N2, theHahn decomposition of the signed measure %1, resp. %2, where with Pi we denote the set of positive mass of %i and with Ni we denote the set of negative mass of %i, fori= 1,2. In view of the %⁺_i ⊥%⁻_i , fori= 1,2 and of the following equalities

Ω = (P1∩P2)∪(P1∩N2)∪(N1∩P2)∪(N1∩N2)

%⁺₁(P1∩P2) =%⁺₁(P1∩P2) +%⁻₂(P1∩P2) =%⁺₂(P1∩P2) +%⁻₁(P1∩P2) =%⁺₂(P1∩P2)

%⁺₁(P1∩N2) +%⁻₂(P1∩N2) =%⁺₂(P1∩N2) +%⁻₁(P1∩N2) = 0 0 =%⁺₁(N1∩P2) +%⁻₂(N1∩P2) =%⁺₂(N1∩P2) +%⁻₁(N1∩P2)

%⁻₂(N1∩N2) =%⁺₁(N1∩N2) +%⁻₂(N1∩N2) =%⁺₂(N1∩N2) +%⁻₁(N1∩N2) =%⁻₁(N1∩N2) we can conclude thatP₁∩P₂,N₁∩N₂is a common Hahn decomposition of the signed measures%₁ and

%2. Therefore

1Aλ−system is also referred to as Dynkin system or d−system or Sierpi´nski class. In [4, Foonote 4, p. 135] the interested reader can find references for this ambiguity in terminology. In [10] is used the termσ−additive class.

2Also known asDynkin’s Lemma.

A.3. AUXILIARY RESULTS OF CHAPTER IV 141

• For every A ∈ {B ⊂(P1∩P2), B ∈ S} we obtain %⁺₁(A) = %⁺₂(A), hence we can conclude that

%⁺₁ =%⁺₂.

• For every A∈ {B ⊂(N1∩N2), B ∈ S} we obtain %⁻₁(A) =%⁻₂(A), hence we can conclude that

%⁻₁ =%⁻₂.

A.3. Auxiliary results of Chapter IV

A.3.1. Moore–Osgood Theorem. The Moore–Osgood Theorem, whose well-known form is The-orem A.8, provides sufficient conditions for the existence of the limit of a doubly-indexed sequence and it can be seen as a special case of Rudin [63, Theorem 7.11]. Here we provide a second form, since the second time³we need to apply the aforementioned theorem we need to relax the existence of the pointwise limits; see Condition (ii) of Theorem A.8. For more comments on the existence of the iterated limits and of the joint limit of a doubly-indexed sequence the interested reader should consult Hobson [37, Chapter VI, Sections 336-338]. Specifically for the validity of the Theorem A.9 see [37, Chapter VI, Section 337, p. 466] and the reference therein.

Theorem A.8. Let (Γ, d_Γ) be a metric space and(γ_k,p)_k,p∈

Nbe a sequence. If (i) lim

p→∞sup

k∈N

dΓ(γk,p, γ_k,∞) = 0 and (ii) lim

k→∞d_Γ(γ_k,p, γ_∞,p) = 0 for allp∈N, then the joint limit lim

k,p→∞γn,k exists. In particular holds lim

k,p→∞γk,p= lim

p→∞γ_∞,p= lim

k→∞γ_k,∞. Theorem A.9. Let (R,| · |)and(ϑk,p)_k,p∈

N be a sequence. If (i) lim

p→∞sup

k∈N

|ϑk,p−ϑ_k,∞|= 0and (ii) lim

p→∞ lim sup

k→∞

ϑk,p−lim inf

k→∞ ϑk,p

= 0, then the joint limit lim

k,p→∞ϑk,p exists.

A.3.2. Weak convergence of measures on the positive real line.

Definition A.10. Let (µ_k)_k∈

Nbe a countable family of measures on R+,B(R+)

. We will say that the sequence (µk)_k∈Nconverges weakly to the measureµ_∞if for everyf : R+,k · k_∞

→ R,| · |

continuous and bounded holds

k→∞lim Z

[0,∞)

f(x)µ_k(dx)− Z

[0,∞)

f(x)µ_∞(dx)

= 0.

We denote the weak convergence of (µk)k∈Ntoµ∞ byµk

−−w→µ∞. In this section we will use the set

W_0,∞⁺ :=

C∈D(R)

C is increasing, withC0= 0 and lim

t→∞Ct∈R .

