Large deviations for Markov-modulated diffusion processes with rapid switching

0.31 MB·English

by  Gang Huang

Large deviations for Markov-modulated diffusion processes with
rapid switching
Gang Huang, Michel Mandjes, Peter Spreij
January 14, 2015
5
1
0
2 Abstract

n
a In this paper,we study smallnoiseasymptotics ofMarkov-modulateddiffusion processes
J in the regime that the modulating Markov chain is rapidly switching. We prove the

3 joint sample-path large deviations principle for the Markov-modulated diffusion process
1
and the occupation measure of the Markov chain (which evidently also yields the large

] deviations principle for each of them separately by applying the contraction principle).
R
The structure of the proof is such that we first prove exponential tightness, and then
P
establish a local large deviations principle (where the latter part is split into proving the
.
h corresponding upper bound and lower bound).
t
a
m Keywords. diffusionprocesses⋆Markovmodulation⋆largedeviations⋆ stochasticexpo-
nentials ⋆ occupation measure
[

1 Address. Korteweg-deVriesInstituteforMathematics,UniversityofAmsterdam,Science
v
Park 904, 1098 XH Amsterdam, the Netherlands.
0
2
Email. {g.huang|m.r.h.mandjes|spreij}@uva.nl
0
3
0
.
1 1 Introduction
0
5
1 The setting studied in this paper is the following. We consider a complete probability space
rXiv: oc(Ωhfa,FiFn,,waPnit)dhw{tirFtahnt}sati∈tfiiRol+tnraiisntitroeingnhs{ittFycotm}ntta∈itnRru+ixo,uQwsh.aeLnredetRsXt+atte:b=esp[a0a,cfi+enS∞ite:)=-.stFa1t,0e ctiomn,teda-hinofsomraoslgloemtnheeeoduPs-nMNul.alrTskeohtves
a { ··· } ∈
Markov-modulated diffusion process is defined as the unique solution to
t t
M = M + b(X ,M )ds+ σ(X ,M )dB ,
t 0 s s s s s
Z0 Z0
whereB isastandardBrownianmotion. Weassumethatthereexisti,xsuchthatσ(i,x) = 0
t
6
throughout this paper. The concept of Markov modulation is also known as ‘regime switch-
ing’; the Markov chain X is often referred to as the‘background process’, or the ‘modulating
t
Markov chain’.
The objective of this paper is to study the above stochastic differential equation under a
particular parameter scaling. For a strictly positive (but typically small) ǫ, we scale Q to
Q/ǫ =: Qǫ, and denote by Xǫ the Markov chain with this transition intensity matrix Qǫ. If
t
1
the expected number of jumps per unit time is y for X , then the time-scaling entails that it
t
is y/ǫ for Xǫ. One could therefore say that the Markov chain has been sped up by a factor
t
ǫ 1, and, as a consequence, Xǫ switches rapidly among its states when ǫ is small. A classical
− t
topic in large deviations theory, initiated by Freidlin and Wentzell [9], concerns small-noise
large deviations. In this paper, we investigate how rapid-switching behavior of Xǫ affects the
t
small-noise asymptotics of Xǫ-modulated diffusion processes on the interval [0,T] (for any
t
fixed strictly positive T).
Let us make the scaling regime considered more concrete now. Importantly, it concerns
a scaling of the function σ( , ) to √ǫσ( , ) in the Markov-modulated diffusion, but at the
· · · ·
same time we speed up the Markovian background process in the way we described above.
The resulting process Mǫ is defined as the unique strong solution to
t
t t
Mǫ = Mǫ+ b(Xǫ,Mǫ)ds+√ǫ σ(Xǫ,Mǫ)dB , (1)
t 0 s s s s s
Z0 Z0
wherewerecallthatXǫ hastransitionintensitymatrixQǫ.Focusingontheregimethatǫ 0,
t →
we call in the sequel Mǫ the Markov-modulated diffusion process with rapid switching. For
t
simplicity, we will assume throughout this paper that Mǫ 0, whereas Xǫ starts at an
0 ≡ 0
arbitrary x S, for all ǫ. When we write e.g. E[Mǫ], this is to be understood as the
∈ t
expectation of Mǫ with the above initial conditions.
