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Teor�� Imov�r. ta Matem. Statist. Theor. Probability and Math. Statist.Vip. 81, 2010 No. 81, 2010, Pages 131–146

S 0094-9000(2011)00815-0Article electronically published on January 20, 2011

FUNCTIONAL LIMIT THEOREMS FOR STOCHASTIC INTEGRALS

WITH APPLICATIONS TO RISK PROCESSES

AND TO SELF-FINANCING STRATEGIES

IN A MULTIDIMENSIONAL MARKET. IUDC 519.21

YU. S. MISHURA, G. M. SHEVCHENKO, AND YU. V. YUKHNOVS’KII

Abstract. We study sufficient conditions for the weak convergence of stochasticintegrals with respect to processes of bounded variation, martingales, or semimartin-gales. A semimartingale theorem is extended to the multidimensional case. We applya limit procedure and pass from processes of bounded variation to risk processes. An“inverse” problem for the weak convergence is also considered.

1. Introduction

In the first part of the paper, we deal with the conditions for the weak convergence ofstochastic integrals ∫ b

0

ξn(t) dXn(t)

with respect to processes {Xn, n ≥ 1} of bounded variation, martingales, or semimartin-gales. We consider the convergence of the corresponding probability measures in theSkorokhod space D[0, b], b > 0. Many papers are devoted to finding conditions for theweak convergence of stochastic integrals; an extensive bibliography and the most generalconditions can be found in the book [5, Chapter IX].

There is a number of differences between conditions introduced in [5] and those con-sidered in this paper. First, a different convergence of the sequence of stochastic pro-cesses ξn(t) is considered in this paper; namely, we study the weak convergence of finite-dimensional distributions of the processes. Further, most of the conditions in [5] imposedon the sequence of semimartingales Xn are related to the triplet of predictable character-istics (B,C, ν). Checking these conditions leads to several complications, since it requiresfinding these characteristics.

In this paper we are not aimed at obtaining a canonical decomposition or the triplet ofits predictable characteristics. The assumptions imposed on the sequence of semimartin-gales Xn are expressed in terms of components of an arbitrary decomposition involvinga square integrable martingale Mn and a process of bounded variation Bn.

Conditions for the weak convergence of stochastic integrals with respect to processesof bounded variation are given in Section 2. We provide an example of the application of

2010 Mathematics Subject Classification. Primary 60G44, 60F05, 60B12.Key words and phrases. Stochastic integrals, functional limit theorems, weak convergence,

semimartingales.The first two authors are grateful to the European Commissions for support in the framework of the

program “Marie Curie Actions”, grant PIRSES-GA-2008-230804.

c©2011 American Mathematical Society

131

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

132 YU. S. MISHURA, G. M. SHEVCHENKO, AND YU. V. YUKHNOVS’KII

the main result of this section to risk processes. Conditions for the convergence of sto-chastic integrals with respect to martingales are presented in Section 3. The convergenceof stochastic integrals with respect to martingales is also considered in the paper [2]. Wedescribe the differences between these two results. The main result on the weak con-vergence of stochastic integrals with respect to semimartingales in the Skorokhod spaceis given in Section 4. We do not provide the most general conditions here; rather, weexhibit conditions that are easy to check. An application of the theorem of Section 5 toself-financing strategies will be given in the second part of this paper.

2. Convergence of integrals over stochastic processes

of bounded variation

Let (Ωn,Fn, (Fn

t )t∈R+,Pn

)be a stochastic basis for every n, and let {μn(t), n ∈ Z+, t ∈ R+} be a sequence ofprocesses whose trajectories almost surely have bounded variation on every boundedinterval [0, b], do not have discontinuities of the second kind, and are right continuous.Also let {ξn(t), n ∈ Z+, t ∈ R+} be a sequence of processes whose trajectories do nothave discontinuities of the second kind and are right continuous. Fix a countable andeverywhere dense set T in R+. Put Tb := T ∩ [0, b).

Denote by LTbthe class of all sequences

αk = {0 = t0k < t1k < · · · < tkbk < b}of finite partitions of the interval [0, b) such that

1) αk ⊆ αk+1 ⊆ Tb;2) for every t ∈ Tb, there exists k(t) such that t ∈ αk for k > k(t).

We say that condition (A) holds if the limit

(1) S(ξn, μn, 0, b) := limk→∞

kb+1∑i=1

ξn(ti−1 k)(μn(ti k)− μn(ti−1 k))

exists with probability one for all b > 0, n ∈ Z+, and for all sequences αk ∈ LTb,

αk = {0 = t0k < t1k < · · · < tkbk < b} such that tkbk < b ≤ tkb+1 k.If condition (A) holds, then the random variable S(ξn, μn, 0, b) is equal to the Rie-

mann–Stieltjes integral∫ b

0ξn(t) dμn(t) if this integral exists (see [8]).

Put

�ikx := x(ti k)− x(ti−1 k), ωi kx = supti−1 k≤s<t≤ti k

|x(t)− x(s)|,

kt = sup{i : tik ≤ t}.In what follows, the symbol “⇒” stands for the weak convergence of finite-dimensional

distributions. We also recall the notion of convergence of probability measures in theSkorokhod space.