Remark A.11. We can extend every element of W_0,∞⁺ , name C its arbitrary element, on [0,∞] such that it is left-continuous at the symbol ∞by definingC_∞:= lim_t→∞Ct.

We provide in the following proposition some convenient equivalence for the weak convergence of finite measures. The statement is tailor-made to our later needs, but the interested reader may consult Bogachev [10, Section 8.1 - Section 8.3]. Then we provide in Theorem A.14 a new, to the best knowledge of the author, characterisation of weak convergence of finite measures to an atomless measure defined on the positive real line, which uses relatively compact sets of the Skorokhod space instead of relatively compact sets of the space of continuous functions defined onR+ endowed with the k · k_∞−topology.

Proposition A.12. Let (µk)_k∈

N be sequence of finite measures on R+,B(R+)

, and denote the as-sociated distribution functions by C^k, for k ∈ N, for which the finiteness of the measure translates to C_∞^k := lim_t→∞C_t^k∈R⁺ for every k∈N. We assume the following

(i) The sequence(µ_k)_k∈Nis bounded, i.e. sup_k∈Nµ_k(R+)<∞. Equivalently,sup_k∈NC_∞^k <∞.

3The first one is in the proof of Theorem IV.10, while the second is in the proof of Proposition A.16

(ii) The sequence (µk)_k∈N is tight. In other words, for every ε >0 there exists a compact setI such that sup_k∈Nµk(R+\I)< ε.⁴ Equivalently,sup_k∈N|C_∞^k −C_N^k|< εfor someN ∈R+.

(iii) µk({0}) = 0 for everyk∈N. Equivalently,C^k ∈ W_0,∞⁺ for everyk∈N.

(iv) µ_∞ is atomless,i.e. µ_∞({t}) = 0for every t∈R+. Equivalently, C^∞ is continuous.

The following are equivalent:

(I) µ_k−−→^w µ.

(II) C_t^k−−−−→

k→∞ C_t^∞, for everyt∈R+. (III) C^k−−−−→^J1(R) C^∞.

(IV) C^k−−→^lu C^∞. (V) sup

I∈I_N

|µk(I)−µ_∞(I)| −−−−→

k→∞ 0 for everyN ∈N, whereIN :=

I⊂[0, N] | I is interval . Proof. The equivalence between (I) and (II) is a classical result,e.g. see Bogachev [10, Proposition 8.1.8]. The equivalences between (II),(III) and (IV) are provided by Jacod and Shiryaev [41, Theorem VI.2.15.c.(i), Proposition VI.1.17 b)]. We obtain the equivalence between (IV) and (V) in view of the validity of the following inequalities for every N∈N:

sup

t∈[0,N]

|C_t^k−C_t^∞| ≤ sup

I∈IN

|µ_k(I)−µ_∞(I)| (becauseC₀^k=C₀^∞= 0)

≤ sup

s,t∈[0,N]

|C_t−^k −C_s^k−C_t^∞+C_s^∞|+ sup

s,t∈[0,N]

|C_t^k−C_s^k−C_t^∞+C_s^∞| + sup

s,t∈[0,N]

|C_t−^k −C_s−^k −C_t−^∞+C_s−^∞|

≤3 sup

t∈[0,N]

|C_t^k−C_t^∞|+ 2 sup

t∈[0,N]

|C_t−^k −C_t−^∞|+ sup

t∈[0,N]

|C_t−^k −C_t^∞|

≤5 sup

t∈[0,N]

|C_t^k−C_t^∞|+ sup

t∈[0,N]

|C_t−^k −C_t^k|+ sup

t∈[0,N]

|C_t^k−C_t^∞| (C^kis c`adl`ag for everyk∈N)

≤6 sup

t∈[0,N]

|C_t^k−C_t^∞|+ sup

t∈[0,N]

|C_t−^k −C_t^k|.

Then Jacod and Shiryaev [41, Lemma VI.2.5] implies that lim sup

k→∞

sup

I∈IN

|µk(I)−µ∞(I)| ≤6 lim

k→∞ sup

t∈[0,N]

|C_t^k−C_t^∞|+ lim sup

k→∞

sup

t∈[0,N]

|C_t−^k −C_t^k|

≤ sup

t∈[0,N]

|C_t−^∞−C_t^∞|= 0.