t
Since Mǫ evolves in the random environment of Xǫ, we need to design a coupling to
t t
separate the effects of the vanishing of the diffusion term and the fast varying of the Markov
chain, but at the same time to keep track of both of them. Since the scaling Q to Q/ǫ is
equivalent to speeding up time by a factor ǫ 1, one could informally say that Xǫ relates to
− t
a faster time scale than Mǫ, and therefore essentially exhibits stationary behavior ‘around’
t
this specific t. Then it is custom to consider the occupation measure of Xǫ, which is defined
t
on Ω [0,T] S as
× ×
t
νǫ(ω;t,i) = 1 ds. (2)
Xǫ(ω)=i
Z0 { s }
As its name suggests, νǫ(;T,i) measures the time Xǫ spends in state i during the time
· t
interval [0,T]. Moreover, we can use the derivative of νǫ(t) to gauge the infinitesimal change
of the occupation measure of Xǫ, at any t [0,T]. We thus construct a coupling (Mǫ,νǫ),
t ∈
which is the main object studied in this paper.
A celebrated result in Donsker and Varadhan [6] concerns the large deviations principle
(LDP) for ν1(ω;t, )/t as t (i.e., the LDP of the fraction of time spent in the individual
· → ∞
states of the background process). The setting of the present paper, however, involves the
sample-path LDP for νǫ on [0,T] as ǫ 0. More precisely, we define the image space M of
T
→
νǫ restricted on[0,T] asthespaceof functionsν on[0,T] Ssatisfyingν(t,i) = tK (s,i)ds,
× 0 ν
where d K (s,i) = 1, K (s,i) > 0 for every i S,s [0,T], and K (s,i) being Borel
i=1 ν ν ∈ ∈ ν R
measurable with respect to s; K is referred to as the kernel of ν. The metric on M is
P ν T
defined as
t t
d (µ,ν) = sup K (s,i)ds K (s,i)ds .
T µ ν
06t6T,i∈S(cid:12)Z0 −Z0 (cid:12)
(cid:12) (cid:12)
We can also view M as a subset of C(cid:12) (Rd) which is the space of (cid:12)Rd-valued continuous
T [(cid:12)0,T] (cid:12)
functions on [0,T]. In addition, the metric d on M is equivalent to the uniform metric on
T T
C (Rd).
[0,T]
2
We also define C as the image space of Mǫ, which is the space of functions f C (R)
T [0,T]
∈
andf(0) = 0equippedwiththeuniformmetricρ (f,g) := sup f(t) g(t).Theproduct
T 06t6T | − |
metric ρ d on C M is defined by
T T T T
× ×
(ρ d )((ϕ,ν),(ϕ ,ν )) := ρ (ϕ,ϕ)+d (ν,ν ), (ϕ,ν),(ϕ,ν ) C M .
T T ′ ′ T ′ T ′ ′ ′ T T
× ∀ ∈ ×
WedenotebyB(C M )theBorelσ-algebrageneratedbythetopology inducedbyρ d .
T T T T
× ×
The main result of this paper is the joint sample-path LDP for (Mǫ,νǫ) on C M . The
T T
×
associated (joint) large deviations rate function is obtained in quite an explicit form. It is
actually thesumoftwoexpressionsthatweintroducelaterinthispaper,viz.(6), i.e.,therate
function I (ϕ,ν) correspondingto Mǫ, and (5), i.e., the rate function I˜ (ν) correspondingto
T T
νǫ. Informedreaders willrecognize thattheseratefunctionsarevariants of thosefordiffusion
processes, as given in e.g. Freidlin and Wentzell [9], and for occupation measures of Markov
processes, as given in e.g. Donsker and Varadhan [6] (where we remark again that the result
in [6] relates to ν1(ω;t, )/t for t large, whereas our statement concerns the sample paths of
·
νǫ).
One method of proving the LDP for a family of probability measures on a metric space,
as was introduced in the seminal papers of Liptser and Pukhalskii [19] and Liptser [18], is to
first prove exponential tightness, and then the local LDP (precise definitions of these notions
will be given in the next section). Our work by and large follows this approach. Importantly,
the model considered in Liptser [18] is similar to ours, in that it also studies the stochastic
differential equation (1), but in the setup of Liptser [18] the process Xǫ is another diffusion
t
process (rather than a finite-state Markov chain). It means that we can roughly follow the
structureoftheproofpresentedin[18](wealsorely onthemethodofstochasticexponentials,
for instance), but there are crucial differences at many places. For instance, as we point out
below, there are several novelties that have the potential of being used in other settings, too.