Denote by D[0, b] the space of functions on the interval [0, b] that have no discontinu-ities of the second kind and are right continuous. We equip this space with the followingmetric d0(·, ·):

d0(x, y) = infλ∈Λ

{sup

0≤t≤b|x(t)− y(λ(t))|+ sup

0≤s<t≤b

∣∣∣∣log λ(t)− λ(s)

t− s

∣∣∣∣}

for x, y ∈ D[0, b], where Λ is the set of strictly increasing continuous mappings of theinterval [0, b] to itself. It is well known that the metric d0(·, ·) is equivalent to the


FUNCTIONAL LIMIT THEOREMS FOR STOCHASTIC INTEGRALS. I 133

Skorokhod metric

d(x, y) = infλ∈Λ

{sup

0≤t≤b|x(t)− y(λ(t))|+ sup

0≤t≤b|λ(t)− t|

},

which generates the Skorokhod topology (see, for example, [3], Chapter 3).Consider a family of probability measures (Qn)n≥1 defined on the space (D[0, b],D),

where D is the σ-algebra of Borel sets generated by the metric d0(·, ·). We say that asequence of measures Qn weakly converges to a probability measure Q on (D[0, b],D) if

limn→∞

∫[0,b]

f(x) dQn =

∫[0,b]

f(x) dQ

for all real functions f = f(x), x ∈ [0, b], bounded and continuous in the topologygenerated by the Skorokhod metric.

The symbol “D[0,b]→ ” stands for the above convergence of probability measures in the

Skorokhod topology on the interval [0, b].We introduce the following notation:

�D(x(·), δ, b) := sup0≤t<t′<t′′<t+δ≤b

(|x(t′′)− x(t′)| ∧ |x(t′)− x(t)|

)

and Tb(Fn) = {0 ≤ τ ≤ b, τ is an F·

n-stopping time}.

Theorem 1. 1. Let condition (A) hold. Assume that

(A1) 1) (ξn(t), μn(t)), t ∈ Tb ⇒ (ξ0(t), μ0(t)), t ∈ Tb, for all b > 0 as n → ∞;2) for all b > 0 and α > 0,

limk→∞

lim supn→∞

Pn

{ kb+1∑i=1

ωi kξn ωi kμn > α

}= 0.

Then

S(ξn, μn, 0, b) ⇒ S(ξ0, μ0, 0, b) for all b > 0.

2. Assume additionally that

(A2) 1) for all b > 0,

limC→∞

lim supn→∞

Pn

{sup

0≤t≤b|ξn(t)| ≥ C

}= 0;

2) for all α > 0 and b > 0,

limδ→0

lim supn→∞

Pn{�D(|μn|(·), δ, b) > α

}= 0

or

(A3) 1) ξn(t) and μn(t) are Fnt -measurable, t ∈ R+;


limδ→0

lim supn→∞

supτ∈Tb(Fn)

Pn

{sup

0≤t≤δ

kτ+t+1∑i=kτ

|ξn(ti−1k)||Δi kμn| > α

}= 0.

Then

S(ξn, μn, 0, ·)D[0,b]→ S(ξ0, μ0, 0, ·), n → ∞,

for all b > 0.



Proof. 1. Consider the following sequence of random variables:

ST,n,k =

kb+1∑i=1

ξn(tik)Δikμn.

If condition (A) holds, then

ST,0,k →∫ b

0

ξ0 dμ0 a.s.

On the other hand, condition (A1) implies that

(2) ST,n,k ⇒ ST,0,k

as n → ∞ for all k.Put

S+(αk, ξn, μn) =

kb+1∑i=1

supti−1 k≤t<ti k

ξn(t)(μn(ti k)− μn(ti−1 k)),

S−(αk, ξn, μn) =

kb+1∑i=1

infti−1 k≤t<ti k

ξn(t)(μn(ti k)− μn(ti−1 k)).

By definition, −∞ < S−(αk, ξn, μn) ≤ S+(αk, ξn, μn) < ∞ P-almost surely, and thesums S+(αk, ξn, μn) (S

−(αk, ξn, μn)) do not increase (do not decrease) with respect to kfor any sequence of partitions αk ∈ LTb

. Thus the limits

S±T (ξn, μn) = lim

k→∞S±(αk, ξn, μn)

exist and are finite almost surely. It is obvious that

(3)S+(αk, ξn, μn) ≥ ST,n,k,

S−(αk, ξn, μn) ≤ ST,n,k.

Using condition (A1) 2), we obtain from (2) and (3) that

limk→∞

limn→∞

Pn{|S±T (ξn, μn)− ST,n,k| > α}

≤ limk→∞

limn→∞

Pn{S+(αk, ξn, μn)− S−(αk, ξn, μn) > α

}(4)

= limk→∞

limn→∞

Pn

{kb+1∑i=1

ωi kξnωi kμn > α

}= 0.(5)

Now the first part of the theorem follows from (2)–(5) (see Theorem 4.2 in [3]).2. Assume that condition (A2) holds. The proof then follows from the easy estimate

�D

(S(ξn, μn, 0, ·), δ, b

)≤ sup

0≤t≤b|ξn(t)| · �D(|μn|(·), δ, b)

and from the weak convergence of the finite-dimensional distributions of S(ξn, μn, 0, ·).Now assume that condition (A3) holds. Then Theorem 1 of [1], Part 2, Chapter 6, §3,

implies that the sequence S(ξn, μn, 0, ·) is relatively compact in the Skorokhod topology.The weak convergence of the finite-dimensional distributions of the sequence of stochasticintegrals implies their convergence in the Skorokhod topology. �

Remark 1. Note that μ0 in Theorem 1 is an arbitrary process (even a process of un-bounded variation) such that the limit S(ξ0, μ0, 0, b) on the right hand side of (1) existsalmost surely for all b > 0.