The following lemma provides a useful criterion to conclude the equivalence betweenδ_k·k∞-convergence andδJ₁(R+)-convergence inD [0,∞]

. Lemma A.13. Let (C^k)_k∈

N⊂ W_0,∞⁺ which satisfies the following properties:

(i) C^k−−−−→^J1(R)

k→∞ C^∞. (ii) C_∞^k −−−−^|·|→

k→∞ C_∞^∞.

(iii) The limit functionC^∞ is continuous.

Then it holdsC^k −−−−−→^k·k∞

k→∞ C.

Proof. Let us fix an arbitraryε >0.We start by exploiting the fact thatC^∞∈ W_0,∞⁺ . Then, there exists K >0 such that

sup

t≥K

(C_∞^∞−C_t^∞)< ε

4. (A.23)

By (ii), there existsk₁=k₁(ε)∈Nsuch that sup

k≥k1

|C_∞^k −C_∞^∞|< ε

4. (A.24)

4Without loss of generality we can assume thatIis of the form [0, N], for someN >0.

A.3. AUXILIARY RESULTS OF CHAPTER IV 143

By (iii) there exists ¯s > K such thatC¯s= ¹₂(CK+C_∞^∞). By (i), (iii) and Proposition I.113, there exists k2=k2(ε,¯s) such that

sup

k≥k2

sup

0≤t≤¯s

|C_t^k−C_t^∞|<1

4(C_∞^∞−CK)< ε

16, (A.25)

where the last inequality is valid in view of (A.23), since ¯s > K.Using the fact that the functions are increasing we derive

sup

k≥k1∨k2

sup

¯s≤t<∞

C_∞^k −C_t^k

≤ sup

k≥k1∨k2

C_∞^k −C_¯s^k

≤ sup

k≥k₁∨k₂

C_∞^k −C_∞^∞

+ sup

k≥k₁∨k₂

C_∞^∞−C_¯s^∞

+ sup

k≥k₁∨k₂

C_¯s^∞−C_¯s^k

(A.24)

<

(A.25)

ε 4 +ε

4+ ε 16 < ε

2. (A.26)

Therefore, by combining the above, we obtain for everyk≥k1∨k2

sup

t∈R+

|C_t^k−C_t^∞| ≤ sup

0≤t≤¯s

|C_t^k−C_t^∞|+ sup

¯s≤t≤∞

|C_t^k−C_t^∞|^(A.25)<

(A.26)

< ε.

Recall that the Kolmogorov metric δ_Kolm(C¹, C²) := kC¹−C²k∞, for C¹, C² ∈ W_0,∞⁺ , dominates the L´evy metric, which metrises the weak topology on the space of finite measures. Hence the previous lemma can be seen as a criterion for the weak convergence µ_Ck

−−w→µ_C^∞, where µ_Ck, for k∈ Nis the finite measures associated to the increasing functionC^k∈ W_0,∞⁺ .

The next step is to provide a characterisation of the aforementioned weak convergence of measures using relatively compact sets of D. Since, however, a relatively compact set of D is, in general, only locally uniformly bounded, see Theorem I.111, we have to restrict ourselves to those relatively compact subsets which are uniformly bounded, so that the integrals make sense. The following theorem can be regarded as an extension of Parthasarathy [56, Theorem 6.8] in the special case that the limiting measure is an atomless measure onR+. Forα∈D(R^p) the modulusw⁰_N has been defined in Definition I.108.

Theorem A.14. Let the measurable space R+,B(R+)

, (µk)_k∈

N be a sequence of finite measures such that µ_∞ is an atomless finite measure andA ⊂D(R). Assume that the following conditions are true

(i) µ_k−−→^w µ_∞.

(ii) Ais uniformly bounded, i.e. sup_α∈Akαk_∞<∞and (iii) Ais relatively compact. In other words, lim

ζ↓0sup

α∈A

w_N⁰ (α, ζ) = 0 for everyN ∈N. Then,

k→∞lim sup

α∈A

Z

[0,∞)

α(x)µk(dx)− Z

[0,∞)

α(x)µ∞(dx)

= 0. (A.27)

Proof. We will decompose initially the quantity whose convergence we intent to prove as follows sup

α∈A

Z

[0,∞)

α(x)µk(dx)− Z

[0,∞)

α(x)µ∞(dx)

≤sup

α∈A

Z

[0,N]

α(x)µk(dx)− Z

[0,N]