Oneofthemethodological novelties isthefollowing. Weexploreaniceconnection between
regularity properties of the rate function I˜ (ν) in the LDP for (Mǫ,νǫ) and a dense subset
T
of the image space M of νǫ. On this dense subset, the optimizer of the integrand of I˜ (ν)
T T
is infinitely differentiable. This eliminates many difficulties in the computation and leads us
to first prove the local LDP on a dense subset of C M . We then extend the local LDP
T T
×
to C M by continuity properties of the rate functions I (ϕ,ν) and I˜ (ν).
T T T T
×
Let U denote the space of functions on [0,T] S being continuously differentiable on [0,T]
×
and infs [0,T],i Su(s,i) > 0. In our analysis in Section 6, we identify the following stochastic
exponen∈tial wh∈ich is directly related to the Markov chain Xǫ and its rate function I˜ (ν) (as
t T
given in (5)):
u(t,Xǫ) t ∂ u(s,Xǫ)+(Qǫu)(s,Xǫ)
t exp ∂s s s ds , u U,
u(0,Xǫ) − u(s,Xǫ) ∈
0 Z0 s !
which plays a key role when proving the local LDP. Here we follow the notational convention
that (Qǫu)(s,i) = d Qǫ u(s,j), for i S.
j=1 ij ∈
As mentioned above, the main result of our paper is the joint sample-path LDP for
P
(Mǫ,νǫ). The LDPs for each component Mǫ and νǫ are then derived as corollaries from our
main result in the standard way, i.e., by an application of the contraction principle. The
small noise LDP for the Markov-modulated diffusion processes (which is Mǫ alone) is also
3
studied in a newly published paper by He and Yin [12] in a setting of multi-dimensional
processes and time-depending transition intensity matrices. In our corresponding result,
which is Corollary 3.2, the rate function for Mǫ is decomposed into two parts that allow an
appealing interpretation: the first part corresponds to the rare behavior of the background
process Xǫ, where the second part corresponds to the rare behavior of Mǫ (conditional on
the rare behavior of Xǫ). The rate function in He and Yin [12] is less explicit, in that
it is expressed in terms of an H-functional in which the aforementioned two parts cannot
be distinguished. The sample-path LDP for occupation measures of rapid switching Markov
chain(whichisνǫ alone)isobtainedinTheorem5.1inHeetal. [13]. Theratefunction,which
is also expressed in terms of an H-functional, coincides with the rate function in our LDP
for νǫ (Corollary 3.3) when the transition intensity matrix is time-homogeneous. However,
focusing on obtaining the LDP for the Markov-modulated diffusion process together with the
background process, our aim and approach in this paper are entirely different from theirs.
The large-deviations analysis for stochastic processes with Markov-modulation is a cur-
rently active research field. Besides the previously mentioned papers of He et al. [13] and
He and Yin [12], we list a few more. Guillin [10] proved the averaging principle (moderate
deviations) of Equation (1) where Xǫ is an exponentially ergodic Markov process and b,σ
t
are bounded functions. He and Yin [11] studied the moderate-deviations behavior of Mǫ in
t
Equation (1), where σ 0 and Xǫ is a non-homogeneous Markov chain with two time-scales.
≡ t
Lasry and Lions [17] and Fourni´e et al. [8] considered large deviations for the hitting times
of Markov-modulated diffusion processes with rapid switching.
Interestingly, the present paper relates to our previous work [14]. For ease ignoring the
initial position, we there considered the Markov-modulated diffusion Mˇǫ described by
t
t t
Mˇǫ = b(Xǫ,Mˇǫ)ds+ σ(Xǫ,Mˇǫ)dB .
t s s s s s
Z0 Z0
In the regime ǫ 0 the solutions of the stochastic differential equation converge weakly to
→
a (non-modulated) diffusion Mˇ satisfying, with π denoting the stationary distribution of Xǫ
t t
(and hence also of X ),
t
1/2
t d t d
Mˇ = b(i,Mˇ )π(i)ds+ σ2(i,Mˇ )π(i) dB .
t s s s
Z0 i=1 Z0 i=1 !