Here is an example of applications of Theorem 1.



Example 1. Let Un(t), n ≥ 0, be the gain process of an insurance company, defined ona probability space (Ω,F,P), and let

Un(t, ω) := πn(t)−Nn(t,ω)∑k=1

X(n)k (ω),

where πn is a sequence of processes with continuous and nondecreasing trajectories (one

can treat πn as the premium income processes);{X

(n)k , k ≥ 1

}, for any n > 0, is a

sequence of independent identically distributed random variables with distribution func-tions Fn; Nn(t, ω) is a sequence of point processes whose trajectories do not have discon-tinuities of the second kind and are right continuous, Nn(0, ω) = 0. One can treat the

sums∑Nn(t,ω)

k=1 X(n)k (ω) as a sequence of insurance premiums. Let

0 < Tn1 (ω) < Tn

2 (ω) < · · ·be a sequence of jumps of Nn, �Nn(T

nk ) = 1. The reserve capital is defined via the

accumulator ϕn as follows:

Rn(t) := ϕn(t)u+

∫ t

0

ϕn(t− s) dUn(s)

= ϕn(t)u+

∫ t

0

ϕn(t− s) dπn(s)−Nn(t)∑k=1

ϕn(t− Tnk )X

nk (ω)

(see [4]), where u > 0 is the initial reserve capital and where ϕn : R+ → R+ is anonrandom continuous nondecreasing function (as a rule, ϕn(t) = exp{δt} for someδ > 0). The following result is a straightforward corollary of Theorem 1 and Theorem 4in [4].

Fix some b > 0 and put

Xn(t) := B−1n

([αnt]∑k=1

X(n)k −An(t)

),

where αn ↑ ∞ and Bn ↑ ∞ as n → ∞ and where An(t) is a nonrandom nondecreasingfunction.

Theorem 2. Assume that

1) ϕn(t) → ϕ0(t) pointwise as n → ∞;2) (πn(t), Xn(t), Nn(t)/αn, t ∈ Tb) ⇒ (π0, X0(t), N0(t), t ∈ Tb);3) S(ϕ0(b− ·), X0(·), 0, b) is well defined;4) for all α > 0,

limk→∞

lim supn→∞

P

{Nn(b)∑i=1

Δlni kϕn(b− ·)|Xn

i | > α

}= 0,

where lni is such that tlni −1k ≤ Tni < tlni k;

5) for all α > 0,

limk→∞

lim supn→∞

P

{ k∑i=1

�i kϕn(b− ·)�i kπn > α

}= 0.

Then the reserve capital processes weakly converge,

Rn(b) ⇒ ϕ0(b)u+

∫ b

0

ϕ0(b− s) dπ0(s)−∫ b

0

ϕ0(b− s) dY0(s),

and the limit process is of the form Y0(t) := X0(N0(t)), t ∈ [0, b].



3. Convergence of stochastic integrals with respect to martingales

As in Section 2, let (Ωn,Fn, (Fnt )t∈R+

,Pn) be a stochastic basis for every n, andlet {Mn(t),F

nt , t ∈ R+, n ≥ 0} be a sequence of square integrable martingales whose

trajectories do not have discontinuities of the second kind and are right continuous. Letμn(t) := 〈Mn〉(t) be the square characteristics of the above martingales. We consider themodifications for which the trajectories do not have discontinuities of the second kind andare right continuous. Also let {ξn(t),Fn

t , t ∈ R+, n ≥ 0} be a sequence of Fn· -predictable

processes.

Theorem 3. Assume that

(A5) 1) (ξn(t),Mn(t), μn(t), t ∈ Tb) ⇒ (ξ0(t),M0(t), μ0(t), t ∈ Tb), for all b > 0;2) for all t ∈ R+,

supn≥0

En

∫ t

0

ξ2n(s) dμn(s) < ∞;

3) for all b > 0,

limc→∞

lim supn→∞

Pn

{sup

0≤t≤b|ξn(t)| ≥ c

}= 0;

4) for all b > 0,

limk→∞

lim supn→∞

Enkb+1∑i=1

ωi kξn ωi kμn = 0;

5) limδ→0 lim supn→∞ supσ∈Tb(Fn) En(μn(σ + δ)− μn(σ)) = 0.

Then, for all b > 0, the sequence of stochastic integrals weakly converges:

∫ ·

0

ξn(t) dMn(t)D[0,b]→

∫ ·

0

ξ0 dM0(t), n → ∞.

Moreover the sequence of their square characteristics also weakly converges:

∫ ·

0

ξ2n(t) dμn(t)D[0,b]→

∫ ·

0

ξ20 dμ0(t), n → ∞.

Proof. Conditions (A5), 1) and 4) together with Theorem 1 imply the convergence offinite-dimensional distributions:

(6)

(∫ t

0

ξ2n(u) dμn(u), t ∈ Tb

)⇒

(∫ t

0

ξ20(u) dμ0(u), t ∈ Tb

), n → ∞.