α(x)µ∞(dx) + sup

α∈A

Z

[N,∞)

α(x)µ_k(dx)

+ sup

α∈A

Z

[N,∞)

α(x)µ_∞(dx)

(A.29)

for someN >0 to be determined and then the first summand of the right-hand side will be decomposed as follows

sup

α∈A

Z

[0,N]

α(x)µk(dx)− Z

[0,N]

α(x)µ_∞(dx)

≤sup

α∈A

Z

[0,N]

α(x)µk(dx)− Z

[0,N]

α(x)µe^α_k(dx) + sup

α∈A

Z

[0,N]

α(x)eµ^α_k(dx)− Z

[0,N]

α(x)eµ^α_∞(dx) + sup

α∈A

Z

[0,N]

α(x)µe^α_∞(dx)− Z

[0,N]

α(x)µ_∞(dx)

(A.31)

where the measureseµ^α_k, fork∈Nandα∈ A, will be constructed given the measureµ_k and the function α. For the second and third summand of (A.29) we will prove that they become arbitrarily small for largeN >0. Then, we will conclude once we obtain the convergence to 0 ask→ ∞of the summands of (A.31). To this end, let us fix anε >0.

Condition (i) implies that sup_k∈Nµk(R+) < ∞as well as the tightness of the sequence by the convergenceµk(R+)−−−−→

k→∞ µ∞(R+). Hence, forM := sup_α∈Akαk∞<∞, which is Condition (ii), there exists N=N(ε, M)>0 such that

sup

k∈N

µ_k [N,∞)

< ε

2M, which implies sup

k∈N,α∈A

Z

[N,∞)

α(x)µ_k(dx)

< ε 2.

In other words, we have proven that the second and third summand of the right-hand side of (A.29) become arbitrarily small for large N.In the following N is assumed fixed, but large enough so that the above hold.

We proceed now to construct for everyk∈Nandα∈ Aa suitable measureeµ^α_k on R+,B(R+) such that we can obtain the convergence of the summands in (A.31). DefineKN := sup_k∈

Nµk([0, N])<∞.

By (iii) there exists ζ=ζ(ε, KN)>0 such that for everyζ⁰ < ζ it holds sup_α∈Aw⁰_N(α, ζ⁰)< _4K^ε

N. By the definition of w⁰ (see Definition I.108) for everyα∈ Aand every ζ⁰ < ζ, there exists aζ⁰-sparse set P_α^N,ζ0, i.e.

P_α^N,ζ0:=n

0 =t(0;P_α^N,ζ0)< t(1;P_α^N,ζ0)< . . . < t(κ^α;P_α^N,ζ0) =N : min

i=1,...,κ^α−1 t(i;P_α^N,ζ0)−t(i−1;P_α^N,ζ0)

> ζ⁰o , such that

i=1,...,κmax^αsupn

|α(s)−α(u)|:s, u∈

t(i−1;P_α^N,ζ0), t(i;P_α^N,ζ0)o

< w⁰_N(α, ζ⁰) + ε 4KN

< sup

α∈A

w_N⁰ (α, ζ⁰) + ε 4KN

< ε 2KN

. (A.32) To sum up, for every (α, ζ⁰)∈ A ×(0, ζ) we can find a partition of [0, N], P_α^N,ζ0, satisfying (A.32). For the following, given a finite measure on R+,B(R+)

, nameλ, and a sparse setP_α^N,ζ0

. we will denoteA(i;P_α^N,ζ0) := [t(i−1;P_α^N,ζ0), t(i;P_α^N,ζ0)) for i= 1, . . . , κ^α and

. we define the finite measureeλ^α: R+,B(R+)

→[0,∞) by eλ^α(·) :=

κ^α

X

i=1

λ A(i;P_α^N,ζ0)

δt(i−1;Pα^N,ζ0)(·),

where δx is the Dirac measure sitting at the pointx. With the above notation we have

Z

[0,N]

α(x)λ(dx)− Z

[0,N]

α(x)λe^α(dx)

=

κ^α

X

i=1

Z

A(i;Pα^N,ζ0)

α(x)λ(dx)−

κ^α

X

i=1

Z

A(i;Pα^N,ζ0)

α(x)λe^α(dx)

≤

κ^α

X

i=1

Z

A(i;Pα^N,ζ0)

α(x)λ(dx)− Z

A(i;Pα^N,ζ0)

