The Skorokhod space - Backward stochastic differential equations with jumps are stable

The above inequality and (I.30) imply the desired sup

k∈N

((S^k((

Ψ≤2psc_Ψsup

k∈N

Ξ^k<∞,

which, in view of Equation (I.31), implies also the ﬁniteness of sup_k_∈NE Ψ(√

2S^k) . Therefore, the sequence

sup_t_∈R

+((E

ξ^kG^kt ((²

is uniformly integrable since it satisﬁes the de La Vall´ee Poussin theorem

for the Young function Φ.

I.6. The Skorokhod space

In this section we present the Skorokhod space D(E) which is the natural path-space for semi-martingales adapted to some ﬁltration that satisﬁes the usual conditions. The weaker of the topologies the Skorokhod space will be endowed with, will be the (Skorokhod) J₁−topology. The J1−topology is metrisable and makes the space to be complete and separable, i.e. D(E) becomes Polish. The main reference of this section is Jacod and Shiryaev [41].

Deﬁnition I.107. (i) The function α: R+ −−→ E will be called c`adl`ag ifαis right-continuous and for every t∈Rthe valueα(t−) exists.

(ii) The E−Skorokhod space is the space of all c`adl`ag, E−valued functions deﬁned on R+ and it will be denoted byD(E),i.e.

D(E) :={α:R+−−→E, αis c`adl`ag}. The zero function will be denoted by 0.

(iii) The ﬁltration D(E) :=

D^t(E)

t∈R+ is the right-continuous ﬁltration obtained by the ﬁltration generated by the canonical projections, i.e.

Dt(E) := )

u>t

{α(s), α∈D(E) ands≤u} Moreover,D(E) :=σ

{α(s), α∈D(E) ands∈R+} .

We will mostly denote the elements ofD(R) with small Greek letters, usually the ﬁrst three one of the Greek alphabet, i.e. α, β andγ.

Deﬁnition I.108. (i) Let λ : R+ −−→ R+ be continuous and increasing such that λ(0) = 0 and limt→∞λ(t) =∞.Thenλwill be calledtime change.

(ii) The set of all time changes will be denoted by Λ,i.e.

Λ :={λ:R+−−→R+, λis continuous and increasing withλ(0) = 0, lim

t→∞λ(t) =∞}. (iii) To eachα∈D(E) and each intervalI⊂R+ we associate themoduli

w(α;I) := sup

s,t∈Iα(s)−α(t)2

and

w_N (α, θ) := inf

maxi≤r w(α; [ti−1, ti)),0 =t₀< t₁<· · ·< tr=N with inf

i<r(ti−ti−1)≥θ '

. Theorem I.109. There is a metrisable topology onD(E), called the Skorokhod J₁−topology, for which the space is Polish. Moreover, the following are equivalent:

(i) The sequence(α^k)k∈N converges to α^∞ under the J₁−topology.

(ii) There exists a sequence(λ^k)k∈N⊂Λ such that sup

s∈R+

|λ^k(s)−s| −−−−→

k→∞ 0 and sup

s≤N

α^k(λ^k(s))−α^∞(s)2−−−−→

k→∞ 0, for all N∈N. (I.32) The metric under which D(E)becomes Polish will be denoted by δ_J₁₍E).²¹ Moreover, the associated Borel σ−algebra equals theσ−algebra D(E), which was deﬁned Deﬁnition I.107.(ii).

21We will omit the state space from the notation, when it is clear to which one we refer.

Proof. See Jacod and Shiryaev [41, Theorem VI.1.14 a)]. For the definition ofδJ₁ see [41, Definition VI.1.26]. In [41, Lemma VI.1.30, Lemma VI.1.31, Corollary VI.1.32] it is proven that δJ₁ is indeed a metric. In [41, Lemma VI.1.33] it is proven that (D(E), δJ₁) is complete. The separability of the space is proven in [41, Corollary VI.1.43 a)].

For (i)⇒(ii) see [41, Lemma VI.1.31], while for the implication (ii)⇒(i) see [41, Lemma VI.1.44].

For the validity of the statement regarding the Borel σ−algebra and D(E) see [41, Lemma VI.1.45,

Lemma VI.1.46].

Remark I.110. Recall the well-known fact that a c`adl`ag functionαhas at most countably many points of discontinuities; see Billingsley [7, Section 12, Lemma 1] for the case the domain is [0,1] and then one can easily conclude for the caseR⁺, as a countable union of finite sets is a countable set. Therefore, the function ∆αis zero except from a set of Lebesgue measure zero.

Theorem I.111. A set D ⊂D(E)is relatively compact for the Skorokhod J1−topology if and only if for every N∈Nboth of the following conditions hold:

sup

α∈D

sup

s≤N

kα(s)k2<∞and lim

θ↓0sup

α∈D

w⁰_N(α, θ) = 0for every N∈N.

Proof. See Jacod and Shiryaev [41, Theorem VI.1.14 b)]. More precisely, [41, Lemma VI.1.39]

proves the necessity and [41, Corollary VI.1.43] provides the sufficiency.

We will say that the sequence (α^k)_k∈N is aδJ₁(E)−convergent sequence if it satisfies Theorem I.109 and we will denote its convergence also by α^k −−−−−→^J¹^(E) α^∞. Apart from the J1−topology we will equip the spaceD(E) with two more topologies, which are presented below.