Further, (A5), 1) implies that

(7)

k∑i=1

ξn(ti−1 k)�i kMn ⇒k∑

i=1

ξ0(ti−1 k)�i kM0, n → ∞.

In its turn, (A5), 2) implies that, for all n ≥ 0,

(8)k∑

i=1

ξn(ti−1 k)Δi kMn →∫ b

0

ξn(s)dMn(s), k → ∞,



in probability. Further, for all α > 0,

limk→∞

lim supn→∞

Pn

{∣∣∣∣∣k∑

i=1

ξn(ti−1 k)Δi kMn −∫ b

0

ξn(s) dMn(s)

∣∣∣∣∣ > α

}

= limk→∞

lim supn→∞

Pn

{∣∣∣∣∣∫ b

0

ϕn(s) dMn(s)

∣∣∣∣∣ > α

}

≤ limC→∞

lim supn→∞

Pn

{sup

0≤t≤b|ξn(t)| ≥ C

}

+ α−2 limC→∞

limk→∞

lim supn→∞

En

∫ b

0

(ϕCn (s)

)2dμn(s)

≤ α−2 limC→∞

(C lim

k→∞lim supn→∞

Enkb+1∑i=1

ωi kξn ωi kμn

)= 0,

(9)

where ϕn(s) := ξn(s)− ξn(ti−1k) and ϕCn (s) := ξn(s) ∧ C − ξn(ti−1k) ∧ C, s ∈ �ik.

Relatons (7)–(9) and Theorem 4.2 in [3] yield the weak convergence of the finite-dimensional distributions:

(10)

∫ b

0

ξn(t) dMn(t) ⇒∫ b

0

ξ0 dM0(t), n → ∞.

Moreover, for all δ > 0, condition (A5), 2) implies that

limC→∞

lim supn→∞

Pn

(supt≤δ

∣∣∣∣∫ t

0

ξn(u) dMn(u)

∣∣∣∣ ≥ C

)

≤ limC→∞

1

C2lim supn→∞

En

(supt≤δ

∣∣∣∣∫ t

0

ξn(u) dMn(u)

∣∣∣∣2)

≤ limC→∞

lim supn→∞

C1

C2En

∫ δ

0

ξ2n(u) dμn(u) = 0.

(11)

Now we consider the following stochastic process:

Zσn(t, C) =

∫ t+σ

0

ξCn (u) dMn(u)−∫ σ

0

ξCn (u) dMn(u),

where t ≥ 0, σ ∈ Tb(Fn), and ξCn (u) = ξn(u) ∧ C. This process is a martingale with

respect to the flow of σ-algebras Fnt+σ, t ∈ R+. Then (A5), 3) and 5) together with

Burkholder’s inequality imply that

limδ→0

lim supn→∞

supσ∈Tb(Fn)

Pn

(supt≤δ

∣∣∣∣∫ t+σ

0

ξn(u) dMn(u)−∫ σ

0

ξn(u) dMn(u)

∣∣∣∣ ≥ η

)

≤ limC→∞

lim supn→∞

Pn(|ξn(t)| ≥ C) +C2

η2limδ→0

lim supn→∞

supσ∈Tb(Fn)

En[Zσn(δ, c)]

2

=C2

η2limδ→0

lim supn→∞

supσ∈Tb(Fn)

En(μn(σ + δ)− μn(σ)) = 0.

(12)

Similarly to (12) one can prove that

(13) limC→∞

lim supn→∞

Pn

(supt≤δ

∣∣∣∣∫ t

0

ξ2n(u) dμn(u)

∣∣∣∣ ≥ C

)= 0



and

(14)limδ→0

lim supn→∞

supσ∈Tb(Fn)

Pn

(supt≤δ

∣∣∣∣∫ t+σ

0

ξ2n(u) dμn(u)−∫ σ

0

ξ2n(u) dμn(u)

∣∣∣∣ ≥ η

)= 0,

η > 0.

Now we obtain from (11)–(10) and Theorem 1 of [1], Part 2, Chapter 6, §3, that

the sequences∫ ·0ξn(t) dMn(t) and

∫ ·0ξ2n(t) dμn(t) are relatively compact in the Sko-

rokhod topology. The weak convergence of the finite-dimensional distributions (see rela-tions (7) and (14)) imply the convergence of these sequences in the Skorokhod topology, aswell. �Remark 2. One can assume that

En

∫ t

0

ξ2n(s) dμn(s) < ∞, t ∈ R+, limC→∞

lim supn→∞

Pn(μn(t) ≥ C) = 0, t > 0,

instead of condition (A5), 2).

Remark 3. As mentioned above, the conditions for the convergence of stochastic integralswith respect to martingales are studied in [2]. The main differences between conditionsof the paper [2] and those of Theorem 2 are listed below.

1) It is not assumed in Theorem 3 that the martingales Mn are continuous, whilethis is one of the assumption in the corresponding theorem in [2].

2) The counterpart of assumption (A5), 4) in [2] is

limk→∞

lim supn→∞

Enkb+1∑i=1

∫ ti k

ti−1 k

�i kξn(t) dμn(t) = 0

for all b > 0.