α(x)eλ^α(dx)

=

κ^α

X

i=1

Z

A(i;Pα^N,ζ0)

α(x)λ(dx)−α t(i−1;P_α^N,ζ0 λ

[t(i−1;P_α^N,ζ0), t(i;P_α^N,ζ0))

=

κ^α

X

i=1

Z

A(i;Pα^N,ζ0)

α(x)−α t(i−1;P_α^N,ζ0 λ(dx)

≤

κ^α

X

i=1

Z

A(i;Pα^N,ζ0)

α(x)−α t(i−1;P_α^N,ζ0 λ(dx)

A.3. AUXILIARY RESULTS OF CHAPTER IV 145

(A.32)

≤ ε

2KN κ^α

X

i=1

Z

A(i;Pα^N,ζ0)

λ(dx) = ε 2KN

λ([0, N]). (A.33)

For every k ∈ Nand α ∈ A we define µe^α_k := (µgk)^α. Now, using the approximation (A.33) for µk, for k∈N, we obtain

sup

k∈N,α∈A

Z

[0,N]

α(x)µ_k(dx)− Z

[0,N]

α(x)eµ^α_k(dx)

< ε

2. (A.34)

To sum up, we have constructed the measuresµe^α_k such that the first and third summand of (A.31) become arbitrarily small uniformly onk∈Nand onα∈ A.

We can conclude (A.27) once we obtain the validity of

k→∞lim sup

α∈A

Z

[0,N]

α(x)eµ^α_k(dx)− Z

[0,N]

α(x)eµ^α_∞(dx)

= 0.

Indeed, for fixedζ⁰∈(0, ζ),

Z

[0,N]

α(x)µe^α_k(dx)− Z

[0,N]

α(x)µe^α_∞(dx)

=

κ^α

X

i=1

Z

A(i;Pα^N,ζ0)

α(x)µe^α_k(dx)−

κ^α

X

i=1

Z

A(i;Pα^N,ζ0)

α(x)eµ^α_∞(dx)

=

κ^α

X

i=1

Z

A(i;Pα^N,ζ0)

α(x)eµ^α_k(dx)− Z

A(i;Pα^N,ζ0)

α(x)eµ^α_∞(dx)

=

κ^α

X

i=1

α t(i;P_α^N,ζ0)h

µk(A(i;P_α^N,ζ0))−µ A(i;P_α^N,ζ0)i

(ii)

≤ kαk_∞

κ^α

X

i=1

µk A(i;P_α^N,ζ0)

−µ_∞ A(i;P_α^N,ζ0)

≤ sup

α∈A

kαk_∞·κ^α· sup

i=1,...,κ^α

µ_k A(i;P_α^N,ζ0)

−µ_∞ A(i;P_α^N,ζ0)

≤ sup

α∈A

kαk_∞·κ^α· sup

I∈IN

µk(I)−µ_∞(I)

. (A.35)

Let us now denotem(P_α^N,ζ0) := min

t(i;P_α^N,ζ0)−t(i−1;P_α^N,ζ0), i= 1, . . . , κ^α > ζ⁰ by the definition of P_α^N,ζ0. Therefore,

sup

α∈A

κ^α≤sup

α∈A

l N m(Pα^N,ζ0)

m5≤lN ζ⁰ m

<∞.

Hence, for every fixed sparse set P_α^N,ζ0 which satisfies (A.32) we obtain sup

α∈A

Z

[0,N]

α(x)µe^α_k(dx)− Z

[0,N]

α(x)µe^α_∞(dx)

≤sup

α∈A

kαk∞·sup

α∈A

κ^α· sup

I∈IN

µk(I)−µ(I)

−−−−(i)→

k→∞ 0, where we have used Proposition A.12.(V) which is equivalent to (i).

Remark A.15. We have presented the previous theorem for A ⊂ D(R). However, the result can be readily adapted forA ⊂D(R^p).

Proposition A.16. Let the measurable space R+,B(R+)

and(µ_k)_k∈

Nbe a sequence of finite measures such that µ_k({0}) = 0 for every k ∈ N and µ_∞ is atomless. Additionally, let A := (α^k)_k∈

N ⊂D(R^p).

Assume the following to be true:

(i) µk

−−→w µ_∞.

(ii) Ais uniformly bounded, i.e. sup_α∈Akαk_∞<∞.