Notation I.112. Letα, β∈D(E).

• Thelocally uniform topology is the one associated with the metricδ_lu, where δ_lu(α, β) =X

k∈N

2^k w(α−β; [0, k]∧1 .

The convergence of a sequence (α^k)_k∈_N toα^∞ under the metricδ_lu will be denoted byα^k−−→^lu α^∞.

• The uniform topology is associated with the norm k · k_∞, wherekαk_∞ := sup_s∈_R

+kα(s)k₂. The convergence of a sequence (α^k)_k∈_Ntoα^∞under the metricδ_k·k_∞ will be denoted byα^k−−−−→^k·k^∞ α^∞.

The next proposition provides the relationship betweenδ_J₁₍_D_(E)) andδ_lu.

Proposition I.113. (i) The J1−topology is weaker than the locally uniform topology.

(ii) Let α^∞∈D(E)be a continuous function. Then

α^k−−−−−−−→^J¹⁽^D^(E)) α^∞ if and only if α^k −−−→^δ^lu α^∞.

Proof. See Jacod and Shiryaev [41, Proposition VI.1.17].

Remark I.114. For the metricsδ_J₁₍_D_(E)),δ_luandδ_k·k_∞ holds

δ_J₁₍_D_(E))(α,0) =δ_lu(α,0) for everyα∈D(E) and

δJ₁(D(E))(α, β)≤δlu(α, β)≤ sup

s∈R+

kα(s)−β(s)k2 for everyα, β∈D(E).

The following lemma is essentially [35, Example 15.11]. We provide an alternative proof in some special cases of the aforementioned result since we will need later to elaborate on similar cases.

Lemma I.115. Let (E,k · k2), (x^k_i)_k∈

N⊂E and(t^k_i)_k∈

N⊂R+, fori= 1,2, such that t^k₁< t^k₂ for every k∈N.

(i) If t^k₁−−→ ∞, then^|·| x^k₁1[t^k₁,∞) J1(E)

−−−−−→0.

(ii) If x^k₁ −−−→^k·k² 0, then x^k₁1[t^k₁,∞) J₁(E)

−−−−−→0.

(iii) If t^k₁−−→^|·| t^∞₁ ∈R+ andx^k₁−−−→^k·k² x^∞₁ , thenx^k₁1[t^k₁,∞) J1(E)

−−−−−→x^∞₁ 1[t^∞₁ ,∞).

I.6. THE SKOROKHOD SPACE 31

(iv) If t^k₁−−→^|·| t^∞₁ ∈R+,t^k₂ −−→ ∞^|·| andx^k₁ −−−→^k·k² x^∞₁ , then x^k₁1[t^k₁,∞)+x^k₂1[t^k₂,∞)

J₁(E)

−−−−−→x^∞₁ 1^[t^∞₁ ,∞).

(v) If t^k₁−−→^|·| t^∞₁ ,t^k₂ −−→^|·| t^∞₂ , wheret^∞₁ < t^∞₂ ∈R+,x^k₁−−−→^k·k² x^∞₁ andx^k₂−−−→^k·k² x^∞₂ then x^k₁1[t^k₁,∞)+x^k₂1[t^k₂,∞)

J₁(E)

−−−−−→x^∞₁ 1[t^∞₁ ,∞)+x^∞₂ 1[t^∞₂ ,∞).

Proof. For (i) and (ii), since the limit function is (trivially) continuous, we can use Proposition I.113 in order to justify that we can choose without loss of generality λ^k = Id for every k ∈ N. In view of Theorem I.109 it is therefore only left to prove that

sup

s≤N

kα^k(s)k2−−→0 for everyN ∈N.

(i) Assume that t^k₁ −−→ ∞. Then for every^|·| N ∈N there exists a k0(N)∈N such thatt^k₁ ≥N for every k≥k0(N).Therefore, for every fixedN ∈N, it holds sup_s≤Nkα^k(s)k2= 0 for everyk≥k0(N).

(ii) Assume now x^k₁ −−−→^k·k¹ 0,i.e. for everyε >0 there exists ak0(ε)∈Nsuch thatkx^kk2 < ε.Let us fix an ε >0 and anN ∈N, then sup_s≤Nkx^kk2< ε for everyk≥k0(ε), which allows us to conclude.

Observe that the choice ofk0 is independent ofN.

For (iii)-(v) see He et al. [35, Example 15.11], which deals with convergence of piecewise constant functions with possibly infinitely many jumps. For this reason recall conditions (i)-(iii) of He et al. [35,

Remark 15.32] for the detailed properties of the jump times.

Remark I.116. The alert reader may have observed the condition imposed in Lemma I.115.(v) for the jump-timest^∞₁ and t^∞₂ , i.e. t^∞₁ < t^∞₂ .Let us examine what may fail in the case t^∞₁ =t^∞₂ . To this end we set the following framework: assume that t^k₁ ↑ ¯tand t^k₂ ↓¯t, i.e. t^k₁ →¯t with t^k₁ ≤¯t andt^k₂ →¯t with t¯≤t^k₂. Then, in view of Lemma I.115.(iii) we have that

α^k₁:=1[t^k₁,∞) J₁(R)

−−−−→1[¯t,∞) andα^k₂ :=−1[t^k₂,∞) J₁(R)

−−−−→ −1[¯t,∞). (I.33) On the other hand, the sequence (α^k₁+α^k₂)_k∈_Ndoes not converge under the J₁−topology, since (1[t^k₁,t^k₂))_k∈_N fails to be aδ_J₁₍_R₎−Cauchy sequence. For the last claim see the comments right after Billingsley [7, Section 12, Lemma 2, p. 126] in conjunction with [7, Example 12.2].