4. Convergence of stochastic integrals with respect to semimartingales

in terms of the canonical decomposition

Now let (Ωn,Fn, (Fnt )t∈R+

,Pn) be a stochastic basis for every n, and let

{Xn(t),Fnt , t ∈ R+, n ∈ Z+}

be a sequence of semimartingales that admits the following decomposition:

(15) Xn(t) = X0n +Mn(t) +Bn(t),

where {Mn(t),Fnt , t ∈ R+, n ∈ Z+} is a sequence of square integrable martingales whose

trajectories do not have discontinuities of the second kind and are right continuous,and {Bn(t), t ∈ R+, n ∈ Z+} is a sequence of processes of bounded variation whosetrajectories do not have discontinuities of the second kind and are right continuous. Wedenote by ΔY (t) = Y (t)− Y (t− 0) the jump of a process Y with such trajectories at apoint t.

Let μn(t) := 〈Mn〉(t) be square characteristics of the corresponding martingales. Wealso consider a sequence {ξn(t),Fn

t , t ∈ R+, n ≥ 0} of Fn· -predictable processes satisfying

the following conditions:

En

∫ b

0

ξ2n(t) dμn(t) < ∞,

∣∣∣∣∣∫ b

0

ξn(t) dBn(t)

∣∣∣∣∣ < ∞ Pn -a.s., b ∈ R+, n ∈ Z+,

where the integral∫ b

0ξn(t) dBn(t) is understood in the Riemann–Stieltjes sense.

Define ∫ t

0

ξn(s) dXn(s) :=

∫ t

0

ξn(s) dMn(s) +

∫ t

0

ξn(s) dBn(s).



Note that the definition of the integral on the left hand side is correct, since the righthand side is invariant with respect to all such decompositions.

In what follows we need the notion of density of a sequence of stochastic processeswhich coincides with the notion of density of a sequence of the corresponding probabilitymeasures. We also need the definition of the C-density.

Definition 1. A sequence of processes Xn is called C-dense if the sequence is dense andevery limit point of the sequence of the corresponding distributions is the distribution ofa continuous process.

Theorem 4. Let {Xn(t),Fnt , t ∈ R+, n ∈ Z+} be a sequence of semimartingales that

admits decomposition (15). Assume that

1) for all b > 0,(ξn(t),Mn(t), Bn(t), μn(t), t ∈ Tb

)⇒

(ξ0(t),M0(t), B0(t), μ0(t), t ∈ Tb

);

2) for all t ∈ R+,

supn≥0

En

∫ t

0

ξ2n(s) dμn(s) < ∞;

3) for all b > 0,

limC→∞

lim supn→∞

Pn

{sup

0≤t≤b|ξn(t)| ≥ C

}= 0;

4) for all b > 0,

limk→∞

lim supn→∞

Enkb+1∑i=1

ωi kξn ωi kBn = 0;


limδ→0

lim supn→∞

Pn{�D(|Bn|(·), δ, b) > α

}= 0;

6) for all b > 0,

limk→∞

lim supn→∞

Enkb+1∑i=1

ωi kξn ωi kμn = 0;

7) limδ→0 lim supn→∞ supσ∈Tb(Fn) En(μn(σ + δ)− μn(σ)) = 0;


lim supn→∞

Pn

{sup

t∈[0,b]

|ΔBn(t)| > α

}= 0.

Then the family of stochastic integrals∫ ·0ξn(t) dXn(t) weakly converges, namely∫ ·

0

ξn(t) dXn(t)D[0,b]→

∫ ·

0

ξ0 dX0(t), n → ∞.

Proof. First we prove the convergence of the finite-dimensional distributions similarly tothe proofs of Theorems 1 and 2. Condition 1) implies that

k∑i=1

ξn(ti−1 k)�i kMn +k∑

i=1

ξn(ti−1 k)�i kBn

⇒k∑

i=1

ξ0(ti−1 k)�i kM0 +

k∑i=1

ξ0(ti−1 k)�i kB0, n → ∞.

(16)



Thus

(17)

k∑i=1

ξn(ti−1 k)Δi kMn +

k∑i=1

ξn(ti−1 k)Δi kBn

→∫ b

0

ξn(s) dMn(s) +

∫ b

0

ξn(s) dBn(s), k → ∞,

in probability, where the convergence of integral sums with Mn follows from 2), whilethe convergence of sums with Bn follows from the definition of the Riemann–Stieltjesintegral.

Further, relations (5) and (9) together with conditions and 4) and 6) imply that

limk→∞

lim supn→∞

Pn

{∣∣∣∣∣k∑

i=1

ξn(ti−1 k)Δi kMn +

k∑i=1

ξn(ti−1 k)Δi kBn

−∫ b

0

ξn(s) dMn(s)−∫ b

0

ξn(s) dBn(s)

∣∣∣∣∣ > α

}

≤ limk→∞

lim supn→∞

Pn

{∣∣∣∣∣k∑

i=1

ξn(ti−1 k)Δi kMn −∫ b

0

ξn(s) dMn(s)

∣∣∣∣∣ >α

2

}

+ limk→∞

lim supn→∞

Pn

{∣∣∣∣∣k∑

i=1

ξn(ti−1 k)Δi kBn −∫ b

0

ξn(s) dBn(s)

∣∣∣∣∣ >α

2

}= 0

(18)

for all α > 0. The convergence of the finite-dimensional distributions∫ t

0

ξn(s) dXn(s) =

∫ t

0

ξn(s) dBn(s) +

∫ t

0

ξn(s) dMn(s)

⇒∫ t

0

ξ0(s) dB0(s) +

∫ t

0

ξ0(s) dM0(s) =

∫ t

0

ξ0(s) dX0(s), n → ∞,

(19)

follows from (16)–(18) and Theorem 4.2 in [3].