X X
This result shows that, when the background chain switches rapidly, it is hard to distinguish
from observed data a Markov-modulated diffusion process from an ‘ordinary’ diffusion. The
work in the present paper, in contrast, indicates that no such property carries over to the
large deviations. The impact of a fast switching background chain does appear in the small
noise asymptotics, as shown in the LDPs in this paper.
We now describe the organization of our paper. The structure of the paper is as follows.
In Section 2, we introduce some preliminary results, definitions, and notation. In Section 3,
we state the paper’s main result and explain the steps of its proof. In Section 4, exponential
tightness of (Mǫ,νǫ) is verified. We identify a dense subset of C M in Section 5, and
T T
×
explore regularity properties of the rate function on it. The upper bound and lower bound of
the local LDP for (Mǫ,νǫ) are proved in Sections 6 and 7, respectively. We present a number
of technical lemmas in the appendix.
4
2 Preliminaries
In this section we first provide the definitions of the LDP, exponential tightness and the local
LDP, and state a set of related theorems that are relevant in the context of the paper. Let
X throughout denote a Polish space with Borel σ-algebra B(X) and a metric ρ.
Definition 2.1 (Varadhan [26]) A family of probability measures Pǫ on (X,B(X)) is said to
obey the LDP with a rate function I() if there exists a function I() : X [0, ] satisfying:
· · → ∞
(1) There exists x X such that I(x) < ; I is lower semicontinuous; for every c < the
∈ ∞ ∞
set x : I(x) 6 c is a compact set in X.
{ }
(2) For every closed set F X, limsup ǫlogPǫ(F) 6 inf I(x).
ǫ 0 x F
(3) For every open set O ⊂X, liminf →ǫlogPǫ(O) > −inf ∈I(x).
ǫ 0 x O
⊂ → − ∈
Definition 2.2 (Den Hollander [5], Puhalskii [24]) A family of probability measures Pǫ on
(X,B(X)) is said to be exponentially tight, if for every L < , there exists a compact set
∞
K X such that
L
⊂
limsupǫlogPǫ(X K )6 L.
L
\ −
ǫ 0
→
Definition 2.3 (Puhalskii [24], Liptser and Puhalskii [18]) A family of probability measures
Pǫ on (X,B(X)) is said to obey the local LDP with a rate function I() if for every x X
· ∈
limsuplimsupǫlogPǫ( y X :ρ(x,y) 6δ ) 6 I(x), (3)
{ ∈ } −
δ 0 ǫ 0
→ →
liminfliminfǫlogPǫ( y X :ρ(x,y) 6 δ ) > I(x). (4)
δ 0 ǫ 0 { ∈ } −
→ →
Since X is a Polish space, Definition 2.1.(1) implies exponential tightness. Also, Definition
2.1.(2)–(3) guarantee that Pǫ satisfies the local LDP. Actually, the converse is also valid and
is the key to prove our main result.
Theorem 2.4 (Puhalskii[24], Liptserand Puhalskii[18]) If a family of probability measures
Pǫ on (X,B(X)) is exponentially tight and obeys the local LDP with a rate function I, then
it obeys the LDP with the rate function I.
The following lemma, which corresponds to Lemma 1.4 in Borovkov and Mogulski˘ı [2],
shows that a local LDP on a dense subset of X is enough for the validation of the local LDP
on X, provided the rate function possesses a regularity property.
Lemma 2.5 (i) If (3) is fulfilled for all x˜ X˜, where X˜ is dense in X and function I(x) is
∈
lower semi-continuous, then it holds for all x X.
∈
(ii) If for every x X with I(x) < there exists a sequence x˜ X˜ converging to x and
n
∈ ∞ ∈
I(x˜ ) I(x), then the fullfillment of (4) for x˜ X˜ implies the same for all x X.
n
→ ∈ ∈
Next we impose some assumptions on the stochastic differential equation (1), as was
defined in the introduction. It is noted that (A.1) (‘Lipschitz continuity’) implies (A.2)
(‘linear growth’); we chose to include (A.2) as well, however, for ease reference in later
sections.
5
(A.1) Lipschitz continuity: there is a positive constant K such that
b(i,x) b(i,y) + σ(i,x) σ(i,y) 6 K x y , i S, x,y R.
| − | | − | | − | ∀ ∈ ∈
(A.2) Linear growth: there exists a positive constant K (which might be different from the
K used in (A.1)) such that
b(i,x) + σ(i,x) 6 K(1+ x ), i S, x R.
| | | | | | ∀ ∈ ∈
(A.3) Independence: the Markov chain Xǫ is independent of the Brownian motion B for all
t t
ǫ.