(iii) Aisδ_J1(Rp)−convergent. In other words,α^{k J1(R}

p)

−−−−−→

k→∞ α^∞. Then,

k→∞lim Z

[0,∞)

α^k(x)µk(dx)− Z

[0,∞)

α^∞(x)µ_∞(dx) ₂

= 0.

5We denote bydxethe least integer greater than or equal tox.

Proof. Let us denote γ^k,m:=

Z

[0,∞)

α^k(x)µ_m(dx)− Z

[0,∞)

α^k(x)µ_∞(dx) ₂

fork∈Nandm∈N.

We are going to apply Theorem A.9 in order to obtain the required result. By Theorem A.14 we have that

m→∞lim sup

k∈N

γ^k,m = 0.

In other words, Condition (i) of Theorem A.9 is satisfied. Let us prove, now, that Condition (ii) of the aforementioned theorem is also satisfied, i.e. we need to prove that

m→∞lim lim sup

k→∞

γ^k,m−lim inf

k→∞ γ^k,m

= 0.

However, it is sufficient to prove that lim_m→∞lim sup_k→∞γ^k,m = 0, since the elements of the doubly-indexed sequence are positive. To this end, we have

lim sup

k→∞

γ^k,m

≤lim sup

k→∞

Z

[0,∞)

α^k(x)µ_m(dx)− Z

[0,∞)

α^∞(x)µ_m(dx) ₂ + lim sup

k→∞

Z

[0,∞)

α^∞(x)µ_m(dx)− Z

[0,∞)

α^∞(x)µ_∞(dx) ₂ + lim sup

k→∞

Z

[0,∞)

α^∞(x)µ_∞(dx)− Z

[0,∞)

α^k(x)µ_∞(dx) ₂

. We are going to prove now that the two last summands of the right-hand side of the above inequality are equal to zero. We start with the second and we realise that we have only to use thatµ_k −−^w→µ_∞ and Theorem A.14 for the special case of a singleton, in order to verify our claim. The third summand is also equal to zero as an outcome of the bounded convergence theorem. Indeed, we have that the the sequence A is uniformly bounded and by Proposition I.121 we have the pointwise convergenceα^k(x)−−→α^∞(x) for k → ∞, at every point xwhich is point of continuity of α^∞. Since the set{x∈R+,∆α^∞(x) 6= 0}

is at most countable, we can conclude that it is a µ_∞−null set. Therefore, since every element of A is Borel measurable, we can conclude.

Finally, we deal with the first summand. Let us provide initially some helpful results. Using again Proposition I.121 we have that

lim sup

k→∞

α^k(x)−α^∞(x) ₂≤

∆α^∞(x)

₂ for everyx∈R+, (A.36) because the only possible accumulation points of the sequence α^k,i(x)

k∈N areα^∞,i(x) and α^∞,i(x−), for everyi= 1, . . . , pand everyx∈R+; recall Remark I.119. Now observe that the functionR+3x7−→

∆α^∞(x)

₂ ∈ R⁺ is Borel measurable and µ_∞−almost everywhere equal to the zero function. This observation allows us to apply Mazzone [51, Theorem 1] in order to conclude that

Z

[0,∞)

∆α^∞(x)

₂µm(dx)−−−−→

m→∞

Z

[0,∞)

∆α^∞(x)

₂µ∞(dx) = 0. (A.37) In view of the above and the boundedness of the sequenceA, we apply Fatou’s lemma and we obtain for every m∈N

lim sup

k→∞

Z

[0,∞)

α^k(x)µm(dx)− Z

[0,∞)

α^∞(x)µm(dx) ₂

≤ Z

[0,∞)

lim sup

k→∞

α^k(x)−α^∞(x) 2

µm(dx)

(A.36)

≤ Z

[0,∞)

∆α^∞(x)

₂µm(dx).

Now, we can conclude that the Condition (ii) of Theorem A.9 is indeed satisfied by combining the above

bound with Convergence (A.37).

Remark A.17. Let us adopt the notation of Theorem A.14 and Proposition A.16 and assume that the measurable space is [0, T],B([0, T])

, for someT ∈R+. We claim that we can adapt the aforementioned results without loss of generality, which can be justified as follows. The limit measureµ_∞is atomless, so the generality is not harmed if in Theorem A.14 the integrandsα^k,for somek∈N, have a jump at pointT.