Conditions 5) and 7) imply that the sequences∫ t

0ξn(s) dBn(s) and

∫ t

0ξn(s) dMn(s)

are dense (this is proved in Theorems 1 and 2, respectively).Conditions 3) and 8) yield

(20) lim supn→∞

Pn

{sup

t∈[0,b]

∣∣∣∣Δ∫ t

0

ξn(s) dBn(s)

∣∣∣∣ > α

}= 0

for all α > 0 and b > 0.Since

∫ t

0ξn(s) dBn(s) is dense, relation (20) implies that the sequence of integrals is

C-dense (see Theorem 3.26 in [5]).

The sequence∫ t

0ξn(s) dBn(s) is C-dense, the sequence

∫ t

0ξn(s) dMn(s) is dense, and

thus the sum of the corresponding sequences∫ t

0

ξn(s) dXn(s)

is also dense (see Theorem 3.33 in [5]), and the proof is complete. �

We generalize Theorem 4 to the multidimensional case. Let

{Xn(t),Fnt , t ∈ R+} =

{(X1

n(t), X2n(t), . . . , X

dn(t)

),Fn

t , t ∈ R+, n ∈ Z+

}



be a sequence of d-dimensional semimartingales whose components admit the followingdecomposition:

(21) Xjn(t) = Xj

n(0) +M jn(t) +Bj

n(t), 1 ≤ j ≤ d,

where {M jn(t),F

nt , t ∈ R+, n ∈ Z+} is a sequence of square integrable martingales whose

trajectories are right continuous and do not have discontinuities of the second kind andwhere {Bj

n(t), t ∈ R+, n ∈ Z+} is a sequence of processes of bounded variation whosetrajectories do not have discontinuities of the second kind and are right continuous. Letμjn(t) := 〈M j

n〉(t) be square characteristics of the corresponding martingales. Also let{ξjn(t),F

nt , t ∈ R+, n ∈ Z+, 1 ≤ j ≤ d

}be a sequence of Fn

· -predictable processes satisfying the following conditions:

En

∫ b

0

(ξjn)2(t) dμj

n(t) < ∞,

∫ b

0

ξjn(t) dBjn(t) < ∞

Pn-almost surely for b ∈ R+, n ∈ Z+, and 1 ≤ j ≤ d.Put ∫ t

0

(ξn(t), dXn(t)) :=d∑

j=1

∫ t

0

ξjn(t) dXjn(t).

Theorem 5. Let {Xn(t),Fnt , t ∈ R+, n ∈ Z+} be a sequence of semimartingales admit-

ting decomposition (21). Assume that

1) for all b > 0 and for all 1 ≤ j ≤ d,

(ξn(t),Mn(t), Bn(t), μn(t), t ∈ Tb) ⇒ (ξ0(t),M0(t), Bn(0), μ0(t), t ∈ Tb);

2) for all t ∈ R+,

supn≥0

En

∫ t

0

(ξjn)2(s) dμj

n(s) < ∞;

3) for all b > 0,

limc→∞

lim supn→∞

Pn

{sup

0≤t≤b|ξjn(t)| ≥ c

}= 0;

4) for all b > 0,

limk→∞

lim supn→∞

Enkb+1∑i=1

ωi kξjnωi kB

jn = 0;


limδ→0

lim supn→∞

Pn{�D

(∣∣Bjn

∣∣ (·), δ, b) > α}= 0;

6) for all b > 0,

limk→∞

lim supn→∞

Enkb+1∑i=1

ωi kξjnωi kμ

jn = 0;

7) limδ→0 lim supn→∞ supσ∈Tb(Fn) En(μjn(σ + δ)− μj

n(σ))= 0;


lim supn→∞

Pn

{sup

t∈[0,b]

∣∣ΔBjn(t)

∣∣ > α

}= 0.



Then the family of stochastic integrals∫ ·0(ξn(s), dXn(s)) weakly converges:

∫ ·

0

(ξn(s), dXn(s))D[0,b]→

∫ ·

0

(ξ0(s), dX0(s)), n → ∞.

Proof. The proof follows the lines of the proof of Theorem 4. Condition 1) implies that

d∑j=1

k∑i=1

ξjn(ti−1 k)�i kMjn +

d∑j=1

k∑i=1

ξjn(ti−1 k)�i kBjn

⇒d∑

j=1

k∑i=1

ξj0(ti−1 k)�i kMj0 +

d∑j=1

k∑i=1

ξj0(ti−1 k)�i kBj0, n → ∞.