Let us explain a bit more this (counter-)example. By Theorem I.109 we have that there exists a sequence of time changes (λ^k_i)k∈N, for i = 1,2, such that the respective convergence in (I.33) hold.

However, the sum of the converging sequences is not converging, because we cannot find a sequence of time changes (λ^k)k∈Nsuch that it isfinally commonfor the sequences (1[t^k₁,∞))k∈N, (1[t^k₂,∞))k∈Nand such that (I.33) holds.

After providing the main results for the Skorokhod J₁−topology, we need to clarify some details.

Remark I.117. (i) The space (D(E), δ_J₁_(E)) is not a topological vector space. This is direct by Remark I.116 which proves that the addition is notδ_J₁−continuous.

(ii) The spaceD(R^p×q) can be identified withD(R)^p×q. Let us endow the space D(R)^p×q :={α:R+−−→R^p×q, α^ij is c`adl`ag for every 1≤i≤p,1≤q}.

with the product topology, which is compatible with the metric δ_Π(α, β) :=

i=1 q

j=1

δ_J₁₍_R₎(α^ij, β^ij).

However, the topological spaces (D(R^p×q), δJ₁(R^p×q)) and (D(R)^p×q, δΠ) do not coincide. Indeed, by Remark I.116 we have

δ_Π (α^k₁, α^k₂),(1[¯t,∞),1[¯t,∞))

−−→0 butδ_J₁₍_Rp×q) (α^k₁, α^k₂),(1[¯t,∞),1[¯t,∞)) 6−−→0.

Now it may be clear to the reader why we have always indicated the state space in the notation of the metric. We will (almost) always indicate the state space in the notation for the sake of clarity.

(iii) For notational convenience we have stated the main results for the space D(E) and not for the spaceD(R^d), wheredis a natural integer, as in Jacod and Shiryaev [41, Chapter VI]. This can be done without loss of generality by defining an isometry from every finite dimensional space (E,k · k₂) to the space (R^p,k · k₂), wherepdenotes the dimension of the space E.

The following are classical criteria for obtaining the joint convergence.

Proposition I.118. Let(α^k)_k∈

N⊂D(E). The convergenceα^k−−−−−→^J¹^(E) α^∞ holds if and only if whenever (t^k)_k∈N⊂R+ be such thatt^k−−→t^∞ the following conditions hold:

(i)

α^k(t^k)−α^∞(t^∞) ₂∧

α^k(t^k)−α^∞(t^∞−)

₂−−→0.

(ii) If

α^k(t^k)−α^∞(t^∞)

₂ −−→ 0 and (s^k)_k∈_N ⊂ R+ be such that t^k ≤ s^k for every k ∈ N and s^k −−→t^∞, then

α^k(s^k)−α^∞(t^∞)

₂−−→0.

(iii) If

α^k(t^k)−α^∞(t^∞−)

₂ −−→ 0 and (s^k)k∈N ⊂ R+ be such that s^k ≤ t^k for every k ∈ N and s^k −−→t^∞, then

α^k(s^k)−α^∞(t^∞−)

₂−−→0.

Proof. See Ethier and Kurtz [33, Proposition 3.6.5].

Remark I.119. Assumeα^k−−−−−→^J¹^(E) α^∞andt^k−−→t^∞. Then by Proposition I.118.(i) we have that the only possible limit points of the sequences α^k(t^k)

k∈N, α^k(t^k−)

k∈Nareα^∞(t^∞) andα^∞(t^∞−).

Proposition I.120. Let(α^k)_k∈

N⊂D(E1)and(β^k)_k∈

N⊂D(E2). The convergence(α^k, β^k)−−−−−−−−→^J¹^(E¹^×E²⁾ (α^∞, β^∞)holds if and only if

(i) α^k −−−−−→^J¹^(E¹⁾ α^∞, (ii) β^k −−−−−→^J¹^(E²⁾ β^∞ and

(iii) for everyt∈R+there exists a sequence(t^k)_k∈_Nwitht^k−−→tand such that∆α^k(t^k)−−−→^k·k² ∆α^∞(t) and∆β^k(t^k)−−−→^k·k² ∆β^∞(t).

Proof. For the implication (i)⇒(ii) apply Jacod and Shiryaev [41, Proposition VI.2.1 a)] and then conclude using Remark I.117.(i). For the implication (ii)⇒(i) apply [41, Proposition VI.2.2 b)] for the spacesD(E1),D(E2) and then conclude using Remark I.117.(i).

Looking back to Remark I.116, it is clear that there exists no sequence (t^k)k∈N that satisfies the condition (iii) of Proposition I.120 for the point ¯t.

The following proposition will be helpful to prove Corollary I.125.

Proposition I.121. Let(α^k)_k∈Nbe a δ_J₁_(E)−convergent sequence and t∈R+.

(i) There exists a sequence (t^k)_k∈_N such that t^k −−→t, α^k(t^k)−−→α^∞(t)and α^k(t^k−)−−→α^∞(t−).

Therefore, the convergence∆α^k(t^k)−−−→^k·k² ∆α^∞(t)also holds.