A.3. AUXILIARY RESULTS OF CHAPTER IV 147

Indeed, by the weak convergenceµk ⇒µ_∞and Proposition A.12 we have that the sequence of distribution functions (C^k)_k∈

N which is associated to the sequence (µ^k)_k∈

N converges uniformly. Therefore, we can easily conclude that

lim sup

k→∞

∆α^k(T)

₂µk({T}) = lim sup

k→∞

∆α^k(T)

₂∆C_T^k ≤

∆α^∞(T)

₂∆C_T^∞= 0.

In other words, we can simply assume that the distribution functions are constant after timeT in order to reduce the general case to the compact-interval case.

The following corollary is almost evident due to the fact that the limit measure is atomless. However we state it in the form we will need it in Subsection IV.5.2 and we provide its complete (rather trivial) proof.

Corollary A.18. Let the measurable space R+,B(R+)

,(µk)_k∈Nbe a sequence of finite measures, where µ^k({0}) = 0for everyk∈Nandµ∞is atomless. Let, moreover,A:= (α^k)_k∈N⊂D(R^p)and the following to be true:

(i) µk

−−→w µ_∞.

(ii) Ais uniformly bounded, i.e. sup_α∈Akαk_∞<∞and (iii) Aisδ_J1(Rp)−convergent.

Then,

Z

[0,·]

α^k(x)µk(dx) ^J1(R

p)

−−−−−→

k→∞

Z

[0,·]

α^∞(x)µ_∞(dx). (A.38)

Proof. LetC^k denote the distribution function associated to the measureµ_k for everyk∈Nand γ^k(t) :=

Z

[0,t]

α^k(x)µ_k(dx) = Z

[0,t]

α^k(s) dC_s^k ∈D(R^p) fork∈N.

We are going to apply Proposition I.118 in order to prove the required convergence. To this end, let us fix a t^∞ ∈ R+ and a sequence (t^k)k∈N such that t^k −−→ t^∞ as k → ∞. We need to prove that the requirements (i) - (iii) of the aforementioned theorem are satisfied.

(i) lim_k→∞

γ^k(t^k)−γ^∞(t^∞) 2∧

γ^k(t^k)−γ^∞(t^∞−) 2

= 0.However,γ^∞is continuous. Therefore, it is sufficient to prove

k→∞lim

γ^k(t^k)−γ^∞(t^∞) ₂= 0.

To this end, we have γ^k(t^k) =

Z

[0,t^k]

α^k(s) dC_s^k= Z

[0,t^k∧t^∞]

α^k(s) dC_s^k+ Z

[t^k∧t^∞,t^∞]

α^k(s) dC_s^k+ Z

[t^∞,t^k∧t^∞]

α^k(s) dC_s^k. The first summand converges toγ^∞(t^∞), because the sequence (C^k1[0,t^k])_k∈

Nisk · k_∞−convergent and (α^k)_k∈

N is uniformly bounded δ_J1(Rp)−convergent. In other words, the conditions of Proposition A.16 are satisfied. For the second summand we have

Z

[t^k∧t^∞,t^∞]

α^k(s) dC_s^k

≤Var Z

[0,·]

α^k(s) dC_s^k

t^∞

−Var Z

[0,·]

α^k(s) dC_s^k

t^k∧t^∞

= Z

[t^k∧t^∞,t^∞]

α^k(s)∨0 dC_s^k+

Z

[t^k∧t^∞,t^∞]

−α^k(s)∨0 dC_s^k

≤sup

k∈N

α^k

_∞µk [t^k∧t^∞, t^∞]

= sup

k∈N

α^k

∞ C_t^k∞−C_t^kk∧t^∞

−−−−→

k→∞ 0,

where we used Proposition A.12 in order to conclude the convergence. Analogously we can prove that the third summand converges also to 0.

(ii) If limk→∞

γ^k(t^k)−γ^∞(t^∞)

₂= 0 and (s^k)k∈Nsuch thatt^k≥s^kfor everyk∈Nands^k−−−−→

k→∞ t^∞, then

k→∞lim

γ^k(s^k)−γ^∞(t^∞) ₂= 0.

We can repeat the exact same arguments as in (i) in order to prove the convergence for every (s^k)_k∈N such thats^k −−−−→

k→∞ t^∞.

Im Dokument Backward stochastic differential equations with jumps are stable (Seite 147-167)