(22)

Condition 2) yields, in particular, that

d∑j=1

k∑i=1

ξjn(ti−1 k)Δi kMjn +

d∑j=1

k∑i=1

ξjn(ti−1 k)Δi kBjn

→∫ b

0

d∑j=1

ξjn(s)dMjn(s) +

d∑j=1

∫ b

0

ξjn(s) dBjn(s), k → ∞,

(23)

in probability for all n ≥ 0.Further, similarly to (18) we derive from 4) and 6) that

limk→∞

lim supn→∞

Pn

{∣∣∣∣∣d∑

j=1

k∑i=1

ξjn(ti−1 k)Δi kMjn +

d∑j=1

k∑i=1

ξjn(ti−1 k)Δi kBjn

−d∑

j=1

∫ b

0

ξjn(s) dMjn(s)−

d∑j=1

∫ b

0

ξjn(s) dBjn(s)

∣∣∣∣∣ > α

}

≤d∑

j=1

(limk→∞

lim supn→∞

Pn

{∣∣∣∣∣k∑

i=1

ξjn(ti−1 k)Δi kMjn −

∫ b

0

ξjn(s) dMjn(s)

∣∣∣∣∣ >α

2d

}

+ limk→∞

lim supn→∞

Pn

{∣∣∣∣∣k∑

i=1

ξjn(ti−1 k)Δi kBjn −

∫ b

0

ξjn(s) dBjn(s)

∣∣∣∣∣ >α

2d

})

= 0 for all α > 0.

(24)

Relations (22)–(24) and Theorem 4.2 of [3] imply the weak convergence of the finite-dimensional distributions:∫ t

0

(ξ0(s), dX0(s)) =

∫ t

0

(ξn(s)dBn(s)) +

∫ t

0

(ξn(s)dMn(s))

=

d∑j=1

∫ t

0

ξjn(s)dBjn(s) +

d∑j=1

∫ t

0

ξjn(s)dMjn(s)

⇒d∑

j=1

∫ t

0

ξj0(s) dBj0(s) +

d∑j=1

∫ t

0

ξj0(s) dMj0 (s)

=

∫ t

0

(ξ0(s), dB0(s)) +

∫ t

0

(ξ0(s), dM0(s)) =

∫ t

0

(ξ0(s), dX0(s)),

n → ∞.

(25)



It follows from conditions 5) and 7) that the sequences∫ t

0

ξjn(s) dBjn(s) and

∫ t

0

ξjn(s) dMjn(s)

are dense for all 1 ≤ j ≤ d.Similarly to the proof of the preceding theorem, the above reasoning and conditions 7)

and 8) imply that the sequences∫ t

0ξjn(s) dB

jn(s) and

∫ t

0ξjn(s) dX

jn(s) are C-dense for all

1 ≤ j ≤ d (see Theorem 3.26 in [5]).

Furthermore, the sequence∫ t

0ξn(s) dXn(s) is C-dense (thus this sequence is dense) in

view of Theorem 3.33 of [5]. This completes the proof of the theorem. �

5. Conditions for the stability of integrands

in the multidimensional case

In this section, we exhibit the conditions on integrators and integrands for the multi-dimensional case that guarantee the weak convergence if the integrals converge. In otherwords, we solve an “inverse” problem for the weak convergence. This problem occurs infinancial mathematics when one needs to investigate the limit behavior of a certain classof strategies if the capitals converge. This problem for the one-dimensional case is solvedin [6]. In this section, we study the multidimensional case. Some applications of theseresults will be presented in the second part of the paper.

Let (Ω,F,P) be a complete probability space.Let a process X(t) be a square integrable semimartingale that admits the following

decomposition:

(26) Xj(t) = Bj(t) +M j(t),

where M j is a square integrable martingale for all 1 ≤ j ≤ d with square characteristicsμj , and where Bj is a predictable process of integrable variation. Assume that thefiltration {Ft, t ∈ [0, T ]} is generated by the process X.

For the integrands we consider Ft-predictable processes

ξ(t) =(ξ1(t), . . . , ξd(t)

)∈ Rd

such that

(27) E

∫ T

0

(ξj(s)

)2dμj(s) < ∞, E

(∫ T

0

∣∣ξj(s)∣∣ d ∣∣Bj∣∣ (s)

)2

< ∞, 1 ≤ j ≤ d.

We assume that the processes Xj are linearly independent. This means that if ζ ∈ Rd

is Ft-predictable andd∑

j=1

∫ t

0

ζj(s) dXj(s) = 0

almost surely for t ∈ [0, T ], then (ζ1(t), . . . , ζd(t)) = 0 almost surely for all t ∈ [0, T ].

As before, we put∫ t

0(ξ(s), dX(s)) :=

∑dj=1

∫ t

0ξj(s) dXj(s).

Lemma 1. Let stochastic processes (ξ(t), X(t)) satisfy the above conditions. Assumethat (

ξ(t), X(t),

∫ t

0

(ξ(s), dX(s)), t ∈ [0, T ]

)d=

(ζ(t), Y (t), Z(t), t ∈ [0, T ]

)

for some processes ζ, Y , and Z.

Then Y is a semimartingale and Z(t) =∫ t

0(ζ(s), dY (s)) for all t ∈ [0, T ].



Proof. Note that if S is an Ft-predictable process such that, for all sequences {Hn,n ≥ 1} of simple Ft-predictable processes converging to H in probability, the sequences∫ T

0Hn(t) dS(t) converge to

∫ T

0H(t) dS(t) (see [9]), then S is a semimartingale with

respect to {Ft, t ∈ [0, T ]}. Thus the semimartingale property with respect to the naturalfiltration depends only on the distribution of the process. Thus Y is a semimartingale.