(ii) Let ∆α^∞(t)6= 0 and(t^k)_k∈_N be a sequence such thatt^k −−→t and∆α^k(t^k)−−→∆α^∞(t). Then, any other sequence (s^k)_k∈N for which holdss^k −−→t and∆α^k(s^k)−−−→^k·k² ∆α^∞(t)coincides with(t^k)_k∈N fork large enough.

Proof. See Jacod and Shiryaev [41, Proposition VI.2.1].

The next classical result informs us that we can obtain the convergence of the sum of jointly convergent sequences.

Corollary I.122. Let (α^k, β^k)

k∈N be aδ_J₁_(E×E)−convergent sequence. Then (α^k, β^k, α^k+β^k)−−−−−−−−−→^J¹^(E×E×E) (α^∞, β^∞, α^∞+β^∞).

Proof. We will prove it forE=R. Let, therefore, (α^k, β^k)_k∈

N⊂D(R) and assume that (α^k, β^k) ^J¹⁽^R

−−−−−→(α^∞, β^∞).

Choose a t ∈R+ such that (∆α^∞(t),∆β^∞(t))6= 0. Then, by Proposition I.121 there exists a sequence (t^k)k∈N such thatt^k −−→t and

(∆α^k(t^k),∆β^k(t^k))−−−→^k·k² (∆α^∞(t),∆β^∞(t)).

In particular, we can conclude that ∆α^k,1(t^k) + ∆α^k,2(t^k) −−→^|·| ∆α^∞,1(t) + ∆α^∞,2(t). Now, we can conclude by Proposition I.120 that

(α^k, β^k, α^k+β^k)−−→(α^∞, β^∞, α^∞+β^∞).

I.6. THE SKOROKHOD SPACE 33

Remark I.123. The result above does not depend on the dimensions of the state spaces, and can be generalized inductively to an arbitrary number of sequences, as long as a common converging sequence exists.

A convenient variation of the Proposition I.121 is provided in Coquet, M´emin, and S lomi´nski [23, Lemma 1], which we state also here.

Lemma I.124. Let (α^k)_k∈N⊂D(R^p). The convergenceα^k ^J¹⁽^R

−−−−−→α^∞holds if and only if the following hold

(i) α^k,i −−−−→^J¹⁽^R⁾ α^∞,i, fori= 1, . . . , p, (ii)

i=1

α^k,i−−−−→^J¹⁽^R⁾

i=1

α^∞,i,forq= 1, . . . , p.

We can provide now a corollary which will be proven quite convenient, since it allows us to conclude the joint convergence of two sequences once we can find a common element between the elements of the two converging sequences.

Corollary I.125. Let (α^k)_k∈

N, resp. (β^k)_k∈

N, be a δ_J₁₍_Rp1)−convergent, resp. δ_J₁₍_Rp2)−convergent, sequence. If there exist 1 ≤ i1 ≤ p1, 1 ≤ i2 ≤ p2 such that α^k,i¹ = β^k,i² for every k ∈ N, then (α^k, β^k) ^J¹⁽^R

p1×R^p²)

−−−−−−−−−→(α^∞, β^∞).

Proof. We will apply Proposition I.120. It is only left to verify that condition (iii) of the aforemen-tioned proposition holds. To this end let us fix at∈R⁺.By (i), resp. (ii), and Proposition I.121 we have that there exists a sequence (t^k₁)_k∈N, resp. (t^k₂)_k∈N, such that

∆α^k(t^k₁)−−→∆α^∞(t), resp. ∆β^k(t^k₂)−−→∆β^∞(t).

However, by (iii) and Proposition I.121.(ii) we have that (t^k₁)k∈N and (t^k₂)k∈N finally coincide. Therefore, we can choose one of the two time sequences, say (t^k₁)k∈N, in order to conclude

∆α^k(t^k₁)−−→∆α^∞(t), resp. ∆β^k(t^k₁)−−→∆β^∞(t),

Proposition I.120.(iii) is satisfied.

We close this part with another convenient result.

Lemma I.126. Let (α^k)_k∈

N⊂D(E) andf : (E,k · k2)−→(R^p,k · k2)be continuous. Ifα^k −−−−→^J¹^(E) α^∞ then f(α^k) ^J¹⁽^R

−−−−−→f(α^∞).

Proof. The continuity off allows us to apply Proposition I.120.

I.6.1. J₁−continuous functions. The aim of this sub–sub–section is the following: for a given δ_J₁_(E)−converging sequence (α^k)_k∈

Nand a given (not necessarily continuous) function g :E −→R, we need to determine suitable conditions for the function gand a setI⊂E such that

0<t≤·

g(∆α^k(t))1^I(∆α^k(t))−−−−→^J¹⁽^R⁾ X

0<t≤·

g(∆α^∞(t))1^I(∆α^∞(t)). (I.34) This is Proposition I.134, which refines the classical result Jacod and Shiryaev [41, Corollary VI.2.8], and it will be crucial for constructing in Chapter III a family of J₁−convergent sequences of submartingales. In order to obtain Corollary I.133, we need to refine another classical result, namely [41, Proposition VI.2.7];

the refinement of the latter is Proposition I.132. For simplicity, we provide the results forE =R^q. Before we proceed we introduce the necessary notation and provide a simple example, which will clarify to a great extend the main idea of Proposition I.132.