(A.4) Irreducibility: the off-diagonal entries of the transition intensity matrix Q are strictly
positive. Hence, the Markov chain Xǫ is irreducible for all ǫ and has an invariant
t
probability measure π = (π(1), ,π(d)).
···
Finally, weintroducesomeextra notation andfunctionspaces. For an arbitrary stochastic
process or a function Y , we denote the runningmaximum process by Y := sup Y . For a
t t∗ s6t| s|
semimartingale Y such that Y = 0, its stochastic exponential is defined as a semimartingale
t 0
E(Y) which is the unique strong solution to
t
t
E(Y) = 1+ E(Y) dY .
t s s
Z0 −
We denote H the Cameron-Martin space of functions ϕ C such that ϕ(t) = tϕ(s)ds
T ∈ T 0 ′
and ϕ is square-integrable on [0,T]. We call ϕ the derivative of ϕ.
′ ′ R
3 Main results
We first introduce the definitions of the rate functions involved in the main result. The rate
function corresponding to νǫ is defined as
T d (Qu)(i)
I˜ (ν) := sup K (s,i) ds, ν M , (5)
T ν T
− u(i) ∈
Z0 u∈U" Xi=1 #
where we recall the notation (Qu)(i) = d Q u(j), for i S, and U denotes the set
j=1 ij ∈
of d-dimensional component-wise strictly positive vectors. We now define the rate function
P
corresponding to Mǫ. For any (ϕ,ν) C M , we define
T T
∈ ×
1 T [ϕ ˆb(ν,ϕ )]2
′t− t dt if ϕ H ,
IT(ϕ,ν) :=  2 Z0 σˆ2(ν,ϕt) ∈ T (6)
 otherwise.
∞
where 
1/2
d d
ˆb(ν,x) := b(i,x)K (t,i), σˆ(ν,x) := σ2(i,x)K (t,i) .
ν ν
!
i=1 i=1
X X
In the above formulae, we follow the conventions that 0/0 = 0 and n/0 = , for all n >
∞
0. When we fix a time T, (Mǫ,νǫ) is understood as a joint process restricted on [0,T].
6
Let P (Mǫ,νǫ) 1 denote P((Mǫ,νǫ) ), which is a family of probability measures on
−
◦ ∈ ·
(C M ,B(C M )). Also, P (Mǫ) 1 and P (νǫ) 1 arefamilies of probability measures
T T T T − −
× × ◦ ◦
on (C ,B(C )) and (M ,B(M )) respectively. The following theorem is our main result
T T T T
which states the joint sample-path LDP of (Mǫ,νǫ) on [0,T], as ǫ 0..
→
Theorem 3.1 ForeveryT > 0, thefamilyP (Mǫ,νǫ) 1 obeystheLDPin(C M ,ρ d )
− T T T T
◦ × ×
with the rate function
L (ϕ,ν) = I (ϕ,ν)+I˜ (ν).
T T T
Proof The proof relies on applying Theorem 2.4. We first need to prove the exponential
tightness of P (Mǫ,νǫ) 1 on (C M ,B(C M )), i.e., for every L < , there exists a
− T T T T
◦ × × ∞
compact set K C M such that
L T T
⊂ ×
limsupǫlogP((Mǫ,νǫ) C M K )6 L.
T T L
∈ × \ −
ǫ 0
→
It is obvious that P (Mǫ,νǫ) 1 is exponentially tight if so are P (Mǫ) 1 and P (νǫ) 1. As
− − −
◦ ◦ ◦
we mentioned earlier, M is a subset of C (Rd). For any ν M , its derivative K (s,i)
T [0,T] T ν
∈
is bounded by 1. Then all ν M have the same Lipschitz constant, and hence M is
T T
∈
equicontinuous. It is easily seen that M is bounded and closed. Then the Arzel`a-Ascoli
T
theorem implies that M is compact. Theexponential tightness of P (νǫ) 1 is satisfied since
T −
◦
we can take K = M . Exponential tightness of P (Mǫ) 1 is verified in Proposition 4.3
L T −
◦
below.