Further, if the processes ξ and X satisfy the above conditions, then the integral∫ t

0(ξ(s), dX(s)) is defined as the limit in probability of the integral sums

S(π, ξ,X) =n∑

k=1

(ξ(tk−1), X(tk)−X(tk−1)

)

as the diameter of the partition π = {0 = t0 < t1 < · · · < tn = t} decreases.

The joint distribution of these sums and of∫ t

0ξ(s) dX(s) is the same as that of the

sums S(π, ζ, Y ) and Z(t). Thus S(π, ζ, Y )P→ Z(t), |π| → 0, whence

Z(t) =

∫ t

0

(ζ(s), dY (s)). �

Let {ξn(t), n ∈ Z+} = {(ξ1n(t), . . . , ξdn(t)), n ∈ Z+} be a sequence of predictable d-dimensional processes satisfying condition (27).

Theorem 6. Let X and ξn satisfy the above conditions. Assume also that

1) for all 0 ≤ t ≤ T ,∫ t

0

(ξn(s), dX(s))P→

∫ t

0

(ξ0(s), dX(s)) as n → ∞;

2) the sequence of measures corresponding to the processes {ξn(·), n ∈ Z+} is weaklyprecompact in the space D[0, T ].

Then ξn(·) → ξ0(·), n → ∞, in D[0, T ].

Proof. We prove that every sequence {ξn, n ≥ 0} contains a subsequence that weaklyconverges to ξ0. Let {ξnk

, k ≥ 0} be a subsequence of {ξn, n ≥ 0}. The weak precom-pactness implies that a certain subsequence of {ξnk

, k ≥ 0} is weakly convergent. Forsimplicity, we assume that the sequence {ξn, n ≥ 0} itself is weakly convergent in D[0, T ].

Moreover, we assume that the sequence of vectors(ξ0(·),

∫ ·0(ξ0(s), dX(s)), ξn(·), X(·)

)weakly converges (recall that the whole sequence is precompact, since {ξn, n ≥ 0} isweakly precompact). Denote its limit by(

ζ0(·), Z(·), ζ(·), Y (·)).

We obtain from Lemma 1 that Z(t) =∫ t

0(ζ0(s), dY (s)).

Now we show that ξ0d= ζ.

According to [7, Theorem 2.2] we obtain the weak convergence in D[0, T ]:(ξ0(·),

∫ ·

0

(ξ0(s), dX(s)), ξn(·), X(·),∫ ·

0

ξn(s) dX(s)

)

⇒(ζ0(·),

∫ ·

0

(ζ0(s), dY (s)), ζ(·), Y (·),∫ ·

0

(ζ(s), dY (s))

).

(28)

In fact, Theorem 2.2 of [7] states only that the three last components, namely the in-tegrands, integrators, and integrals are jointly convergent. Nevertheless, we obtain theconvergence of extended vectors, since if the additional components are attached to theintegrands, while the integrator is appended with zeros, then the stochastic integral hasthe same value.



The convergence in probability means that, for all ε > 0,

P

(∣∣∣∣∫ t

0

(ξn(s), dX(s))−∫ t

0

(ξ0(s), dX(s))

∣∣∣∣ > ε

)→ 0, n → ∞.

On the other hand, the weak convergence (28) implies that

P

(∣∣∣∣∫ t

0

(ζ(s), dY (s))−∫ t

0

(ζ0(s), dY (s))

∣∣∣∣ > ε

)= 0.

Therefore ∫ t

0

(ζ(s), dY (s)) =

∫ t

0

(ζ0(s), dY (s))

for all t.The condition that the processes Xj are “linearly independent” is expressed in terms

of Ft-predictable processes. On the other hand, the filtration {Ft, t ∈ [0, t]} is generatedby the process X; thus, this condition remains true if we use another process Y withthe same distribution. In particular, the latter equality implies that ζ(t) = ζ0(t) almost

surely for all t ∈ [0, T ]. Hence ξ0d= ζ and the weak convergence ξn → ζ, n → ∞,

completes the proof of the theorem. �

Remark 4. Theorem 6 may not hold if one assumes the weak convergence instead of theconvergence in probability stated in condition 1). Indeed, if d = 1, X = B is a Brownianmotion, and ξn ≡ −1, ξ0 ≡ 1, then{∫ t

0

ξn(t) dX(t), t ∈ [0, T ]

}d=

{∫ t

0

ξ0(t) dX(t), t ∈ [0, T ]

},

while ξn does not converge to ξ in any sense.

6. Concluding remarks

Sufficient conditions for the weak convergence of stochastic integrals with respect toprocesses of bounded variation as well as with respect to martingales or semimartingalesare studied in the paper. In particular, we proved the convergence of the correspondingprobability measures in the Skorokhod space D[0, b], b > 0, and for the multidimensionalcase.

The conditions on the integrators are imposed on components of an arbitrary fixeddecomposition (not necessarily canonical decomposition), on the square integrable mar-tingale, and on the process of bounded variation. An example of possible applicationsof the theorem on the convergence of stochastic integrals with respect to processes ofbounded variation is presented in the case of risk processes. We also solved the “in-verse” problem for the weak convergence that occurs in financial mathematics. Thisproblem concerns the behavior of a certain class of strategies if the capitals converge.This application will be discussed in the second part of the paper in more detail.

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Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for

Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov

Avenue 2, Kiev 03127, Ukraine




E-mail address: [email protected]




E-mail address: [email protected]

Received 10/JUL/2009

Translated by N. SEMENOV












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