Definition I.127. Forβ∈D(R^q) andγ∈D(R) we introduce the sets U(β) :={u >0,∃t >0 withk∆β(t)k2=u}, W(γ) :={u∈R\ {0},∃t >0 with ∆γ(t) =u}

and

I(γ) :={(v, w)⊂R, v < wwithvw >0 andv, w /∈W(γ)}.

Forα∈D(R^q) we define the set J(α) :=nY^q

i=1

I_i, whereI_i∈ I(αⁱ)∪

R for every i= 1, . . . , qo

\ R^q .

The setW(γ), which is at most countable, collects the heights of the jumps of a real-valued function γ. The setI(γ) collects all the open intervals of R\ {0} with boundary points of the same sign, which, moreover, do not belong toW(γ).

Remark I.128. For the rest of this subsection we will use additionally the notationα_sfor the value of the arbitrary α∈D(R^q) at the points.

Notation I.129. Letα∈D(R^m) andI:=Qm

i=1I_i∈ J(α).

(i) We define the time points

t⁰(α, θ) := 0, tⁿ⁺¹(α, θ) := inf{t > tⁿ(α, θ),

∆α_t

2> θ}, n∈N. If{t > tⁿ(α, θ),

∆α_t

₂> θ}=∅, then we settⁿ⁺¹(α, θ) :=∞.

(ii) To the set I we associate the set of indices J_I :=

i∈ {1, . . . , m}, I_i6=R}. (I.35) (iii) For the pair (α, I) we define the time points

s⁰(α, I) := 0, sⁿ⁺¹(α, I) := inf{s > sⁿ(α, I),∆αⁱ_s∈Ii for everyi∈JI}, n∈N. (I.36) If{s > sⁿ(α, I),∆αⁱ_s∈I_i for every i∈J_I}=∅, then we setsⁿ⁺¹(α, I) :=∞.

The value of sⁿ(α, I) marks the n−th time at which the value of ∆α lies in the set I and it is well-defined since JI 6=∅forI∈ J(α).

Example I.130. Let (t^k)_k∈

N ⊂ R+, (x^k)_k∈

N ⊂ R be such that t^k −−→ t∞ and x^k ↓ x^∞ ∈ R\ {0}.

Define γ_·^k :=x^k1[t^k,∞)(·), for every k∈N. By Lemma I.115.(iii), we have γ^k −−−−→^J¹⁽^R⁾ γ^∞.On the other hand, for I := (¹₂x^∞,³₂x^∞) holds s¹(γ^k, I) = t^k for all but finitely many k and s¹(γ^∞, I) = t^∞, i.e.

s¹(γ^k, I)−−−−→

k→∞ s¹(γ^∞, I). Moreover, for w > x^∞ we also have

∆γ_t^kk1(x^∞,w)(∆γ_t^kk) =x^k1(x^∞,w)(x^k) =x^k for all but finitely manyk∈N, and

∆γ_t^∞_∞1(x^∞,w)(∆γ_t^∞_∞) =x^∞1(x^∞,w)(x^∞) = 0.

Therefore, forR3x7−→^g x1(x^∞,w)(x)

g(∆γ_s^k1(γ^k,I)) = ∆γ^k_tk1(x^∞,w)(∆γ_t^kk)−−−−→

n→∞ x^∞6= 0 = ∆γ_t^∞_∞1(x^∞,w)(∆γ_t^∞_∞) =g(∆γ_s^∞1(α^∞)), and for this reason wecannot obtain the convergence

∆γ_t^kk1(x^∞,w)(∆γ_tⁿk)1(x^∞,w)(·)−−−−→^J¹⁽^R⁾ ∆γ_t^∞_∞1(x^∞,w)(∆γ^∞_t_∞)1(x^∞,w)(·).

We can remedy the problem appearing in the previous example by not allowing elements ofW(γ^∞) to be endpoints of the interval appearing in the indicator. Recall thatW(γ) is at most countable, therefore we are allowed to choose values for the endpoints from a dense subset ofR\ {0}. More precisely, we will choose intervals fromI(γ) so that only finitely many jumps occur on any every compact time-interval.

For the rest of this subsection we adopt a convenient abuse of notation and we will assume the extended positive real line [0,∞] endowed with a metric ˜δ_|·| which extends the usual metric ofR+ and for which ˜δ_|·|(∞,∞) = 0. Moreover, the convergence of a sequence (t^k)_k∈_N ⊂[0,∞] to the symbol ∞ will be understood as follows: either t^k = ∞ for every k ∈ N or for the subsequence (t^k^l)_l∈N, where (kl)_l∈N:={k∈N, t^k^l <∞}, t^k^l −−→ ∞ in the usual sense. In this case, we will denote the convergence of a sequence (t^k)_k∈N⊂[0,∞] to the symbol∞as usually by t^k −−→ ∞.

Proposition I.131. Fixq, n∈N.

(i) The functionD(R^q)3α7−→tⁿ(α, θ)∈R+ is continuous at each point αsuch that θ /∈U(α).

(ii) If tⁿ(α, θ) <∞, then the function D(R^q)3 α7−→ ∆α_tn(α,θ) ∈R^q is continuous at each point α such that θ /∈U(α).

I.6. THE SKOROKHOD SPACE 35

Proof. See [41, Proposition VI.2.7].

The following proposition refines the above result.

Proposition I.132. Fixq, n∈N.