Secondly, we proceed to prove that P (Mǫ,νǫ) 1 obeys the local LDP with the rate
−
◦
function L (ϕ,ν). That is, for every (ϕ,ν) C M , we need to obtain the upper bound
T T T
∈ ×
limsuplimsupǫlogP(ρ (Mǫ,ϕ)+d (νǫ,ν)6 δ) 6 L (ϕ,ν),
T T T
−
δ 0 ǫ 0
→ →
and the lower bound
liminfliminfǫlogP(ρ (Mǫ,ϕ)+d (νǫ,ν) 6 δ) > L (ϕ,ν).
T T T
δ 0 ǫ 0 −
→ →
The core of the proof is proving the local LDP on a dense subset of C M . The upper
T T
×
bound is validated in Proposition 6.4. The lower bound is first proved in Proposition 7.3
given the condition inf σ2(i,x) > 0. Then the condition is lifted in Proposition 7.5 by a
i,x
perturbation argument. (cid:3)
TheLDPforP (Mǫ) 1 only(ratherthanforP (Mǫ,νǫ) 1)isthenderivedfromTheorem
− −
◦ ◦
3.1 by the contraction principle in Dembo and Zeitouni [4]. We follow the convention that
inf( ) = .
∅ ∞
Corollary 3.2 The family P◦(Mǫ)−1 obeys the LDP withthe rate functioninfν∈MT LT(ϕ,ν).
At an intuitive level, I˜ (ν) can be interpreted as the ‘cost’ of forcing νǫ to behave like ν
T
on [0,T]. The other term I (ϕ,ν), can be seen as the ‘cost’ of the sample paths of Mǫ being
T
close to ϕ conditional on νǫ behaving like ν on [0,T]. Then infν MLT(ϕ,ν) indicates the
∈
minimal ‘cost’ of the sample paths of Mǫ being close to ϕ on [0,T].
Suppose F is a closed or an open subset of C . We can also interpret Corollary 3.2 as the
T
concentration of theprobability P (Mǫ) 1(F), whichisthesetofsamplepathsof Mǫ, onthe
−
◦
7
‘mostlikely path’arginfϕ F(infν MT[IT(ϕ,ν)+I˜T(ν)]).Sotherearetwosourcescontributing
∈ ∈
to the large deviations behavior of Mǫ: I (ϕ,ν) represents the contribution resulting from
T
the small noise, and I˜ (ν) represents the one from the rapid switching of the modulating
T
Markov chain.
Again by the contraction principle, P (νǫ) 1 obeys the LDP in (M ,d ) with the rate
− T T
◦
ftu∈nc[t0i,oTn]inanfϕd∈CaTllIνT(∈ϕM,νT),+itI˜Tim(νm).edSiiantceelythfoelrleowexsistthsaat iϕnf∈ϕ∈HCTT IsTu(cϕh,tνh)a=t ϕ0′t.=Hˆbe(nνc,eϕ,tw)efohraavlel
the following corollary.
Corollary 3.3 The family P (νǫ) 1 obeys the LDP in(M ,d ) with the rate function I˜ (ν).
− T T T
◦
4 Exponential tightness
We show the exponential tightness of P (Mǫ) 1 by Aldous-Pukhalskii-type sufficient condi-
−
◦
tions, as dealt with in e.g. Aldous [1], Liptser and Pukhalskii [19]. The following criterion for
exponential tightness in C , as well as an auxiliary lemma, are adapted from Theorem 3.1
T
andLemma3.1inLiptserandPukhaskii[19](whichconsiderc`adla`gprocesseswithjumps)to
our setting of continuous processes. Let Γ (F ) denote the family of stopping times adapted
T t
to F taking values in [0,T].
t
Theorem 4.1 Let Ytǫ :(Ω,{Ft}t6T,P)→ CT. If
(i)
lim limsupǫlogP Yǫ > K = ,
K′ ǫ 0 T∗ ′ −∞
→∞ →
(cid:0) (cid:1)
(ii)
limlimsupǫlog sup P sup Yǫ Yǫ >η = , η > 0,
δ→0 ǫ→0 τ∈ΓT(Ft) (cid:18)t6δ | τ+t− τ| (cid:19) −∞ ∀
then P (Yǫ) 1 is exponentially tight.