(i) The functionD(R^q)3α7−→sⁿ(α, I)∈R+ is continuous at each point (α, I)∈D(R^q)× J(α).

(ii) If sⁿ(α, I)<∞, I ∈ J(α), then the function D(R^q)3α7−→∆αsⁿ(α,I)∈R^q is continuous.

Proof. Let (α^k)_k∈

Nbe such thatα^k ^J¹⁽^R

−−−−−→α^∞andI:=Qq

i=1I_i∈ J(α^∞).Observe thatJ_I 6=∅, by definition ofJ(α^∞). We defines^k,n:=sⁿ(α^k, I), fork∈Nandn∈N.

(i) The convergence s^k,0 −−−−→

k→∞ s^∞,0 holds by definition. Assume that the convergence s^k,n −−−−→

k→∞

s^∞,nholds for some arbitraryn∈N. We will prove that the convergences^k,n+1−−−−→

k→∞ s^∞,n+1holds also.

For the following, fix a positive numberθ_I^∞such thatθ^∞_I ∈ ∪/ _i∈J_I{|u|, u∈W(α^∞,i)} and θ_I^∞< 1

qmin∪i∈JI{|v|, v∈∂Ii}.²² Recalling the Notation I.129.(i), we have for every i∈JI that

t∈R+,|∆α^∞,i|> θ_I^∞ =n

t^l α^∞,i, θ^∞_I

<∞, l∈N o

. (I.37)

In other words, the set

t^l α^∞,i, θ^∞_I

< ∞, l ∈ N is another way to write the set of times that α^∞,i exhibits a jump of height greater thanθ_I^∞.

Case 1: s^∞,n+1<∞.

By property (I.37), there exist uniquel^i,n, l^i,n+1∈Nwithl^i,n< l^i,n+1 such that

t^l^i,n(α^∞,i, θ^∞_I ) =s^∞,nand t^l^i,n+1(α^∞,i, θ^∞_I ) =s^∞,n+1 for everyi∈J_I. (I.38) By Proposition I.131 and the above identities we obtain

t^l^i,n(α^k,i, θ_I^∞)−−−−→

k→∞ s^∞,n and ∆α^k,i

t^li,n(α^k,i,θ_I^∞)−−−−→

k→∞ ∆α^∞,i_s∞,n for everyi∈J_I, (I.39) as well as

t^l^i,n+1(α^k,i, θ_I^∞)−−−−→

k→∞ s^∞,n+1 and ∆α^k,i

t^li,n⁺¹(α^k,i,θ_I^∞)−−−−→

k→∞ ∆α^∞,i_s∞,n+1 for every i∈JI. (I.40) Recall, now, the induction hypothesis, i.e. the convergence s^k,n −−−−→

k→∞ s^∞,n is true. The Proposi-tion I.121.(ii) guarantees that every sequence (r^k)_k∈_N which converges to the time points^∞,n and it is such that ∆α^k,i_r_k −−−−→

k→∞ ∆α^∞,i_s∞,n, for some i= 1, . . . , q, then it finally coincides with (s^k,n)_n∈_N.Therefore, in view of the Convergence (I.39) we can obtain that

t^l^i,n(α^k,i, θ_I^∞)−s^k,n

k∈N∈c₀₀(N), for every i∈J_I, (I.41) where c00(N) :={(x^m)_m∈N⊂R^N,∃m0∈Nsuch thatx^m= 0 for everym≥m0}.Define fori∈JI

k^n,i₀ := max{k∈N, t^l^i,n(α^k,i, θ_I^∞)6=s^k,n}<∞.

SinceJI is a finite set, the number

¯k₀ⁿ:= max

k^n,i₀ , i∈J_I is well–defined and finite, therefore

s^l^i,n(α^k,i, U) =s^k,n, for every k >¯kⁿ₀ and for every i∈JI. (I.42) Now, in view of Convergence (I.40) we can conclude the induction step once we prove the analogous to (I.41) forn+ 1 in place ofn,i.e.

(t^l^i,n+1(α^k,i, θ_I^∞)−s^k,n+1)k∈N∈c00(N), for everyi∈JI. (I.43) At this point we distinguish two cases:

Case 1.1: For everyi∈J_I, holdsl^i,n+1= 1 +l^i,n.

22Recall that∂Adenotes the| · |−boundary of the setA⊂R.

By Proposition I.121.(ii), Convergence (I.40) and the convergenceα^k ^J¹⁽^R

−−−−−→α^∞, we can conclude that

t^l^i,n+1(α^k,i, θ^∞_I )−t^l^j,n+1(α^k,j, θ^∞_I )

∈c00(N), for everyi, j∈JI. (I.44) Therefore, we can fix hereinafter an index from J_I and we will do so for µ := minJ_I, i.e. µ is the minimum element ofJ_I. Define

k^n+1,i₀ := max{k∈N, t^l^i,n+1(α^k,i, θ_I^∞)6=t^l^µ,n+1(α^k,µ, θ^∞_I )}fori∈J_I\ {µ}.

By property (I.44), we obtain thatk₀^n+1,i<∞for everyi∈JI\ {µ}. SinceJI\ {µ} is a finite set, the number

¯k₀ⁿ⁺¹:= max

k^n+1,i₀ , i∈JI\ {µ}

is well–defined and finite. Observe that

t^1+l^i,n(α^k,i, θ^∞_I ) =t^l^i,n+1(α^k,i, θ_I^∞) =t^l^µ,n+1(α^k,µ, θ_I^∞), fork >¯kⁿ⁺¹₀ and fori∈J_I, (I.45) where the first equality holds by the assumption of this sub-case. Moreover, by Convergence (I.40) we obtain that

1 = Y

i∈JI

1^Ii(∆α^k,i

t^1+li,n(α^k,i,θ_I^∞)), for all but finitely manyk, (I.46) since ∆α^∞,i_s∞,n+1 lies in the interior of the open intervalIi, for everyi∈JI.

For notational convenience, we will assume that the above convergence holds fork >¯kⁿ⁺¹₀ . Therefore, fork >k¯₀ⁿ∨k¯₀ⁿ⁺¹

s^k,n+1 = inf{s > s^k,n,∆α^k,i_s ∈Ii for everyi∈JI}= inf \

i∈JI

s > s^k,n,∆α^k,i_s ∈Ii (I.41)

k>k¯ⁿ₀

inf \

i∈JI

s > t^l^i,n(α^k,i, θ_I^∞),∆α^k,i_s ∈Ii

= inf \

i∈JI

ns∈

t^l^i,n(α^k,i, θ_I^∞), t^l^µ,n+1(α^k,i, θ^∞_I )i

,∆α^k,i_s ∈I_io

∧inf \

i∈J_I

s > t^l^µ,n+1(α^k,i, θ^∞_I ),∆α^k,i_s ∈Ii (I.45)

k>¯kⁿ⁺¹₀

inf \

i∈JI

n s∈

t^l^i,n(α^k,i, θ_I^∞), t^1+l^i,n(α^k,i, θ^∞_I )i

,∆α^k,i_s ∈Ii

∧inf \

i∈JI

s > t^1+l^i,n(α^k,i, θ^∞_I ),∆α^k,i_s ∈I_i

= \

i∈J_I

t^1+l^i,n(α^k,i, θ^∞_I ) ∧inf \

i∈J_I

s > t^1+l^i,n(α^k,i, θ^∞_I ),∆α^k,i_s ∈Ii

= \

i∈JI

t^1+l^i,n(α^k,i, θ^∞_I ) ^(I.45)= t^l^µ,n+1(α^k,i, θ^∞_I ) = t^l^i,n+1(α^k,i, θ_I^∞), i.e. Property (I.43) indeed holds.

Case 1.2: There exists i∈JI for which holdsl^i,n+1>1 +l^i,n.

Define ¯JI : {i∈JI, l^i,n+1 >1 +l^i,n} and fixξⁱ ∈(l^i,n, l^i,n+1)∩N, for everyi∈J¯I.Recall that by Proposition I.131 holds

k→∞lim t^ξⁱ(α^k,i, θ^∞_I ) =t^ξⁱ(α^∞,i, θ^∞_I ) and lim

k→∞∆α^k,i

t^ξi(α^k,i,θ^∞_I )= ∆α^∞,i

t^ξi(α^∞,i,θ^∞_I ). (I.47) We can conclude that property (I.43) holds, if for every ¯s∈(s^∞,n, s^∞,n+1) such that

k→∞lim t^ξⁱ(α^k,i, θ^∞_I ) = ¯sfor every i∈J¯I (I.48) holds

0 = Y

i∈JI\J¯_I

1^Ii ∆α_s^k,i_¯ Y

i∈J¯_I

1^Ii ∆α^k,i

t^ξi(α^k,i,θ^∞_I )

for all but finitely manyk. (I.49)

I.6. THE SKOROKHOD SPACE 37

assume to the contrary that

1 = Y

i∈JI\J¯_I

1^Ii ∆α_s^k,i_¯ Y

i∈J¯_I

1^Ii ∆α^k,i

t^ξi(α^k,i,θ^∞_I )

for all but finitely manyk,

then in view of definition of s^∞,n+1, we would also have s^∞,n+1 =t^ξⁱ(α^∞,i, θ_I^∞) for every i∈J¯_I. But, this contradicts property (I.38). The contradiction arises in view of

t^ξⁱ(α^∞,i, θ^∞_I )< t^l^i,n+1(α^k,i, θ_I^∞), sinceξⁱ< l^i,n+1.

Case 2: s^∞,n+1=∞.

We distinguish two cases:

Case 2.1: s^∞,n<∞

Using the same arguments as the ones used in Property (I.38) fors^∞,n, we can associate totⁿ(α^∞,i, θI) a unique natural number l^i,n such that Convergence (I.39) holds. Moreover, by definition ofs^∞,n+1 we obtain that

{s > s^∞,n,∆α^∞,i_s ∈I_i for everyi∈J_I}=∅, or, equivalently,

for everys > s^∞,n there exists ani∈J_I such that ∆α^∞,i6∈I_i. (I.50) Assume, now, that

lim inf

k→∞ s^k,n+1= ¯s, for some ¯s∈(s^k,n,∞).

Therefore, there exists (kl)_l∈N such thats^k^l^,n+1−−−→

l→∞ s. Equivalently, ∆α¯ _s^k_kl,n^l^,i +1 ∈Ii for all but finitely many l for every i ∈ JI. By convergence α^k ^J¹⁽^R

−−−−−→ α^∞, Proposition I.120.(iii) and convergence s^k^l^,n+1 −−−→

l→∞ s, we conclude that ∆α¯ ^∞,i_¯_s ∈ Ii, for every i ∈ JI. But this contradicts the assumption s^∞,n+1=∞in view of its equivalent form (I.50).