−
◦
Lemma 4.2 Let Y = (Yt)t>0 be a continuous semimartingale with Y0 = 0. Let D denote
the part corresponding to a predictable process of locally bounded variation, and V the part
corresponding to the quadratic variation of the local martingale. Assume that for T > 0 there
exists a convex function H(λ),λ R with H(0) = 0 and such that for all λ R and t 6 T
∈ ∈
λD +λ2V /2 6 tH(λξ), a.s.,
t t
where ξ is a nonnegative random variable defined on the same probability space as Y. Then,
for all c > 0 and η > 0,
P(Y > η) 6 P(ξ > c)+exp sup[λη TH(λc)] .
T∗ − −
(cid:26) λ R (cid:27)
∈
We are now ready to prove the exponential tightness claim. The technique borrows ele-
ments from Liptser [18].
Proposition 4.3 ForeveryT > 0, thefamilyP (Mǫ) 1 isexponentially tighton(C ,B(C )).
− T T
◦
8
Proof Firstly, weverify thecondition (i)ofTheorem4.1fortheprocessMǫ . Forany T > 0,
T∗
evidently,
T t
Mǫ 6 b(Xǫ,Mǫ)ds+sup √ǫ σ(Xǫ,Mǫ)dB , a.s..
T∗ Z0 | s s | t6T (cid:12) Z0 s s s(cid:12)
(cid:12) (cid:12)
We denote Cǫ := √ǫ tσ(Xǫ,Mǫ)dB . By (A.2(cid:12)), (cid:12)
t 0 s s s (cid:12) (cid:12)
R T T
Mǫ 6 K (1+Mǫ )ds+Cǫ = KT +Cǫ +K Mǫ ds, a.s..
T∗ s∗ T∗ T∗ s∗
Z0 Z0
Since KT +Cǫ is nonnegative and non-decreasing in T, Gronwall’s inequality implies
T∗
Mǫ 6 eKT [KT +Cǫ ], a.s.. (7)
T∗ T∗
Now define jK′ := K′exp( KT) KT. Then (7) entails that for sufficiently large K′ such
− −
that jK′ > 0,
P(MTǫ∗ > K′) 6 P(CTǫ∗ > jK′) 6 jK−1′/ǫE (CTǫ∗)1/ǫ ,
h i
using Chebyshev’s inequality. We thus conclude
ǫlogP(MTǫ∗ > K′) 6 −logjK′ +ǫlogE (CTǫ∗)1/ǫ . (8)
h i
We assume that 1/ǫ > 2 in the rest of the proof (justified by the fact that we consider the
limit ǫ 0). Since Cǫ is a local martingale, the process Cǫ 1/ǫ has a unique Doob-Meyer
→ t | t|
decomposition; let Cˇǫ denotetheuniquepredictableincreasingprocessinthisdecomposition.
t
Applying a local martingale maximal inequality (see e.g. Liptser and Shiryaev [20, Thm.
1.9.2]) to Cǫ, we have for the running maximum process that
t
1/ǫ
1
E (Cǫ )1/ǫ 6 E Cˇǫ . (9)
T∗ 1 ǫ T
h i (cid:18) − (cid:19)
(cid:2) (cid:3)
In order to obtain an explicit expression for Cˇǫ, we apply Itˆo’s formula to Cǫ 1/ǫ. This
t | t|
means that, for any t [0,T],
∈
1 t 1 ǫ t
|Ctǫ|1/ǫ = √ǫ |Csǫ|1/ǫ−1sign(Csǫ)σ(Xsǫ,Msǫ)dBs+ 2−ǫ |Csǫ|1/ǫ−2σ2(Xsǫ,Msǫ)ds.
Z0 Z0
We notice that the first part is a local martingale and the second part is a predictable
increasing process. As a consequence,
1 ǫ T
CˇTǫ = 2−ǫ |Csǫ|1/ǫ−2σ2(Xsǫ,Msǫ)ds. (10)
Z0
Invoking(A.2) again, wehavethatσ2(Xǫ,Mǫ) 6 K2(1+Mǫ )2.Since(7)remainsvalidwhen
s s s∗
replacing T by s, for any s 6 T, we find
Cǫ 1/ǫ 2σ2(Xǫ,Mǫ) 6 (Cǫ )1/ǫ 2K2 1+eKs(Ks+Cǫ ) 2
| s| − s s s∗ − s∗
6 (Csǫ∗)1/ǫ−2K2(cid:2)1+eKT(KT +Csǫ∗(cid:3)) 2
6 (Csǫ∗)1/ǫ−2K2[(cid:2)2(1+eKTKT)2+2e(cid:3)2KT(Csǫ∗)2].