Case 2.2: s^∞,n=∞.

By definition ofs^k,n+1holds

s^k,n< s^k,n+1 whenevers^k,n<∞, fork∈N, n∈N, and

s^k,n+1=∞whenevers^k,n=∞, fork∈N, n∈N. Therefore, induction hypothesiss^k,n−−−−→

k→∞ s^∞,nand the above yield s^k,n+1−−−−→

k→∞ ∞=s^∞,n+1.

(ii) Assume that there exist n ∈ N and I ∈ J(α) such that s^∞,n < ∞. By (i), we have that s^k,n −−−−→

k→∞ s^∞,n, which in conjunction with convergence α^k ^J¹⁽^R

−−−−−→ α^∞ and Proposition I.120 implies that

∆α^k_sk,n−−−−→

k→∞ ∆α^∞_s∞,n.

Corollary I.133. Let α^k ^J¹⁽^R

−−−−−→α^∞ andI∈ J(α^∞). Define ˆ

n:=

(max{n∈N, sⁿ(α^∞, I)<∞}, if {n∈N, sⁿ(α^∞, I)<∞} is non–empty and finite,

∞, otherwise.

Then for every functiong:R^q →Rwhich is continuous on C:=

i=1

Ai, whereAi :=

(W(α^∞,i), ifi∈J_I,

W(α^∞,i)∪ {0}, ifi∈ {1, . . . , q} \JI, and for every 0≤n≤nˆ holds

g(∆α^k_sn(α^k,I))−−−−→

k→∞ g(∆α^∞_sn(α^∞,I)).

In particular, for the c`adl`ag functions

β_·^k:=g(∆α_s^kn(α^k,I))1[sⁿ(α^k,I),∞)(·), fork∈N, the convergence β^k −−−−→^J¹⁽^R⁾ β^∞ holds.

Proof. Fix ann∈Nsuch thatn≤ˆn.By Proposition I.132.(ii) holds

∆α^k_sn(α^k,I)−−−−→

k→∞ ∆α^∞_sn(α,∞,I), where ∆α^∞,i_sn(α^∞,I)∈

(W(α^∞,i), ifi∈J_I,

W(α^∞,i)∪ {0}, ifi∈ {1, . . . , q} \JI. Therefore, by definition of the timesⁿ(α^∞, I) and of the setCholds

g(∆α^k_sn(α^k,I))−−−−→

k→∞ g(∆α^∞_sn(α^∞,I)). (I.51)

By Proposition I.132.(i), the above convergence and Lemma I.115 we obtain the convergence β^k−−−−→^J¹⁽^R⁾ β^∞.

Proposition I.134. Fix some subsetI:=Qq

i=1Ii of R^q and a function g:R^q →R. Define the map D(R^q)3α7−→α[g, I] := (αˆ ¹, . . . , α^q, α^g,I)∈D(R^q+1),

where

α_·^g,I:= X

0<t≤·

g(∆αt)1^I(∆αt). (I.52)

Then, the mapα[g, I]ˆ isJ1−continuous at each pointαfor whichI∈ J(α)and for each functiongwhich is continuous on the set

C:=

i=1

Ai, whereAi:=

(W(αⁱ), if i∈JI,

W(αⁱ)∪ {0}, if i∈ {1, . . . , q} \JI. (I.53) Proof. The arguments are similar to those in the proof of [41, Corollary VI.2.8], but for the conve-nience of the reader we will present the proof below.

Let (α^k)_k∈N⊂D(R^q) be such thatα^k ^J¹⁽^R

−−−−−→α^∞,I:=Qq

i=1Ii∈ J(α^∞) and a functiong:R^q →R which is continuous on the set

C:=

i=1

A_i, whereA_i:=

(W(α^∞,i), ifi∈J_I,

W(α^∞,i)∪ {0}, ifi∈ {1, . . . , q} \JI.

For notational convenience, denote s^k,p :=s^p(α^k, I), for k∈ Nand p∈N, and R^q 3x7−→^g^I g(x)1I(x).

Recall that for everyi∈J_I holds∂I_i∩W(α^∞,i) =∅, thereforeg_I remains continuous onC.

Step1: Let ˆ

p:=







0, if{p∈N, s^∞,p<∞}=∅,

max{p∈N, s^∞,p<∞}, if{p∈N, s^∞,p<∞} is non–empty and finite,

∞, if{p∈N, s^∞,p<∞} is non–empty and infinite,

• If ˆp= 0, then by convergenceα^k −−−−→^J¹⁽^R⁾ α^∞ holds in particular α^k₀ −−−−→

k→∞ α^∞₀ . Since s⁰(α^k, I) = s⁰(α^∞, I) = 0 by definition, we can conclude.

• If ˆp <∞, using Corollary I.133 we get that for everyp∈Nwithp≤pˆit holds that g_I(∆α^k_sk,p)1[s^k,p,∞)(·)−−−−→^J¹⁽^R⁾

k→∞ g_I(∆α^∞_s∞,p)1[s^∞,p,∞)(·).

By Proposition I.132 we obtain that forp >pˆit holdss^k,p−−−−→

k→∞ ∞. Therefore, by Lemma I.115.(i) we conclude that for everyp >pˆ