9
Let L = 2K2max (1+eKTKT)2,e2KT . Then
T,K
(cid:8) (cid:9)
Cǫ 1/ǫ 2σ2(Xǫ,Mǫ) 6 (Cǫ )1/ǫ 2L 1+(Cǫ )2 6 L 1+(Cǫ )1/ǫ ,
| s| − s s s∗ − T,K s∗ ′T,K s∗
h i
(cid:2) (cid:3)
where L L is a positive constant not depending on K (nor ǫ). We plug (10) and the
≡ ′T,K ′
above estimate into (9), so as to obtain
1 1/ǫ 1 ǫ T
E[(CTǫ∗)1/ǫ] 6 1 ǫ E 2−ǫ |Csǫ|1/ǫ−2σ2(Xsǫ,Msǫ)ds
(cid:18) − (cid:19) (cid:20) Z0 (cid:21)
6 1 1/ǫ−1 L T + T E (Cǫ )1/ǫ ds
1 ǫ 2ǫ s∗
(cid:18) − (cid:19) (cid:18) Z0 h i (cid:19)
1/ǫ 1 1/ǫ 1
1 − LT 1 − LT
6 exp ,
1 ǫ 2ǫ 1 ǫ 2ǫ
" #
(cid:18) − (cid:19) (cid:18) − (cid:19)
the last inequality following from Gronwall’s inequality. Now observe that (1 ǫ)1 1/ǫ is
−
−
decreasing on ǫ [0, 1), with limiting value e as ǫ 0. As a result, we have the following
∈ 2 →
upper bound on the exponential decay rate of E[(Cǫ )1/ǫ]:
T∗
1/ǫ 1 1/ǫ 1
limsupǫlogE (Cǫ )1/ǫ 6 limsup ǫlog 1 − +ǫlog LT + 1 − LT
T∗ 1 ǫ 2ǫ 1 ǫ 2
ǫ→0 h i ǫ→0 " (cid:18) − (cid:19) (cid:18) − (cid:19) #
eLT
6 < .
2 ∞
Hence, by (8), for all T > 0, condition (i) of Thm. 4.1 follows for the process Mǫ :
T∗
lim limsupǫlogP Mǫ > K = . (11)
K′ ǫ 0 T∗ ′ −∞
→∞ →
(cid:0) (cid:1)
Secondly, we verify condition (ii) of Theorem 4.1. To this end, note that for arbitrary
T > 0, δ 6 1, and stopping time τ Γ ( ),
T t
∈ F
P sup Mǫ Mǫ > η 6 P sup(Mǫ Mǫ) > η +P sup(Mǫ Mǫ ) > η . (12)
| τ+t− τ| τ+t− τ τ − τ+t
(cid:18)t6δ (cid:19) (cid:18)t6δ (cid:19) (cid:18)t6δ (cid:19)
We can see that Mτǫ+t−Mτǫ is a semimartingale with respect to the filtration {Fτ+t}t>0. For
any τ Γ ( ), we denote
T t
∈ F
τ+t τ+t
Dǫ := b(Xǫ,Mǫ)ds, Vǫ := ǫ σ2(Xǫ,Mǫ)ds.
t s s t s s
Zτ Zτ
By (A.2), we have, for all λ R, t 6 δ 6 1 and τ 6 T,
∈
λ2 λ2ǫ
λDǫ+ Vǫ 6 λ K(1+Mǫ )t+ K2(1+Mǫ )2t, a.s..
t 2 t | | T∗+1 2 T∗+1
We define
λ2ǫ
H(λ) := λ + , ξ := K(1+Mǫ ).
| | 2 T∗+1
Then Mǫ Mǫ satisfies the conditions of Lemma 4.2 and, for all c > 0,η > 0,
τ+t− τ
P sup(Mǫ Mǫ) > η 6 P(ξ > c)+exp sup[λη δH(λc)] .
(cid:18)t6δ τ+t − τ (cid:19) (cid:26)−λ R − (cid:27)
∈
10